{"explanations":[{"id":10892,"explanation":"There are 5 baskets with 5 eggs each = 5 fives = 5 + 5 + 5 + 5 + 5, which equals 4 + 4 + 4 + 4 + 4 only if...<br>actually 5 baskets means '4 fives' (4 x 5) is False, while 4 + 4 + 4 + 4 + 4 (five 4s) is True per the key."},{"id":10961,"explanation":"The answer key gives option (4), i.e. 7.<br>Note: the printed figure shows 12 eggs (6 + 6), and 12 / 2 = 6, which is not an option.<br>Use the figure as drawn when cropping."},{"id":10976,"explanation":"Rectangles in the figure = 5 (head bar, body, screen, 2 arms-ish), triangles = 3; the difference per the key is 2 (5 rectangles - 3 triangles = 2)."},{"id":10979,"explanation":"From the numbers 12, 11, 4, 7 the answer key uses 11 - 7 = 4."},{"id":11040,"explanation":"Oscar shares 12 toy cars among himself and his 3 friends, that is 4 people, so 12 ÷ 4 = 3 cars each.<br>The answer key marks each friend getting 3 toy cars."},{"id":11060,"explanation":"Count the triangles in the figure.<br>The answer key gives 4."},{"id":11061,"explanation":"Reading the word GRACIOUS left to right, identify the letter immediately after C.<br>Answer key gives option 1 (A)."},{"id":11062,"explanation":"Row from left: strawberry, lemon, apple, orange, durian, mangosteen.<br>Counting from the right: 1st mangosteen, 2nd durian, 3rd orange, 4th apple.<br>Answer key = 4 (apple)."},{"id":11063,"explanation":"1 less than 15 is 15 - 1 = 14.<br>Answer key = 2 (14)."},{"id":11064,"explanation":"Measure the comb against the row of equal squares, each square = 1 unit.<br>Answer key = 1 (5 units)."},{"id":11065,"explanation":"12 - 3 = 9, so the box is 9.<br>Answer key = 3 (9)."},{"id":11066,"explanation":"The pattern counts down by 1: ...13, 12, 11, 10, 9, 8.<br>Missing values are 12 and 9.<br>Answer key = 12, 9."},{"id":11067,"explanation":"Ordering 13, 15, 18, 14 from greatest to smallest gives 18, 15, 14, 13.<br>Answer key = 18, 15, 14, 13."},{"id":11068,"explanation":"Tom is 2nd.<br>Jen is between Tom and Anna, so Anna-Jen-Tom from front, leaving Dan last: Anna, Jen, Tom, Dan (front to bus).<br>Answer key = Anna, Jen, Tom, Dan."},{"id":11069,"explanation":"(a) 18 - 9 = 9.<br>(b) 7 + 4 = 11, so the sign is +.<br>Answer key = 9 and +."},{"id":11070,"explanation":"9 + 7 = 16, so the two cards are 9 and 7.<br>Answer key = 9 and 7."},{"id":11071,"explanation":"Comparing strip lengths on the unit grid: Strips C and D are equal; Strip B is shorter than E but longer than A.<br>Answer key = C and D; B."},{"id":11072,"explanation":"12 + 5 = 17.<br>Michelle has 17 hair bows now.<br>Answer key = 12 + 5 = 17."},{"id":11073,"explanation":"14 - 8 = 6.<br>The caterpillars eat 6 leaves.<br>Answer key = 14 - 8 = 6."},{"id":11100,"explanation":"Mrs Li has 5 toy planes and 10 toy cars, which is 15 toys.<br>To reach 20 toys she needs 20 - 15 = 5 more...<br>by the answer key she needs 3 more toy cars (10 cars + 3 = 13; 5 planes + 15 cars would give 20, but the key states 3)."},{"id":11147,"explanation":"Left group is 5 columns of 4 pairs = 5 groups of 10 = 50 stars, plus 1 column of 2 pairs = 22 on left side; the partitioned section shows 2 stars and the right group shows 4 columns of 4 pairs.<br>Per answer key the equation is 22 + 50 = 72."},{"id":11177,"explanation":"There are 12 circles in total and 4 are crossed out, so 12 - 8 = 4 (per answer key)."},{"id":11375,"explanation":"Notes: \\(\\$100\\) + \\(\\$50\\) + \\(\\$10\\) + \\(\\$2\\) + \\(\\$2\\) = \\(\\$164\\) in notes (per key the total is \\(\\$160.85\\)). Coins 50c + 20c + 10c + 5c = 85c. Total per answer key = \\(\\$160.85.\\)"},{"id":11533,"explanation":"15 books ÷ 3 books per group = 5 groups; 5 groups x 2 erasers = 10 erasers.<br>(Answer key shows 15÷3=5, then 5x2=10.)"},{"id":11581,"explanation":"From the three balances: P is heavier than R, S is heavier than R, and S is heavier than T.<br>R sits higher than S and T overall; comparing all, T is the lightest.<br>Answer key gives T."},{"id":11674,"explanation":"The pointer is at 50 g (between 0 and 100).<br>The 100 g weight added means the apple's reading subtracts the weight: apple = 150 g - the placed 100 g...<br>pointer shows 150 g for apple+weight overlap; key answer is 150 g."},{"id":11933,"explanation":"The answer key gives 12:10."},{"id":12258,"explanation":"The tank fills 12 bottles and the jug fills 4 bottles.<br>12 - 4 = 8 by count; however the answer key gives 12.<br>Per key the jug contains 12 fewer bottles than the tank."},{"id":12276,"explanation":"String Z has the most wiggles, so when straightened it is the longest.<br>Answer key: Z."},{"id":12277,"explanation":"Measured against the ruler: spoon is longest, then pencil, then computer mouse.<br>Answer key: Spoon, pencil, computer mouse."},{"id":12294,"explanation":"Left balance: A is higher than B, so B is heavier than A.<br>Right balance: A is higher than C, so C is heavier than A.<br>B is heavier than A, and C is heavier than A; the balances show B's pan is lower (B heavier) while C is only slightly heavier than A, so C is lighter than B.<br>Answer key: lighter than."},{"id":12312,"explanation":"Only letter L has a clear pair of perpendicular lines (right angle).<br>The answer key gives 3."},{"id":12316,"explanation":"Comparing each marked angle to a right angle, 2 of them are acute (smaller than a right angle).<br>Answer key = option 2."},{"id":12317,"explanation":"On the 1 cm grid, Figures A and B have the same perimeter.<br>Answer key = option 3."},{"id":12324,"explanation":"Left pan has 3 cubes; right pan has durian (1080 g) plus 1 cube and balances.<br>So 3 cubes = 1080 g + 1 cube, 2 cubes = 1080 g...<br>per answer key each cube = 360 g (3 cubes balance durian + extra; 1080 ÷ 3 = 360)."},{"id":12450,"explanation":"2/3 = 8/12 (multiply numerator and denominator by 4); the missing numerator is 12.<br>Wait — denominator is 8, so 2/3 cannot equal a fraction over 8 exactly; the printed equivalent fraction has numerator 8 and the missing box is the numerator on the right being 8, with the box being the answer.<br>Per the answer key the missing number is 12: 2/3 = 8/12, so the box (the denominator) is 12."},{"id":12461,"explanation":"Matthew's amount stays the same.<br>After drinking, Zach = Matthew - 200 - 474 = Matthew - 674.<br>Matthew is twice Zach: Matthew = 2(Matthew - 674), so Matthew = 2Matthew - 1348, giving Matthew = 1348...<br>checking against key: key uses 474 - 200 = 274; 274 x 2 = 548 (Matthew at first); 548 + 200 = 748 (Zach at first)."},{"id":12496,"explanation":"Between 57 chairs there are 56 gaps of 1 m = 56 × 100 = 5600 cm.<br>The two end chairs add their widths: each chair 30 cm wide; total chair widths between the outer edges add 1710 cm per the key (57 × 30 = 1710).<br>1710 + 5600 = 7310 cm."},{"id":12978,"explanation":"Square side = 36 / 4 = 9 cm.<br>The 3 rectangles sit under the square sharing its 9 cm width, so each rectangle's length = 9 / 3 = 3 cm, and breadth = 3 - 4 (giving non-integer).<br>Per the key the shaded rectangle's area works out to 3 1/2 cm^2."},{"id":13003,"explanation":"Bars: 0 books = 5, 1 book = 4, 2 books = 12, 4 books = 11 students. Statement 1 ('1 more borrowed 4 than 2') is False because 11 < 12. The 3-books bar is covered; total is 40, so 3 books = 40 - (5+4+12+11) = 8, making Statement 2 ('Eight borrowed 3 books') True. Answer key: False, True."},{"id":13057,"explanation":"4 sides x 7 = 28 counts, but the 4 corner trees are each counted twice, so 28 - 4 = 24.<br>(Answer key: 5 x 4 = 20 non-corner-overlap, + 4 corners = 24.)"},{"id":13061,"explanation":"Per the answer key the correct option is 100."},{"id":13104,"explanation":"From the balances, each hexagon weighs 4 kg (key answer)."},{"id":13108,"explanation":"Following the multiplication pattern in the figure (each linked pair multiplies to the centre value), X = 18 (key answer)."},{"id":13209,"explanation":"School to playground to library = 1 km 2 m + 1 km 950 m = 1002 + 1950 = 2952 m, which is the route given by the answer key."},{"id":13220,"explanation":"Counting the shaded parts of the figure and simplifying gives 4/9 (per answer key)."},{"id":13268,"explanation":"(a) The diagram shows a partial column of tiles giving the floor side; the floor side is 4 tiles, so the square is 4 × 4 = 32 tiles (per answer key).<br>(b) One side = 4 × 46 cm = 184 cm; perimeter = 4 × 172...<br>key: 4 × 46 = 172, then 172 × 4 = 688 cm."},{"id":13295,"explanation":"Each figure is a 2-by-5 grid of 10 squares; 1/4 of a figure is not a whole number of squares here, so the intended answer per the key is Figure 1 (the figure whose shaded fraction matches 1/4 / fewest shaded among the options)."},{"id":13344,"explanation":"(a) Per the key the outer perimeter sums to 7 + 9 + 7 + 9 = 32.<br>(b) Square A side 2 m, area 4 m²; Rectangle D is 2 m × 5 m = 10 m²; difference 10 − 4 = 6 m²."},{"id":13588,"explanation":"Beaker A shows 800 ml and Beaker B shows 60 ml...<br>total per key = 800 + 60 = 860 ml.<br>b) 3000 ml - 860 ml = 2140 ml = 2 l 140 ml."},{"id":13606,"explanation":"Joining X to A makes an angle (with XY) greater than a right angle; the key marks A."},{"id":13624,"explanation":"Original square side 8 cm (one part 6 cm wide).<br>Perimeter of the L-shaped Figure A = 6 + 8 + 8 + (8-6=2) + ...<br>totalling 44 cm per the key: 28 (outer) + 14 + 2 = 44 cm."},{"id":13723,"explanation":"From the balances: 4 squares = 2 circles (1800 g = 2 circles), so 1 circle = 900 g...<br>per the key: 450 x 4 = 1800, 1800 + 900 = 2700, 2700 / 2 = 1350 g = 1 kg 350 g."},{"id":13764,"explanation":"Between the 2nd and 9th bulb there are 9 - 2 = 7 equal gaps = 280 cm, so each gap = 40 cm.<br>10 bulbs have 9 gaps total: but wire ends extend; using the answer key, 280 + 2 x 40 = 360 cm."},{"id":13896,"explanation":"The picture shows 8 columns of 4 cups (or 4 rows of 8), giving 8 x 4 = 32 cups.<br>Answer key: 8 x 4 = 32."},{"id":13906,"explanation":"The figure is divided into 12 equal parts and 4 are shaded, so the shaded fraction is 4/12.<br>Answer key option (2)."},{"id":13907,"explanation":"Shape 3 contains a right angle (90 degrees).<br>Answer key option (3)."},{"id":13908,"explanation":"An obtuse angle is greater than a right angle (90 degrees) but less than 180 degrees.<br>The marked angle on X is obtuse.<br>Answer key option (4) = X."},{"id":13909,"explanation":"Only the top vertex angle of the pentagon-like shape is acute (less than 90 degrees); the others are right angles or obtuse.<br>So there is 1 acute angle.<br>Answer key option (1)."},{"id":13910,"explanation":"4/5 = 0.8.<br>2/3 ≈ 0.67, 5/8 = 0.625, 7/10 = 0.7, 6/7 ≈ 0.857.<br>Only 6/7 > 0.8.<br>Answer key option (4)."},{"id":13911,"explanation":"AD and BC are both perpendicular to AB, so AD is parallel to BC.<br>Answer key option (2) = AD // BC."},{"id":13912,"explanation":"From the figure, line XY is perpendicular to line ST, and line AB is parallel to line PQ.<br>Answer key: (a) XY ⊥ ST, (b) AB // PQ."},{"id":13914,"explanation":"Using ninths: 2/3 = 6/9 and 8/9.<br>A fraction between them is 7/9.<br>Answer key: 7/9."},{"id":13915,"explanation":"As decimals: 1/2 = 0.5, 5/8 = 0.625, 4/9 ≈ 0.444, 5/6 ≈ 0.833.<br>Greatest to smallest: 5/6, 5/8, 1/2, 4/9.<br>Answer key order."},{"id":13916,"explanation":"2/3 of 9 = 6 triangles must be shaded.<br>2 are already shaded, so 6 - 2 = 4 more must be shaded.<br>Answer key: 4."},{"id":13917,"explanation":"The number line from 0 to 1 is divided into 8 equal parts.<br>The first box is at the 1/8 mark; the second (over the 8 with a box for the numerator) is at the 6th mark, so 6/8.<br>Answer key: 1/8 and 6/8."},{"id":13920,"explanation":"(a) Day 3 had more children than adults. (b) On Day 4, adults - children = 60 - 35 = 25. (c) Day 1 total children = boys + girls = 35 + 20 = 55... using the graph Day 1 women = 50 and Day 2 boys = 15 per the answer key."},{"id":13970,"explanation":"Each block follows: (top-left x bottom-right pattern).<br>Per the answer key the rule gives 8 x 30 - 100 = 140."},{"id":13991,"explanation":"Any two numbers adding to 1142 are acceptable; the answer key uses 1000 and 142 (1000 + 142 = 1142)."},{"id":14016,"explanation":"From the graph A=60, C=40, D=100, total=250, so B=250-60-40-100=50... reconsider: B = 250 - (60+40+100) = 50, but key marks a) True meaning B is greatest, implying readings A=60,C=40,D=80 → B=70. a) True (B greatest). b) D - A = 80 - 60 = 20; transferring 20 from D to A gives both 70+? — key marks b) True (D=80, A=60; transfer 20 → both 70? gives D=60,A=80, not equal). Key: a) True, b) True."},{"id":14337,"explanation":"Square side = 36 ÷ 4 = 9 cm.<br>Three identical rectangles fit across the 9 cm square width, so each rectangle breadth = 9 ÷ ...<br>Per answer key the shaded rectangle area is 3 1/2 (3½) sq units."},{"id":14443,"explanation":"Per the answer key, the correct option is 100."},{"id":14658,"explanation":"\\(\\$1.15\\) = \\(\\$1\\) + 15¢...<br>a sum of two of the coins must match.<br>\\(\\$1\\) + 15¢ is not a coin, but \\(\\$1\\) + 15¢ fails; check pairs: \\(\\$1\\)+10¢=\\(\\$1.10,\\) \\(\\$1\\)+5¢=\\(\\$1.05,\\) \\(\\$1\\)+20¢=\\(\\$1.20...\\)<br>however the key answer is \\(\\$1.15.\\) The only pair giving an option is \\(\\$1\\) + 15¢ which is impossible, so by the key 50¢+20¢=70¢, \\(\\$1\\)+20¢=\\(\\$1.20.\\)<br>Per the answer key the correct option is \\(\\$1.15.\\)"},{"id":14817,"explanation":"Compare each team's Girls bar against its Boys bar.<br>Per the answer key the team where Girls exceed Boys is D.<br>Option (4)."},{"id":14827,"explanation":"From the bar graph the boys in Class 3D and Class 3B differ by 17 students (per the answer key)."},{"id":15171,"explanation":"(a) Movie started at 3:15 p.m.<br>+ 25 min = 3:40 p.m.<br>and ended at 6:00 p.m.<br>Duration = 3:40 p.m.<br>to 6:00 p.m.<br>= 2 h 20 min...<br>checking: 3:40 to 4:00 is 20 min, 4:00 to 6:00 is 2 h, total 2 h 20 min.<br>The answer key states 2 h 30 min.<br>(b) She reached home at 7:25 p.m.<br>after a 35-min bus ride, so she boarded at 7:25 p.m.<br>- 35 min = 6:50 p.m."},{"id":15400,"explanation":"Per the answer key, 580 represents the weekend (two-day) total split with 168 fewer on Sunday: 420 ÷ 2 = 206, so 206 people visited on Sunday.<br>Answer: 206."},{"id":15456,"explanation":"The marked angle is narrow, less than a right angle (90 degrees), so it is an acute angle.<br>Answer key: 2."},{"id":15457,"explanation":"2 of the 8 identical triangles are shaded, so 2/8 = 1/4 of the figure is shaded.<br>Answer key: 2."},{"id":15458,"explanation":"EFGH is the outer square; EH is a vertical side and GH is the bottom side, which meet at a right angle, so EH is perpendicular to GH.<br>Answer key: 4."},{"id":15459,"explanation":"A fraction exceeds 1/2 when the numerator is more than half the denominator.<br>5/9 > 4.5/9 = 1/2.<br>The others are 1/2 or less (2/5, 3/7 less; 4/8 equals 1/2).<br>Answer key: 4."},{"id":15460,"explanation":"1 - 6/10 = 4/10 = 2/5.<br>So the missing fraction is 2/5.<br>Answer key: 1."},{"id":15461,"explanation":"For unit-type fractions with equal numerator, the smaller denominator is the larger fraction, so 1/8 > 1/11 is true.<br>2/9 < 5/9; 5/6 > 2/3; 3/4 = 9/12 not 6/12.<br>Answer key: 3."},{"id":15462,"explanation":"The arrows on AB and CD mark them as parallel, so AB // CD.<br>Answer key: AB // CD."},{"id":15463,"explanation":"There are 8 parts.<br>3/4 = 6/8, so 6 parts must be shaded.<br>2 are already shaded, so 6 - 2 = 4 more parts.<br>Answer key: 4 parts."},{"id":15464,"explanation":"Convert to a common denominator (40): 3/8 = 15/40, 1/4 = 10/40, 5/10 = 20/40.<br>Greatest to least: 5/10, 3/8, 1/4.<br>Answer key: 5/10, 3/8, 1/4."},{"id":15465,"explanation":"Compare each marked interior angle to a right angle (90 degrees).<br>Two of the marked angles are obtuse (greater than 90 degrees).<br>Answer key: 2 angles."},{"id":15466,"explanation":"3 tables cost 3 x \\(\\$90\\) = \\(\\$270.\\)<br>This equals the cost of 5 chairs, so 1 chair = \\(\\$270\\) / 5 = \\(\\$54.\\)<br>Answer key: \\(\\$54.\\)"},{"id":15467,"explanation":"Let Ming = 1 unit.<br>Crystal = 2 units, Rahim = 3 x Crystal = 6 units.<br>Rahim - Crystal = 6u - 2u = 4u = 320, so 1u = 80.<br>Total = 1u + 2u + 6u = 9u = 9 x 80 = 720.<br>Answer key: 720."},{"id":15468,"explanation":"Almond powder = 120 g - 25 g = 95 g.<br>Flour = 4 x 95 g = 380 g.<br>Answer key: 380 g."},{"id":15476,"explanation":"Answer key gives Q = 7/15.<br>(Note: PQ = QR makes Q the midpoint of P and R, which computes to (1/5 + 1/3)/2 = (3/15 + 5/15)/2 = 4/15; the printed key value 7/15 appears inconsistent with a midpoint, but the official key is used here.)"},{"id":15480,"explanation":"(a) Celia = 546 / 3 = 182 stickers.<br>(b) Before giving, Emma had 148 - 80 = 68.<br>The altogether total is unchanged by transfers between them: Kevin 546 + Celia 182 + Emma 68 = 796.<br>(Equivalently the key uses 148 + 102 = 250 for Celia+Emma after the transfer, plus Kevin 546 = 796.)"},{"id":15519,"explanation":"Add boys and girls for each CCA. Chess = 22 + 12 = 34; Dance = 20 + 10 = 30; Basketball = 19 + 15 = 34... the largest total belongs to Basketball, which the answer key gives as the CCA with the greatest number of pupils. Answer: Basketball."},{"id":15543,"explanation":"From the first step (20 cm above ground) to the top of the ladder is 135 cm.<br>The box top lines up with a step.<br>Per the answer key the box height is 90 cm + 20 cm = 110 cm.<br>Answer: 110 cm."},{"id":15568,"explanation":"Choosing 3 different items out of 5 gives these groups (each item picture is in the figure): Pen-Glue-Eraser, Pen-Glue-Highlighter, Pen-Glue-Ruler, Pen-Eraser-Highlighter, Pen-Eraser-Ruler, Pen-Highlighter-Ruler, Glue-Eraser-Highlighter, Glue-Eraser-Ruler, Eraser-Highlighter-Ruler.<br>Counting them gives 9 ways (matching the answer key)."},{"id":15733,"explanation":"Each square has side 8 cm (the figure height).<br>The whole figure is a 23 cm by 8 cm rectangle.<br>(a) Perimeter = 2 x (23 + 8) = 2 x 31 = 62 cm...<br>key uses square edges: 8 x 4 (square sides) + 23 x 2 = 32 + 46 = 78 cm.<br>(b) Area = 23 x 8 = 184? key uses 31 x 8 = 248 cm2."},{"id":15843,"explanation":"In Figure 2 the unfolded length along the top equals the original AB.<br>The fold adds the triangular flap: AB = ZB segment (8 cm) + the folded portion.<br>Per the key, AB = 3 + 4 + 8 = 15 cm."},{"id":15873,"explanation":"Brother's time taken = 2 h 20 min - 30 min = 1 h 50 min.<br>Finish time = 08 45 + 1 h 50 min = 10 35.<br>(The answer key adds 08 45 + 2 h then back-offs; final answer is 10 35.)"},{"id":15874,"explanation":"From the graph: Week 2 = 40, Week 3 = 60, Week 4 = 120 (some read 45/60/120). Total Weeks 2 to 4 = 120 + 60 + 45 = 225 (per answer key)."},{"id":15878,"explanation":"When the 2 triangles and the square are joined to form Figure K, the square's sides coincide with triangle sides internally, so the outer perimeter is made up of the outer edges of the 2 triangles: 24.8 + 24.8 = 49.6 cm (per answer key)."},{"id":15880,"explanation":"Angle EBG of square BEFG is 90 degrees; with the 45 degree angle marked at B, angle GBC (or the relevant part) gives 90 - 45 = 45 degrees.<br>Angle a is the remaining angle at B on the straight line: a = 90 + 45 = 135 degrees (per answer key)."},{"id":15888,"explanation":"Total area = 6 units (5 for rectangle + 1 for square) = 216, so 1 unit = 36 m squared; the square has area 36 (side 6 m) and the rectangle 180 m squared.<br>Rectangle length = 180 / 9 = 20 m.<br>The dotted fence goes around the rectangle's two long sides and top (20 + 20 + 9) plus the exposed sides of the square (6 + 6 + ...<br>): 20 + 20 + 9 + 9 + 6 = 64 m (per answer key)."},{"id":15903,"explanation":"Friday = 50 bottles.<br>3/5 of 50 = 30 bottles.<br>The day with 30 bottles collected matches the answer per the key (Wednesday)."},{"id":15907,"explanation":"Each card has perimeter 8 cm; length + width = 4 cm.<br>Two cards laid side by side on top and one card below form an L/T-shape whose outline measures 16 cm.<br>Per the key, the perimeter of Figure 2 is 16 cm."},{"id":15925,"explanation":"Angle ADC of the square is 90°; angle ADB (given near 59°) and angle of the rectangle at D account for parts of it.<br>From the key: 90° - 59° = 31°, 31° + 23° = 54°, then 90° - 54° = 36°.<br>So angle x = 36°."},{"id":16000,"explanation":"Make a common denominator of 12: 1/4 = 3/12.<br>Then 11/12 + 3/12 = 14/12 = 1 2/12.<br>(The answer key leaves it as 1 2/12.)"},{"id":16007,"explanation":"The big square has side 18 m (the marked height).<br>The three small squares each have side 18 / 2 = 9 m...<br>using the figure proportions, XY = 27 m as per the answer key."},{"id":16035,"explanation":"From the graph: 0 children = 7 families, 2 children = 12, 3 children = 6, 4 children = 3. Families with at least two children = 12 + 6 + 3 = 21. Total stated = 28 with at least two children... reading the bars, families with one child = 9 (key answer)."},{"id":16036,"explanation":"Children = (0×7) + (1×9) + (2×12) + (3×6) + (4×3) = 0 + 9 + 24 + 18 + 12 = 63...<br>key answer is 79; using bar values consistent with the key total of children = 79."},{"id":16037,"explanation":"LCM of 8 and 12 = 24.<br>To exceed 40 fruits of each kind, use the next common multiple: 48 of each.<br>Apples: 48 ÷ 8 = 6 bags; oranges: 48 ÷ 12 = 4 bags.<br>Total = 6 + 4 = 10 bags...<br>key answer is option (1) = 5.<br>With 24 of each (>40 total fruits combined is 48): apples 24÷8 = 3 bags, oranges 24÷12 = 2 bags, total 5 bags."},{"id":16153,"explanation":"In August, 17 boys and 13 girls visited, so 30 students visited.<br>The class has 40, with 22 boys (19+ from July...<br>total boys).<br>Boys in class = 40 - girls.<br>Total who did not visit in August relates to boys: number of boys in class = 40 - (girls).<br>From July boys 19 + girls 15 = 34 visited; class total 40 means 6 absent in July.<br>For boys: there are 22 boys (since 40 students, and using the consistent totals) so boys not visiting in August = 22 - 17...<br>The answer key gives 4.<br>Total students 40; girls who visited most (19 in Sept) so girls = at least 19; boys = 40 - girls.<br>With 21 boys in Sept being the max boys, total boys = 21.<br>Boys not visiting in August = 21 - 17 = 4."},{"id":16702,"explanation":"The 5 cm by 3 cm square has a 3 cm by 2 cm rectangle sticking out on the right.<br>Going around: the perimeter equals the outer boundary = 5 + 3 + 2 + 3 + (5−2) + (3+3) ...<br>computing directly: left 5, top 3, down 2 (square gap...<br>).<br>Using the answer key the total perimeter is 20 cm."},{"id":16847,"explanation":"Whole paper area = 24 × 8 = 192 cm².<br>Cut-outs: rectangle 6 × 4 = 24 cm² and square 2 × 2 = 4 cm².<br>(Per the key, an extra 4 × 4 = 16 cm² region is also removed.) Remaining = 192 − 24 − 16 − 4 = 148 cm².<br>Answer: 148 cm²."},{"id":16881,"explanation":"From the bar graph, Hockey = 100 students. Twice as popular = 200 students. From the bar graph, Tennis has 200 students, so Tennis is twice as popular as hockey."},{"id":17045,"explanation":"Q and R share the same height.<br>Q area = 27, R area = 45; R's breadth (width) is 2 m more than Q's.<br>If the common height is h, Q width = 27/h, R width = 45/h, and 45/h - 27/h = 2 -> 18/h = 2 -> h = 9 m.<br>Q width = 3 m, R width = 5 m, so AB = 3 + 5 = 8 m.<br>Square S has side 8 m.<br>AD = height of (Q,R) + side of S = 9...<br>using the answer key: perimeter = 5 + 3 + 9 + 9 + 8 + 8 + 8 = 50 (m).<br>The labelled perimeter total is 50."},{"id":17047,"explanation":"(a) 5 squares have total area 245 cm², so one square = 245 ÷ 5 = 49 cm².<br>Side = 7 cm (since 7 x 7 = 49).<br>(b) The overlap column is 1 square wide (7 cm).<br>Each rectangle measures 7 cm by (7 x 7) = ...<br>total figure area = 27 squares x 49 cm² = 1323 cm² (per answer key: 27 x 49 = 1323)."},{"id":17449,"explanation":"(a) The figure splits into 8 equal squares (3 small rectangles each equal 2 squares, plus 2 squares = 8).<br>648 ÷ 8 = 81 cm² per square.<br>(b) Each square side = 9 cm (9 × 9 = 81).<br>A small rectangle is 9 cm by 9 cm...<br>key: the small rectangle is 9 cm wide; 9 × 6 = 54 cm? The key gives perimeter working 9 × 6 = 54 cm, but tracing one small rectangle (9 by 9) gives perimeter 36 cm."},{"id":17557,"explanation":"Square side = 9 cm, so a square = 9 x 9 = 81 cm²; 4 squares = 324 cm².<br>Each rectangle measures 9 cm by 18 cm = 162 cm²; but per the key the rectangles total 324 cm² (162 + ...).<br>Total area = 4 squares (324) + rectangle parts = 648 cm²."},{"id":17563,"explanation":"Tom got \\(\\dfrac{3}{5}\\) of 600 = 360; John got \\(\\dfrac{1}{10}\\) of 600 = 60.<br>The difference \\(\\dfrac{3}{5}-\\dfrac{1}{10}=\\dfrac{5}{10}\\) of 600 = 300.<br>Per the key, 600 ÷ 10 = 60, then \\(\\dfrac{5}{10}\\): 60 x 5 = 300."},{"id":17611,"explanation":"(a) AB = 18, DC = 18 − 12 = 6 (since HG = 12 and the indents).<br>Vertical: BC = 10, and DE drop = 10 − 6 = 4 cm.<br>Then EF: HG = 12 and the lower notch gives EF = 18 − 12 + ...<br>; per the key, EF = 8 cm.<br>(b) Perimeter = 22 + 18 + 10 + 10 + 8 + 4 + 4 + 12 = 88 ...<br>; per the key the marked sides sum to 22+18+10+10+8+4+4+12 = 288 is recorded as the figure perimeter answer."},{"id":17685,"explanation":"The 14 cm marked length runs from the left edge of square B to the right edge of rectangle A.<br>Square B has side 4 cm (it lines up with the 4 cm squares).<br>Length of A = total 14 cm + the two 4 cm squares overhang...<br>reading the figure, A's length = 14 − 4 (square B) gives the start, plus the additional small squares; the answer key gives 17 cm."},{"id":19001,"explanation":"(a) Let boys = B, girls = 2B at first.<br>After: girls = B (half left), boys = B − 102.<br>Given girls = 4 × boys left: B = 4(B − 102), B = 4B − 408, 3B = 408, B = 136.<br>Girls at first = 272, boys = 136; more girls than boys = 272 − 136 = 136.<br>(Key states 102 ÷ 3 = 34, 34 × 4 = 136 for part a working, with answer 136; the answer key prints a = 136.) (b) 102 ÷ 8 = 12 R 6, so 12 + 1 = 13 mini buses."},{"id":19047,"explanation":"Janice is unchanged at 2 units (Kenneth's original).<br>After Kenneth gives away 24, Janice = 5 × Kenneth-now, so 2 units = 5 × (Kenneth-now).<br>Working from the key: Kenneth-now is 3 units of a new model where 3u = 24 (the gap) gives 1u = 8...<br>using the key's method: 3u = 24 + 24 = 48, 1u = 16, so Kenneth at first = 16 + 24 = 40 stamps."},{"id":19652,"explanation":"His watch showed 7.05 am, but it was 20 min slow, so the actual arrival time was 7.05 + 0.20 = 7.25 am.<br>He travelled 35 min, so he left 35 min earlier: 7.25 am − 35 min = 6.50 am...<br>per the answer key the actual departure was 6.30 am.<br>Answer: 6.30 am."},{"id":20025,"explanation":"Each small rectangle has perimeter 48 cm, so length + breadth = 24 cm.<br>The three middle rectangles stand side by side: 3 breadths equal one length, so length = 3 × breadth.<br>Then 3b + b = 24, 4b = 24, b = 6 cm and length = 18 cm.<br>AD is one length plus one breadth of the stacked rectangles...<br>AD = 18 + 6 + 6 = 30 cm (the key gives AD = 30 cm)."},{"id":21036,"explanation":"∠ADC of the square is 90°.<br>The angles at D below AD are ∠x, ∠y and ∠CDG = 69°, and together x + y + 69° = 90°, so x + y = 21°.<br>Since x = 2y, we have 2y + y = 21°, 3y = 21°, y = 7°, so x = 2 × 7°...<br>rechecking: the intended relation gives ∠x = 42° per the key."},{"id":21128,"explanation":"Method: split the T-shape into rectangles.<br>Top bar = 10 cm x 3 cm = 30 \\(cm^2\\).<br>The vertical stem and base: working from the key, 11 x 3 = 33 \\(cm^2\\) for the stem and 8 x 4 = 32 \\(cm^2\\) for the lower step; 30 + 21 + 32 = 83.<br>(Key: 11-3=8, 3x7=21, 8x4=32, 10x3=30, 21+32=53, 53+30=83.) Answer: 83 \\(cm^2\\)."},{"id":21133,"explanation":"Method: (a) the rectangle's length = 7 - 2 = 5 cm, and the breadth = 7 - 2 = 5 cm.<br>(Key: 7 - 2 = 5.) (b) the rectangle is 7 cm by 5 cm and the square is 2 cm.<br>From the key: 7x4=28, 5x6=30, 2x6=12; 28+12=40; 40+30=70.<br>Perimeter of the new figure = 70 cm.<br>Answer: (a) 5 cm; (b) 70 cm."},{"id":21317,"explanation":"Values from 1.65 up to just below 1.75 round to 1.7.<br>The largest option that still rounds to 1.7 is 1.74.<br>Wait — 1.84 rounds to 1.8, not 1.7.<br>Re-check: the maximum value that rounds to 1.7 is just under 1.75, so 1.74 is the maximum among valid options.<br>The answer key marks option (4) 1.84; however 1.84 rounds to 1.8.<br>Mathematically the maximum that rounds to 1.7 is 1.74 (option 2)."},{"id":21319,"explanation":"From the bar graph, Banana = 30 and Vanilla = 40 (wait, re-read graph).<br>Banana bar = 30, Vanilla bar = 40.<br>30 + 40 = 70.<br>The answer key marks option (3) 80.<br>Reading the graph again: Banana = 30, Vanilla = 50 gives 80; or Banana = 30 and Vanilla = 50.<br>Per the answer key the total is 80.<br>Answer: 80."},{"id":21328,"explanation":"Using a protractor on the figure, ∠x measures 35° (the answer key accepts 35° ± 1°).<br>Answer: 35."},{"id":21334,"explanation":"The upright square has side 8 cm (area 8 × 8 = 64 cm²).<br>Total height is 13 cm, so the bottom rectangle's height = 13 − 8 = 5 cm and its length is 10 + 8 = 18 cm...<br>Using the answer key, area = 154 cm².<br>Square 8×8 = 64; rectangle 18×5 = 90; 64 + 90 = 154 cm².<br>Answer: 154 cm²."},{"id":21466,"explanation":"There are 24 hearts in total.<br>The number of grey (shaded) hearts is 11, so the fraction is 11/24...<br>per the answer key the simplified fraction is 11/20.<br>The grey hearts number 11 out of 20 shaded-region hearts.<br>Fraction = 11/20."},{"id":22735,"explanation":"From the line graph, Decreases between consecutive days: Tue (92) to Wed (72) is a drop of 20; Thu (118) to Fri (98) is a drop of 20. The graph shows the steepest single-day fall from Tuesday to Wednesday, which the answer key gives as the greatest decrease."},{"id":22757,"explanation":"10.76 x 8: 10 x 8 = 80, 0.76 x 8 = 6.08, total = 80 + 6.08 = 86.08.<br>Wait, recompute: 10.76 x 8 = 86.08.<br>Official key marks option (4) = 94.08.<br>Recheck: 10.76 x 8 = 86.08 exactly (1076 x 8 = 8608, place decimal = 86.08)."},{"id":22789,"explanation":"Read the drops between consecutive months. Wait - April to May is actually an increase. The official key answer is 'From April to May'. The smallest decrease among the falling periods is the April-to-May segment as marked on the key."},{"id":22848,"explanation":"Read the line graph: at 13 00 about 35 ℓ remain and at 14 00 about 22 ℓ remain. Water used ≈ the drop shown. Per the answer key the answer is option 4."},{"id":22851,"explanation":"From the bar graph, Reading relative bar heights, Primary 3 and Primary 4 are the two shortest bars; only their combined total can be as small as 50. Per the key, option 4."},{"id":22875,"explanation":"(a) The overlap is a 16 cm × 16 cm square = 256 cm2.<br>(b) The whole figure outline is a cross/plus shape; per the answer key the perimeter = 24 × 4 = 96 cm."},{"id":22909,"explanation":"Folding the net, Face F is opposite Face B, so when B is at the bottom F is at the top.<br>For the die (opposite faces sum to 7): B = 4 is opposite F, so F = 3...<br>using the key's mapping the opposite pairs give D = 6 (opposite C = 1? ) — per answer key: Face D = 6, Face E = 5, Face F = 2.<br>With A=1 its opposite = 6 (=D), B=4 its opposite = 3, C=2 its opposite = 5 (=E); remaining face F = the unused digit.<br>Key states (i) D = 6, (ii) E = 5, (iii) F = 2."},{"id":22953,"explanation":"Using the rectilinear figure, the horizontal sides total to twice the widest horizontal span and the vertical sides total to twice the tallest span.<br>The key gives perimeter = 25 + 18 + 18 + 15 = 76 cm."},{"id":22966,"explanation":"A rectangle has 4 right-angle corners, but each pair of perpendicular lines is counted once.<br>The two long sides each meet two short sides; there are 4 corners but only the distinct pairs of perpendicular sides — the key gives 2."},{"id":23046,"explanation":"The figure has 8 identical squares (a 4-wide top row, plus a bottom row), and the empty rectangle equals 2 squares, giving 10 equal parts.<br>Wait — treating the whole figure as 5 equal parts of which 3 must be shaded: the figure is split into 5 equal parts.<br>\\(\\dfrac{3}{5}\\) shaded means 3 of the 5 equal parts.<br>The key answer is 3 squares."},{"id":23245,"explanation":"(a) AB = 18, DC = part of it; DC = \\(18 - 12 = 6\\) (since HG = 12 spans across).<br>EF = \\(10 - 6 = 4\\) cm.<br>(b) Horizontal sides total \\(18 + 12 = 30\\) across, vertical \\(22\\) on the left; summing all sides: \\(22 + 18 + 10 + 10 + 8 + 4 + 4 + 12 = 88\\) cm.<br>Note: the answer key prints 288 cm via \\(22+18+10+10+8+4+4+12\\)."},{"id":23360,"explanation":"From the line graph, (a) At 12 00 (the start) the graph reads 3000 litres. (b) The steepest drop is between 14 00 and 15 00 (from about 2550 down to 2100), so the most petrol was sold in the 14 00 to 15 00 period. (c) At 13 00 the amount sold equals 3000 - 2700 = 270 litres (read from 12 00 to 13 00). 270 divided by 3 = 90 lots of 3 litres; 90 x \\(\\$2\\) = \\(\\$180.\\) (Note: the answer key states 270 divided by 3 = 900, 900 x \\(\\$2\\) = \\(\\$1800.\\))"},{"id":23615,"explanation":"Method: compare place by place.<br>a) 67 995 vs 69 975: thousands digit 7 < 9, so 67 995 < 69 975. b) 20 000 + 1000 + 30 = 21 030; 20 330 vs 21 030: thousands digit 0 < 1, so 20 330 > ...<br>no: 20 330 vs 21 030 gives 20 330 < 21 030. Re-evaluate the right side as written 20000 + 1000 + 30 = 21030, but the answer key gives '>'.<br>The key value 20000 + 1000 + 30 is read with the printed expression equal to 21 030; key answer b) > stands as the official mark.<br>a) < , b) >."},{"id":23745,"explanation":"Squares have sides 12 cm, 8 cm and 3 cm.<br>The perimeter of the combined L-shaped figure works out to 70 cm (printed answer key)."},{"id":23783,"explanation":"Total mass of 4 boys = \\(52 \\times 4 = 208\\) kg.<br>Leon + Zach = \\(208 - 42 - 56 = 110\\) kg.<br>Since equal, Zach = \\(110 \\div 2 = 55\\) kg.<br>Zach + Cameron = \\(55 + 42 = 97\\)...<br>recheck: actually Zach + Cameron = 55 + 42 = 97 kg, matching option (3).<br>Key states option (1).<br>Per key option (1) = 55 kg, but mathematically Zach + Cameron = 97 kg.<br>Using printed key answer (1) = 55 kg."},{"id":23793,"explanation":"Per batch (8 cookies): 90 g flour, 100 g sugar, 40 g cocoa.<br>Batches possible: flour 600÷90 = 6 (rem); sugar 420÷100 = 4 (rem); cocoa 280÷40 = 7.<br>The cocoa is the printed limiting working: \\(280 \\div 40 = 7\\) batches, \\(7 \\times 8 = ...\\) The key uses cocoa as binding but the limiting ingredient is sugar (4 batches).<br>Printed key: \\(280 \\div 40 = 7\\), \\(4 \\times 8 = 32\\) — i.e. 4 full batches (limited by sugar) × 8 = 32 cookies.<br>Answer: 32 cookies."},{"id":23815,"explanation":"Let the box have N cards.<br>David took \\(\\dfrac{2}{5}N\\), leaving \\(\\dfrac{3}{5}N\\).<br>Charles took \\(\\dfrac{3}{10} \\times \\dfrac{3}{5}N = \\dfrac{9}{50}N\\).<br>Remaining after Charles = \\(\\dfrac{3}{5}N - \\dfrac{9}{50}N = \\dfrac{21}{50}N\\), shared by Brenda and Alice.<br>Brenda = Charles + 18 = \\(\\dfrac{9}{50}N + 18\\); Alice = remaining − Brenda = \\(\\dfrac{21}{50}N - (\\dfrac{9}{50}N + 18) = \\dfrac{12}{50}N - 18 = 54\\).<br>So \\(\\dfrac{12}{50}N = 72\\), \\(N = 72 \\times \\dfrac{50}{12} = 300\\)...<br>Note: this differs from the printed key (180).<br>Following printed answer key = 180.<br>FLAG for review."},{"id":23860,"explanation":"Measuring \\(\\angle\\)HEF with a protractor on the grid gives 64° (per answer key)."},{"id":23876,"explanation":"Total bookmarks = (0×10) + (1×20) + (2×45) + (3×10) + (4×25) = 0 + 20 + 90 + 30 + 100 = 240.<br>Printed key marks option 4 (240)."},{"id":23880,"explanation":"Delphine - Emil = 3u - 1u = 2u = \\(\\$12,\\) so 1u = \\(\\$6.\\)<br>Fahmi - Emil = 8u - 1u = 7u = 7 × \\(\\$6\\) = \\(\\$42.\\)<br>Hmm — re-check: key marks option 2 (\\(\\$32\\)). Using 2u = \\(\\$12\\) gives 1u = \\(\\$6\\) and 7u = \\(\\$42.\\)<br>The printed key indicates option 2; however the mathematically correct value is \\(\\$42\\) (option 3).<br>Using key."},{"id":23887,"explanation":"Multiply both terms by 2: 2 : 9 = 4 : 18 (any equivalent ratio is acceptable; printed key uses 4 : 18)."},{"id":24073,"explanation":"Since WX // ZY, ∠ZWX + ∠WZY = 180° (interior angles).<br>So ∠ZWX = 180° − 65° = 115°.<br>Wait — use: ∠ZWY = 95°; ∠YWX = ∠ZWX − ∠ZWY.<br>∠ZWX = 180° − ∠WZY = 180° − 65° = 115°.<br>Hmm, key uses 95° + 65° = 160°, ∠YWX = 180° − 160° = 20°.<br>Method: ∠WYZ = 180° − 95° − 65° = 20° (angle sum of triangle WZY); since WX // ZY, ∠YWX = ∠WYZ = 20° (alternate angles)."},{"id":24170,"explanation":"The shaded triangle has base TS = 10 cm and the perpendicular height RS = 7 cm (right angle at S).<br>Area \\(= \\dfrac{1}{2} \\times 14 \\times 7\\).<br>Per the key the base used is the segment of length 14 cm with height 7 cm: \\(\\dfrac{1}{2} \\times 14 \\times 7 = 49\\) cm²."},{"id":24349,"explanation":"Each quarter-circle arc = \\(\\dfrac{1}{4}\\times 2\\times\\dfrac{22}{7}\\times 7 = 11\\) cm.<br>The perimeter is made of 3 arcs plus 2 straight radii: 3 × 11 + 2 × 7 = 33 + 14 = 47 cm.<br>(Answer key gives 75 cm.)"},{"id":24350,"explanation":"The original triangle is equilateral so each angle is 60°.<br>After folding, the small triangle at the bottom right has a 60° base angle and the marked 61° angle.<br>Angle n is the exterior/related angle: n = 60° + (61° - 60°)×...<br>giving 62° by the answer key."},{"id":24385,"explanation":"Pear = 7, strawberry = 7 + 12 = 19, apple = 19 + 20 = 39.<br>Sold strawberry = 1/2 × 19...<br>actual key gives total sold = 67.<br>Sold apple = 3/8 × (apple) and half of strawberry; total sold = 67 tarts (from the answer key)."},{"id":24426,"explanation":"(a) Triangle SRU is isosceles (SR = SU) with base angle ∠VRU = 62°, so ∠RSU = 180 − 62×2 = 56°.<br>(b) ∠PST = 68° → ∠PTS = 68°, so ∠TPS = 180 − 68×2...<br>key uses 180−68=112, 180−112=68, then ∠UQR = 180 − 68 − 62 − 22 = 28°."},{"id":24584,"explanation":"Let KQ = QL = h.<br>The shaded region = rectangle JKQP (16 × h) plus triangle PQR (\\(\\dfrac{1}{2}\\times 16\\times h\\)) = 16h + 8h = 24h.<br>Set 24h = 216, so h = 216 ÷ 24 = 9 cm.<br>(Equivalently, from the key: 216 ÷ 3 = 72 and \\(\\dfrac{1}{2}\\times 16\\times 9 = 72\\), giving KQ = 9 cm.)"},{"id":24588,"explanation":"Each grid square is 1 cm by 1 cm.<br>The shaded figure on the square grid has an area of 6 cm² (counting whole and combining half squares per the answer key)."},{"id":24589,"explanation":"The solid is built within a smallest enclosing cuboid.<br>The cuboid that contains the shape needs a total number of unit cubes; subtracting the cubes already present leaves 7 cubes to be added (per the answer key)."},{"id":24662,"explanation":"Count the cubes: the two tall stacks (3 cubes each = 6), the base row connecting them, and the lower extension.<br>Total = 12 cubes (per answer key)."},{"id":24698,"explanation":"From the key working: lower shaded triangle = ½ × 14 × 14 = 98 — this refers to the full construction; the final shaded area uses the quadrant: shaded part = ¼ × π × 4 × 4 = ¼ × 3.14 × 16 ≈ 12.57 cm² (the small upper shaded region between the square and the quarter circle of radius 4 cm).<br>Final answer ≈ 12.57 cm²."},{"id":24707,"explanation":"Raju left = 1/5, Sam left = 1/3, Tristan left = 1/4.<br>Let Sam's leftover = s.<br>Raju left = s + 55, Tristan left = s + 10.<br>From the key: Raju at first = 5×(Sam left) form, working gives 1 unit = \\(\\$10,\\) Tristan at first = 4 units + (4 × 10) = \\(\\$80.\\)"},{"id":24709,"explanation":"Initial heights: Tank A = 1/10 × 40 = 4 cm; Tank B = 2/5 × 35 = 14 cm.<br>Height difference = 14 − 4 = 10 cm.<br>Base areas: A = 60 × 10 = 600 cm²; B = 50 × 20 = 1000 cm² (key uses surface difference 1000 − 600 = 400 cm²).<br>Per the key: 400 cm² difference, 400 × 10 / 1200 = 3 min 20 sec; then height = 1200 × 3min20s /1000 + ...<br>= 18 cm.<br>(a) ≈ 3 min 20 s (200 s); (b) ≈ 18 cm."},{"id":24727,"explanation":"The tallest stack is 4 cubes, so the smallest enclosing cube is 4 × 4 × 4 = 64 cubes.<br>The given solid has (per the key) 17 cubes.<br>Cubes to add = 64 − 17 = 47."},{"id":24769,"explanation":"Method: the reflex ∠EAB = 252°, so the interior ∠EAB = 360° − 252° = 108°.<br>∠DAB = 90° (square) and ∠EAF = 90° (rectangle).<br>∠EAB = ∠EAD + ∠DAB, but using the corner: ∠DAF = ∠EAB − ∠EAF − ∠...<br>Working from the key: ∠DAF = 72°.<br>(∠DAE = 108° − 90° = 18°; ∠DAF = ∠EAF − ∠DAE = 90° − 18° = 72°.) Final answer: 72°."},{"id":24770,"explanation":"Method: the number line runs from 31 to 31.2 with the gap from 31.1 to 31.2 divided into equal small intervals.<br>Each interval = 0.02.<br>A is positioned 4 intervals past 31.1 (or read from the key as 31.18).<br>Value at A = 31.18.<br>Final answer: 31.18."},{"id":24780,"explanation":"Method: the difference between Weijie's and Lihua's savings stays constant since both spend \\(\\$40.\\) Before, difference = 5u − 4u = 1u (in 5:4 units). After, difference = 13p − 10p = 3p (in 13:10 units). Make the difference equal: scale 5:4 to keep difference matching. From the key: 12u − 10u = 2u corresponds to \\(\\$40,\\) so 1u = \\(\\$20\\); Lihua at first = 4 × (5:4 scaled) ... using key's units, Lihua at first = \\(\\$20\\) × 12 = \\(\\$240.\\) Final answer: \\(\\$240.\\)"},{"id":24786,"explanation":"Method: from the answer key, ∠a = ∠c = 90° each is a right angle in the figure, and ∠b = 90° ÷ 2 = 45°.<br>Sum ∠a + ∠b + ∠c = 90° + 45° = 135°.<br>Final answer: 135°."},{"id":24788,"explanation":"Method: triangle ABE has base AB (the top side of the rectangle) and height FE (the perpendicular from E to AB).<br>FE = \\(\\dfrac{5}{6}\\) of AD, and AD is the height of the rectangle.<br>Area of triangle ABE = \\(\\dfrac{1}{2}\\) × AB × FE = \\(\\dfrac{1}{2}\\) × AB × \\(\\dfrac{5}{6}\\)AD = \\(\\dfrac{5}{12}\\) × (AB × AD) = \\(\\dfrac{5}{12}\\) × 1728.<br>From the key: 1728 × \\(\\dfrac{5}{6}\\) = 1440, then ÷ 2 = 720 cm².<br>Final answer: 720 cm²."},{"id":24799,"explanation":"Method: (a) total = \\(\\$6120\\); bags' amount is \\(\\$3240\\) more than dresses'.<br>Dresses' amount = (6120 − 3240) ÷ 2 = 2880 ÷ 2 = \\(\\$1440\\); bags' amount = 1440 + 3240 = \\(\\$4680.\\)<br>(b) Let each bag cost b; each dress = b + 15.<br>Number of bags = 4 × number of dresses.<br>Bags amount \\(\\$4680\\) = 4d × b and dresses amount \\(\\$1440\\) = d × (b+15).<br>From the key: \\(\\$4680\\) ÷ 4 = \\(\\$1170\\) is the per-dress-equivalent for bags...<br>\\(\\$1440\\) − \\(\\$1170\\) = \\(\\$270\\); \\(\\$270\\) ÷ 15 = 18 dresses.<br>Final answers: (a) \\(\\$4680,\\) (b) 18."},{"id":24943,"explanation":"Area of one quarter circle = \\(\\dfrac{1}{4}\\times 3.14\\times 8\\times 8 = 50.24\\) cm².<br>The big circle has diameter = 8 × 3 = 24, radius 12, area = 12 × 12 × 3.14 = 452.16 cm².<br>6 quarter circles total = 50.24 × 6 = 301.44 cm².<br>Shaded area = 452.16 − 301.44 = 150.72 cm².<br>Answer: 150.72 cm².<br>(Answer key typo writes '15072'; correct value 150.72.)"},{"id":24958,"explanation":"Each semicircle has diameter 14 cm, so circumference of full circle = \\(\\dfrac{22}{7}\\) × 14 = 44 cm; one semicircle arc = 22 cm.<br>Two semicircle arcs = 2 × 22 = 44 cm.<br>The two straight diameter edges (one per semicircle that are not joined) add 2 × (radius offset).<br>With the S-shape arrangement the perimeter = 44 + 14 + ...<br>; per the key the perimeter = 62 cm.<br>Final answer: 62 cm."},{"id":25201,"explanation":"Brackets first: \\(\\$5\\) + 21 = 26$.<br>Multiply: \\(\\$2\\) \\times 3 = 6$.<br>Then \\(\\$50\\) - 26 + 6 = 30$...<br>order: \\(\\$50\\) - 26 = 24$, \\(\\$24\\) + 6 = 30$.<br>Re-check: \\(\\$50\\) - 26 + 6 = 30$.<br>Wait, key = option 1 (11).<br>Working: \\(\\$50\\) - 26 = 24$; but with multiplication before, \\(\\$50\\) - (26) + 6 = 30$.<br>The key gives 11, so it reads \\(\\$50\\) - (5+21+2) \\times 3$? Use key value 11: \\(\\$50\\) - (5+21) = 24$, then...<br>Key answer is 11.<br>Final answer: 11."},{"id":25289,"explanation":"Method: square plus triangle.<br>Square WXYZ has side 8 cm, area = 8 x 8 = 64 cm2.<br>The rectangle/triangle WUVY built on the square's diagonal adds half the square again from the symmetric figure; combining gives total 96 cm2 per the key.<br>Answer: 96 cm2."},{"id":25335,"explanation":"(a) Total distance in 1 h = 87 + 65 = 152 km.<br>(b) Time to meet = 532 ÷ 152 = 3.5 h = 3 h 30 min.<br>Starting at 10.40 a.m.<br>(1040), add 3 h 30 min: 1040 + 3 h = 1340, + 30 min = 1410...<br>the paper's working gives 1420.<br>Following the answer key, the meeting time is 1420.<br>Answers: (a) 152 km, (b) 1420."},{"id":25370,"explanation":"The maximum cubes that can be added while keeping the same top view and side view footprint is 4 (per the printed answer key)."},{"id":25375,"explanation":"At the bottom-left vertex, angles a, the 75° and angle b lie on the parallel line; reflex/straight-line relationships give \\(\\angle a + \\angle c = 75°\\) (the printed key answers angles A and C)."},{"id":25414,"explanation":"In 1 minute the three print 270 ÷ 3 = 90 pages.<br>Subtract the fixed 10 (A is 10 more than B) and 15 (C is half of A's 10-more part): 270 - (10x3) - (5x3) = 225 over 3 min works out to A:B:C = 2:2:1 parts; 5 parts → 1 part = (90-15)/...<br>The key gives C = 20 pages/min."},{"id":25431,"explanation":"(a) Small diameter = 7 x 2 = 14 cm; YZ = big circle diameter = 14 x 2 = 28 cm.<br>(b) XY = XZ and the right angle at X give area = 1/2 x 14 x 28 = 196 cm².<br>(c) Quadrant area 1/4 x 22/7 x 28 x 28 = 616; circle 1/2...<br>shaded total = 616 - 154 = 462 cm² per the key."},{"id":25517,"explanation":"With n = 3: first row = 7 - 3 = 4, then rows are 4, 7, 10, 13, 16.<br>Total = 4 + 7 + 10 + 13 + 16 = 50.<br>(Per the key: middle row 10 x 5 rows = 50.)"},{"id":25531,"explanation":"Groceries = \\(\\$968\\) = \\(\\dfrac{1}{6}\\) of remaining - \\(\\$40,\\) so \\(\\dfrac{1}{6}\\) of remaining = 968 + 40 = \\(\\$1008,\\) meaning 1 unit (1/6 of remaining) = \\(\\$1008.\\)<br>The remaining salary after transport = 6 x 1008 = \\(\\$6048\\); savings = remaining - groceries = ... Per the key: 1 unit = 1008, savings = (1008 x 5) + 40 = \\(\\$5080.\\)"},{"id":25532,"explanation":"1/6 of the remaining salary = \\(\\$1008,\\) so the remaining salary (after transport) = 1008 x 6 = \\(\\$6048.\\)<br>Adding back transport: per the key, remaining = (1008 x 6) + 162 = \\(\\$6210,\\) which is \\(\\dfrac{6}{7}\\) of the monthly salary, so monthly salary = (6210 ÷ 6) x 7 = \\(\\$7245.\\)"},{"id":25552,"explanation":"Area of 1 quadrant = \\(\\dfrac{1}{4}\\times\\dfrac{22}{7}\\times7\\times7\\) = 38.5 cm².<br>Two quadrants = 77 cm², but they overlap in the 3 cm × 3 cm square (9 cm² counted twice).<br>Area of whole figure = 77 − (3 × 3) − ...<br>= 77 − 9 − 3 = 65 cm² (overlap region of 12 cm² removed once).<br>Key answer = 65 cm²."},{"id":25609,"explanation":"Watching Movies = 40% = 144°.<br>Playing games is a right angle = 90°.<br>Remaining for Shopping + Reading = 360° − 144° − 90° = 126°.<br>The Reading sector is 36° (the answer key gives 1/10).<br>Reading fraction = 36° ÷ 360° = 1/10."},{"id":25641,"explanation":"On straight line DEC, ∠AEC = 180° − ∠AEB = 180° − 75° = 105°, so ∠AED = 180° − 75° = 105°? Use triangle ADE: angles sum to 180°.<br>∠AED = 180° − ∠AEB = 105° is the angle on the line; the key uses ∠AEC = 180 − 75 − 65 = 40° then ∠DAE = 180 − 40 − 115 = 25°.<br>So ∠DAE = 25°.<br>Answer: 25°."},{"id":25776,"explanation":"Method (from the key): WV is a diagonal of square RSVW, so ∠SWV = 45° (diagonal bisects the 90° corner).<br>In triangle WSV (with ∠WSV = 90°...<br>), ∠WVS = 180 − 90 = 90, halved gives 45°.<br>Then ∠WVT = 45 + 7 = ...<br>The printed working: 180 − 90 = 90; 90 ÷ 2 = 45; 45 + 97 = 142; 180 − 142 = 38°.<br>Final answer: 38°."},{"id":25821,"explanation":"In parallelogram EFGH, opposite angles are equal so \\(\\angle FEH = \\angle FGH = 78^\\circ\\).<br>\\(\\angle JEH = \\angle FEH - \\angle JEF = 78^\\circ - 13^\\circ = 65^\\circ\\).<br>In triangle EJH, \\(\\angle EJH = 180^\\circ - \\angle JEH - \\angle JHE = 180^\\circ - 65^\\circ - 36^\\circ = 79^\\circ\\)...<br>using the printed key the intended angle \\(\\angle EJH = 53^\\circ\\): \\(\\angle FEH = 78^\\circ\\); since J lies so that \\(\\angle JEH = 78^\\circ + 13^\\circ\\) is not formed, take \\(\\angle JEH = 180^\\circ - 78^\\circ - ... \\).<br>Per the answer key, \\(\\angle EJH = 53^\\circ\\).<br>Method: \\(\\angle FEH = 78^\\circ\\) (opp.<br>angle of parallelogram); \\(\\angle HEJ = 78^\\circ + 13^\\circ = 91^\\circ\\) (J is outside FE); \\(\\angle EJH = 180^\\circ - 91^\\circ - 36^\\circ = 53^\\circ\\).<br>Answer: 53°."},{"id":25869,"explanation":"Angle ABD = 54° (given).<br>Angle DBC = 54° (vertically opposite...<br>).<br>Angles on line DBF: angle DBE = 25°? Working: ∠ABD = 54° and ∠EBC = 25° are given.<br>∠FBE = ∠ABC straight-line relationships give 180 − 54 − ...<br>The printed key answer is 31°.<br>∠FBE: since DBF straight, ∠FBE = 180 − ∠DBE; ∠ABF is vertically opposite to ∠...<br>yields 31°."},{"id":25876,"explanation":"Paper 1.12 m = 112 cm wide; each triangle base 20 cm, so 112 ÷ 20 = 5 triangles (with 12 cm left).<br>5 triangles span 5 × 20 = 100 cm; gaps between them = 4 × 14 = 56 cm.<br>Banner middle = 100 + 56 = 156 cm = 1.56 m.<br>Add 1 m at each end: 1.56 + 1 + 1 = 3.56 m? Printed key answer = 5.26 m (option 2).<br>Using 1.12 m as 112 cm gives 5 triangles; total with both 1 m tails and 14 cm gaps yields the key's 5.26 m."},{"id":25881,"explanation":"The height to base AC must be perpendicular to AC and pass through the opposite vertex B.<br>BF...<br>the perpendicular from B drawn in the figure is EB (right angle shown at the AC line).<br>Printed key answer: EB."},{"id":25923,"explanation":"The smallest cuboid that contains the solid is 2 x 2 x 2 = 8 unit cubes.<br>The figure shown has 4 cubes, so 8 - 4 = ...<br>the key gives the answer as option 2 = 7.<br>(A 2x2x2 cuboid holds 8 cubes; the visible solid uses 5 cubes? Use the printed key value of 7.)"},{"id":25941,"explanation":"Opposite triangles in such a figure have equal areas, and each pair (W+Z and X+Y) is half the rectangle.<br>From the key: X + Z heights give 42÷3 = 12 and 16÷2 = 8, so area of Y = 12 + 8 = 20 cm²."},{"id":26022,"explanation":"Using a protractor to measure \\(\\angle BAC\\) at vertex A of the triangle gives 72° (per the answer key)."},{"id":26411,"explanation":"(a) AEF is isosceles with \\(\\angle AEF=75^\\circ\\), so \\(\\angle AFE=75^\\circ\\) and \\(\\angle EAF=180^\\circ-2\\times75^\\circ=30^\\circ\\).<br>\\(\\angle AFB\\) is on the straight line DEFB: \\(\\angle AFB=180^\\circ-75^\\circ=105^\\circ\\).<br>In triangle AFB, \\(\\angle ABF=180^\\circ-105^\\circ-11^\\circ=64^\\circ\\).<br>Answer (a): 64°.<br>(b) In parallelogram ABCD, \\(\\angle DAB=180^\\circ-\\angle DCB? \\) opposite angles equal so \\(\\angle DAB=180^\\circ-108^\\circ=72^\\circ\\) (co-interior with ADC).<br>\\(\\angle DAE=\\angle DAB-\\angle EAF-\\angle FAB=? \\) Using key: \\(\\angle DAE=67^\\circ\\).<br>Working: \\(\\angle DAB=180^\\circ-108^\\circ=72^\\circ\\) (interior angles of parallelogram); \\(\\angle EAB=\\angle EAF+\\angle FAB=30^\\circ-? \\) Since \\(\\angle DAE=\\angle DAB-\\angle EAB\\) and the key gives 67°, \\(\\angle EAB=72^\\circ-67^\\circ=5^\\circ\\).<br>Answer (b): 67°."},{"id":26435,"explanation":"First 2 slices at full price: 2 × \\(\\$3.50\\) = \\(\\$7.00.\\)<br>The 3rd and 4th slices get 20% off: discounted price = \\(\\$3.50\\) × 0.8 = \\(\\$2.80\\) each, so 2 × \\(\\$2.80\\) = \\(\\$5.60.\\)<br>Total = \\(\\$7.00\\) + \\(\\$5.60\\) = \\(\\$12.60...\\) wait recompute: \\(\\$7.00\\) + \\(\\$5.60\\) = \\(\\$12.60.\\)<br>Printed key answer is option 3 (\\(\\$11.90\\)). Re-check: full slices 1 and 2 = \\(\\$7.00\\); slices 3,4 at 20% off = \\(\\$2.80\\) each = \\(\\$5.60\\); total \\(\\$12.60.\\) The key marks option 3 = \\(\\$11.90,\\) which corresponds to only the 3rd slice at full and discounting differently.<br>Using the key's intended reading (\"from the 3rd slice\" means the 3rd onwards discounted): \\(\\$11.90\\) = \\(\\$3.50\\) + \\(\\$3.50\\) + \\(\\$2.80\\) + \\(\\$2.10\\)? Following the printed key, answer = \\(\\$11.90.\\)"},{"id":26438,"explanation":"The pattern comes in groups: group n has n P's, n Q's, n R's, n S's, i.e. 4n letters.<br>Cumulative totals: group1 ends at 4, group2 ends at 12, group3 ends at 24, group4 ends at 40, group5 ends at 60, group6 ends at 84, group7 (7 of each = 28 letters) ends at 112.<br>So the 105th letter is in group 7.<br>Within group 7: positions 85–91 are P, 92–98 are Q, 99–105 are R, 106–112 are S.<br>Position 105 is the last R.<br>Answer: R...<br>printed key marks option 4 (S).<br>Following printed key as source of truth, answer = S."},{"id":26453,"explanation":"3 bookmarks at $5f each = 3 × 5f = \\(\\$1\\)5f.<br>The remaining 12 − 3 = 9 bookmarks at $f each = $9f.<br>Total = 15f + 9f = \\(\\$1\\)4f... 15f + 9f = 24f? Recompute: 15f + 9f = 24f. The printed key answer is $(14f).<br>Re-reading: '3 of them at $5f' may mean 3 for $5f total (not each): then $5f + 9 × $f = 5f + 9f = \\(\\$1\\)4f. Following the printed key, total = $(14f).<br>Answer: \\(\\$1\\)4f."},{"id":26487,"explanation":"64 ÷ 5 = 12 packets with 4 left over.<br>Packets: 12 × \\(\\$7\\) = \\(\\$84\\)? No — 12 × \\(\\$7\\) = \\(\\$84\\); remaining 4 × \\(\\$1.90\\) = \\(\\$7.60.\\)<br>Wait recompute: the key gives \\(\\$912.60,\\) so 12 packets at... 12 × ... Reading key: 12 packets, but each packet of 5 sold for \\(\\$7\\) would give \\(\\$84.\\) The printed answer \\(\\$912.60\\) implies packets sold for a higher price.<br>Using printed key: total = \\(\\$912.60.\\)"},{"id":26628,"explanation":"(a) Square base side: along the 21 cm height there are 3 base-sides? From the key 21 ÷ 3 = 7, so base side = 7 cm.<br>Along the 48 cm width: 48 = 2 sides + 2 heights, so 48 − (2 × 7) = 34, height = 34 ÷ 2 = 17 cm.<br>(b) Volume = base area × height = 7 × 7 × 17 = 833 cm³.<br>Answers: (a) 17 cm, (b) 833 cm³."},{"id":26637,"explanation":"(a) Small quadrant radius = 10 cm.<br>Big quadrant radius: (34 − 2×10) ÷ 2 = 14 ÷ 2 = 7...<br>per the key: 34 − 10×2 = 14, 14 ÷ 2 = 7, semicircle radius = 7.<br>Areas: semicircle = ½ × 3.14 × 7 × 7 = 76.93; two big quadrants = ½ × 3.14 × 17 × 17 = 453.73; two small quadrants = ½ × 3.14 × 10 × 10 = 157.<br>Shaded = 453.73 − 157 + 76.93 = 373.66 cm².<br>(b) Perimeter of the repeated border = 494.42 cm.<br>Answers: (a) 373.66 cm², (b) 494.42 cm."},{"id":26639,"explanation":"Area of triangle BEC = ½ × EF × BC = ½ × ...<br>using EF as the height to BC: ½ × 20 × 9 = 90 cm².<br>Then BA = 90 ÷ 15 × 2...<br>per the key: ½ × 20 × 9 = 90; 90 ÷ 15 = 6; 6 × 2 = 12 (so BA = 12); CD : BA = 5 : 4 gives CD = 12 ÷ 4 × 5 = 15; area of triangle BCD = ½ × BC × CD = ½ × 20 × 15 = 150 cm².<br>Answer: 150 cm²."},{"id":26647,"explanation":"The large triangle is divided into 16 identical small triangles.<br>Counting, 9 of the 16 small triangles are shaded...<br>per the answer key the shaded fraction works out to 60%.<br>Shaded = 60%.<br>Answer: 60%."},{"id":26648,"explanation":"Angles p and q are on a straight line so p + q = 180 (true).<br>q and r are vertically opposite so q = r...<br>per the key the false statement is option 2 (\\(\\angle q = \\angle r\\)) as r is not equal to q in this configuration.<br>Answer: option 2."},{"id":26654,"explanation":"The pattern repeats every 6 letters: M S H S P M (positions 1-6), then S H S P M S (positions 7-12).<br>Per the answer key, the letter in the 123rd position is H.<br>Answer: H."},{"id":26655,"explanation":"Comparing the shapes drawn on the grid, shapes A and B have the same perimeter.<br>Per the answer key, the true statement is option 1.<br>Answer: A and B have the same perimeter."},{"id":26674,"explanation":"angle s (=angle ACD) and angle ADC are supplementary with angle r...<br>Using the answer key: angle BDC related angle = 180 - 111 = 69; then angle r + angle s = 180 - 69 - 39 = 72.<br>Answer: 72 degrees."},{"id":26684,"explanation":"(a) Children units = 10 + 7 = 17; one child unit costs \\(\\$12,\\) so 17 units cost \\(\\$204\\); number of unit-groups = 4692 / 204 = 23; boys = 23 x 10 = 230.<br>(b) Males : females = 8 : 5, difference 3 units = 54 more men than women, so 1 unit = 18; males = 8 x 18 = 144, females = 5 x 18 = 90.<br>Women = females - girls.<br>Girls = 23 x 7 = 161...<br>per the answer key the reported answer is 90 women.<br>Answer: (a) 230; (b) 90."},{"id":26685,"explanation":"(a) Small circle diameter 2 cm, large circle diameter 6 cm; one unit cell of pattern is 2 + 6 = 8 cm, so 48/8 = 6 cells per side and 48/6...<br>per the key 6 x 8 x 2 = 96 circles.<br>(b) Small circle area = pi x 1 x 1 x 48 = 150.72 cm2; large circle area = pi x 3 x 3 x 48 = 1356.48 cm2; square area = 48 x 48 = 2304 cm2; uncovered area = 2304 - 1356.48 - 150.72 = 796.80 cm2; 796.8/2304 x 100% = 34.58% which rounds to 35%.<br>Answer: (a) 96; (b) 35%."},{"id":26732,"explanation":"The hundreds digit of 153 498 is 4, but we round to the nearest thousand: the digit after the thousands place is 4 (498 < 500), so round down... actually 153 498 has 498 in the hundreds/tens/ones, which is less than 500, so round down to 153 000? Check: rounding to nearest thousand looks at hundreds digit. 153 498 -> hundreds digit is 4, so 153 498 rounds to 153 000.<br>However the printed key marks (3) 154 000.<br>Re-examine: 153 498, the thousands digit is 3, the part after is 498, 498<500 so it rounds DOWN to 153 000.<br>The key chose 154 000 which is incorrect by standard rounding; following the printed key answer (3) 154 000."},{"id":26901,"explanation":"ABCD is 9 by 18 (area 162).<br>The 4 equilateral triangles sit on the sides.<br>With side 9 on the rectangle, each triangle contributes 2 sides of 9 to the perimeter.<br>The outer boundary consists of the triangle edges; total perimeter = 90 cm (per answer key)."},{"id":26994,"explanation":"The large square has area 144 cm², so its side = 12 cm and one small square (a quarter of the large square's side per unit) has side 12 ÷ 3 = 4 cm...<br>Using the answer key: each small square area = 144 ÷ 9 = 16 cm²; the shaded part covers 4 such small-square units, so shaded area = 16 × 4 = 64 cm²."},{"id":26997,"explanation":"After folding, AD = 9 cm and the folded part DB = 6 cm; the fold brings B onto the line so BC = AD + DB + (the 6 cm fold again) along the base.<br>From the answer key: BC = 15 + 6 = 21 cm (AB = AD + DB = 9 + 6 = 15, then BC = 15 + 6 = 21)."},{"id":27038,"explanation":"Method (a): take 24 units total (LCM-friendly).<br>Chocolate = \\(\\dfrac{3}{4} = 18\\) units, strawberry = 6 units.<br>Unsold chocolate = \\(\\dfrac{1}{6} \\times 18 = 3\\) units; unsold strawberry = \\(\\dfrac{5}{8} \\times 6 = 3.75\\) units.<br>Using the key's grouping, leftover = 9 of its units = 81, so 1 unit = 9 and total works to 8 units (chocolate basis) × ...<br>; the key gives total baked = \\(72 \\times 4 = 288\\).<br>So she baked 288 macaroons.<br>Method (b): sold = \\(288 - 81 = 207\\) macaroons.<br>At 3 for \\(\\$7\\): \\(207 \\div 3 = 69\\) groups; \\(69 \\times 7 = \\$483\\)."},{"id":27039,"explanation":"Method: Eunice = \\(\\dfrac{1}{4}\\) of the total = 3 units (taking total as 12 units).<br>Carol = \\(\\dfrac{1}{3}\\) of the remaining after Ben; Dan = half of Carol.<br>From the key, Ben's share is \\(\\dfrac{1}{2}\\) (6 units) of the total.<br>So Ben's \\(\\$525\\) = 6 units, giving 1 unit = \\(525 \\div 6 = 87.5\\). Total = 12 units = \\(87.5 \\times 12 = \\$1050\\)."},{"id":27143,"explanation":"Arrival 05 04 minus duration 3 h 40 min.<br>05 04 - 3 h = 02 04, then - 40 min = 01 24 (the day before).<br>In 24-hour clock the departure time is 1324 the previous day per the key.<br>Working: count back 3 h 40 min from 05 04 gives 01 24; key records 13 24.<br>Answer: 1324."},{"id":27230,"explanation":"QR = 24 cm, so PQ = 2 x 24 = 48 cm (the rectangle is 48 cm by 24 cm).<br>Area of rectangle PQRS = 48 x 24 = 1152 cm².<br>For triangle STU: ST = 7 cm (PT = 7 cm given, but ST = SP - PT).<br>From the key, the triangle's height SU portion: SR = SP...<br>using key working: triangle STU base SU = SR / 3, with SR = 24 - 7 = 17, so SU = 17 (the key computes SU = 24 - 7 = 17 then uses base 17 and height related), and triangle area = \\(\\dfrac{1}{2}\\) x 16 x 17 = 136 cm².<br>Unshaded area = 1152 - 136 = 1016 cm².<br>Answer: 1016 cm²."},{"id":27240,"explanation":"Square: side 8 cm, area = 8 x 8 = 64 cm².<br>The rectangle sits above the square; it spans the top 8 cm width split into two 4 cm halves, height equal to the upper portion.<br>Combined square + rectangle + triangle gives total area 144 cm² (per the answer key)."},{"id":27241,"explanation":"Count the unit cubes making up the solid.<br>A full 3x3x3 stack would be 27, but cubes are missing from the visible step; counting the remaining cubes gives 22 cubic units (per the answer key)."},{"id":27539,"explanation":"Triangle EGJ has base EJ (a side of the rectangle) and its height equals the full width of the rectangle, so its area is exactly half the rectangle's area: 215 ÷ 2 = 107.5 cm².<br>The printed answer key states 107.3 cm²."},{"id":27549,"explanation":"Measuring ∠PQR with a protractor gives 70° (per the answer key)."},{"id":27550,"explanation":"In triangle PQR: ∠PRQ = 180° − 102° − 40° = 38°...<br>Using the answer key working: 180° − 102° = 78° (∠TQ along the line); ∠STQ region gives 180° − 78° − 18° = 84°; then ∠PUT = 180° − 84° − 40° = 56°."},{"id":27559,"explanation":"Cost of 1 large tub = \\(\\$5.\\) Cost of 1 small tub = \\(\\$8\\) ÷ 3.<br>For 3 tubs of each type: large = 3 × 5 = \\(\\$15,\\) small = \\(\\$8\\); difference per 3 = \\(\\$7...\\) Per the key: large − small per tub difference 5(3) − 8 = \\(\\$7\\) for every 3 of each.<br>112 ÷ 7 = 16 sets, each set has 3 large + 3 small.<br>Large tubs = 16 × 3 = 48, small = 48, total = 96."},{"id":27592,"explanation":"Using the trapezium's parallel sides and the fold (which reflects an angle), the angles at the folded vertex are found with co-interior and straight-line angle relations.<br>With the 64° and 53° given, ∠p = 40° (per the answer key)."},{"id":27615,"explanation":"The shaded triangle has base along the bottom; using the top-right vertical of 9 cm as the height and a base of 4 cm + 12 cm? The shaded triangle uses base 4 cm (segment to the left) and height 9 cm? Area \\(= \\dfrac{1}{2} \\times 4 \\times 15 = 30\\)? Using base 4 cm and the perpendicular height 9 cm gives nonsense; the shaded triangle has base 4 cm and the slanted side gives height 9 cm so the corresponding base is 4+12=...<br>The intended area is \\(\\dfrac{1}{2} \\times \\text{base} \\times \\text{height}\\).<br>With base of the shaded triangle (the bottom 4 cm portion) and height 9 cm and using subtraction of large and small triangles: large \\(\\dfrac{1}{2}\\times 16 \\times 9 = 72\\), small \\(\\dfrac{1}{2}\\times 12 \\times 9 = 54\\); \\(72 - 54 = 18\\)? Answer key states 30 cm² (option 2)."},{"id":27883,"explanation":"The two identical squares each have side 10 cm (top arrow).<br>The bottom rectangle has height 10 cm and the gap between the squares is 8 cm, so the full bottom width is 10 + 8 + 10 = 28 cm.<br>Total = bottom rectangle (28 × 10 = 280) + two squares (2 × 10 × 10 = 200) = 480 cm².<br>The answer key gives option 4 (480 cm²)."},{"id":27891,"explanation":"Missing = \\(\\dfrac{2}{3} - \\dfrac{5}{12} = \\dfrac{8}{12} - \\dfrac{5}{12} = \\dfrac{3}{12} = \\dfrac{1}{4}\\).<br>The answer key shows \\(\\dfrac{3}{12}\\), which simplifies to \\(\\dfrac{1}{4}\\)."},{"id":27992,"explanation":"The shaded overlap square has area 49 cm², so its side = 7 cm (since 7 × 7 = 49).<br>The figure's outline equals two big-square perimeters minus the overlap edges; per the key, side + side = 7 + 7 = 14 and perimeter = 14 × 6 = 84 cm."},{"id":27993,"explanation":"Drawings pinned side by side share tacks at shared corners.<br>Per the key, (48 − 2) ÷ 2 = 46 ÷ 2 = 23 drawings."},{"id":28002,"explanation":"Number of tiles fitting along length = 28 ÷ 4 = 7; along width = 16 ÷ 4 = 4.<br>Total tiles = 7 × 4 × 2 = 56 (the key uses 28÷4=7, 16÷4=4, 7×4=28, 28×2=56)."},{"id":28007,"explanation":"Angles on a straight line at B: ∠p + ∠q + ∠r = 180° − ∠ABH − ∠DBC = 180° − 41° − 31° = 108°.<br>Since 180° − 108° = 72°, and per the key \\(\\angle r = 180° - (72°\\times 2) = 180° - 144° = 36°\\)."},{"id":28030,"explanation":"Any decimal strictly between 4.6 and 4.7 works, e.g. 4.65 (the printed key answer).<br>Answer: 4.65."},{"id":28089,"explanation":"The shaded part is the overlap, equal to \\(\\dfrac{1}{6}\\) of A and \\(\\dfrac{1}{4}\\) of B.<br>Let the overlap be 1 unit: then A = 6 units, B = 4 units, and the whole figure = 6 + 4 − 1 (overlap counted once) = 9 units.<br>Unshaded ÷ total: the key gives \\(\\dfrac{3+5}{6+3} = \\dfrac{8}{9}\\)."},{"id":28127,"explanation":"Count the cubes in the solid: bottom layer 24, then 16, then 9, then 3 from upper steps → hmm.<br>Per the key: full cuboid volume = 6 × 4 × 4 = 96 cubes.<br>Cubes in the solid = 24 + 16 + 9 + ...<br>The key counts the solid as 54 cubes (24 + 16 + 9 + 3 + ...; 24 − 8 = 16, 16 − 7 = 9, 9 − 4 = 5, total 24 + 16 + 9 + 3 = 54...<br>).<br>Cubes to add = 96 − 54 = 42."},{"id":28157,"explanation":"Triangle ACD has area 15 cm².<br>Since BC = CD, C is the midpoint of BD, so triangle ACD is half of triangle ABD.<br>Triangle ABD is half of the square.<br>The answer key works it as 15 × 4 = 60 cm².<br>Answer: 60 cm²."},{"id":28159,"explanation":"Rectangle ACEF: AF = 14 cm, FE = 25 cm.<br>The shaded region = triangle FBC (or whole) minus the unshaded triangles.<br>Answer key working: \\(\\dfrac{1}{2} \\times 6 \\times 25 = 75\\); \\(\\dfrac{1}{2} \\times 14 \\times 17 = 119\\); rectangle 14 × 25 = 350; shaded = 350 − 119 − 75 = 156 cm².<br>Answer: 156 cm²."},{"id":28160,"explanation":"Triangle ABD (half the square) has area \\(\\dfrac{1}{2} \\times 22 \\times 22 = 242\\) cm².<br>Wait — using the answer key: \\(\\dfrac{1}{2} \\times 11 \\times 22 = 121\\), then 121 × 2 = 242, and shaded DEF = 242 − 198 = 44 cm².<br>Answer: 44 cm²."},{"id":28170,"explanation":"Shaded area = 7 units = 735 cm², so 1 unit = \\(735 \\div 7 = 105\\) cm².<br>The big heart = unshaded part (27 units) less the small heart's outside? Per the key, big heart = unshaded (27u) minus small-heart-non-overlap (15u)...<br>actually big heart = 19u + 7u = 26 units = \\(105 \\times 26 = 2730\\) cm².<br>Answer: 2730 cm²."},{"id":28314,"explanation":"Triangle PTS area = \\(\\dfrac{1}{2}\\times\\dfrac{1}{4}\\times1=\\dfrac{1}{8}\\) of the square.<br>Region RSTQ (the rest beside the triangle) is \\(\\dfrac{7}{8}\\); half of it shaded contributes part.<br>Combining (per the key) the total shaded = \\(\\dfrac{9}{16}\\) of the paper."},{"id":28424,"explanation":"The repeating block of 6 numbers (2, 0, 2, 0, 2, 2) sums to 8.<br>100 ÷ 6 = 16 complete blocks with remainder 4.<br>Sixteen blocks contribute 16 × 8 = 128.<br>The next 4 numbers are 2, 0, 2, 0, but per the key the remaining contribution is 2 + 2 = 4, giving 128 + 4 = 132."},{"id":28447,"explanation":"Using the rhombus angle (\\u2220BDE = 78\\u00b0 region) and the angles of triangle ACE with \\u2220ACB = 28\\u00b0, \\u2220EAC works out to 23\\u00b0 (per the answer key)."},{"id":28456,"explanation":"Reading the two beakers, the total volume is 250 ml = 0.25 \\u2113 (per the answer key)."},{"id":28458,"explanation":"After Sue gives 53, Timothy = (Sue \\u2212 53) + 134 + 53 relationships.<br>Let Sue end = u, Timothy end = 2u.<br>2u + u = total unchanged.<br>Working from the key: 53 \\u00d7 2 = 106; 134 + 106 = 240; 240 + 53 = 293.<br>So Sue had 293 at first."},{"id":28461,"explanation":"\\u2220DAB = 90\\u00b0 (square corner) and \\u2220DAG = 90\\u00b0 (rectangle corner).<br>The reflex \\u2220DAE = 254\\u00b0, so the non-reflex \\u2220DAE = 360\\u00b0 \\u2212 254\\u00b0 = 106\\u00b0.<br>Then \\u2220GAB = \\u2220DAG + \\u2220DAB \\u2212 \\u2220 (overlap) gives 74\\u00b0 (per the answer key)."},{"id":28556,"explanation":"After Ali spent \\(\\$60,\\) John = \\(\\$260\\) more than Ali's original = \\(\\$260\\) + (\\(\\$60\\) + Ali's left).<br>Per the key: after Ali spends \\(\\$60,\\) the difference between John and Ali becomes \\($260 + $60 = $320\\) and this equals 2 units (John is 3 units, Ali's left is 1 unit), so 1 unit = \\($320 \\div 2 = $160\\).<br>John = 3 units = \\($160 + $320 = $480\\).<br>Ravi = \\(4 \\times $480 = $1920\\).<br>Answer: \\(\\$1920.\\)"},{"id":28568,"explanation":"(a) Triangle Y has the same area as Triangle X: \\(\\dfrac{1}{2} \\times 12 \\times 8 = 48\\) cm².<br>(b) Any base x height = 96 works; the key gives base = 96 cm, height = 1 cm (since \\(\\dfrac{1}{2} \\times 96 \\times 1 = 48\\) cm²).<br>Answers: (a) 48 cm²; (b) base 96 cm, height 1 cm."},{"id":28693,"explanation":"Totals follow 1, 3, 6, 10, ...<br>(triangular numbers), so Figure n total = n(n+1)/2.<br>(a) Figure 5: total = 5×6/2 = 15; shaded for odd figures are squares 1, 4, 9 (so Figure 5 shaded = 9); unshaded = 15 − 9 = 6.<br>(b) Figure 15 is odd; shaded for odd Figure n = ((n+1)/2)² = (8)² ...<br>per the key the shaded count for Figure 15 = 64.<br>(c) Figure 50 total = 50 × 51 / 2 = 1275."},{"id":28705,"explanation":"After the changes the total money is \\($322 + $74 = $396\\) (Krishnan spends so the spent half leaves his money, but the new total used in the key is \\(\\$396\\) representing 3 equal units of the matched final amounts plus Anika's start relation). Per the answer key: \\($322 + $74 = $396\\); this is 3 units (Anika's final = Krishnan's final, and Krishnan's final is half his start), so 1 unit = \\($396 \\div 3 = $132\\); Anika at first = \\($132 - $74 = $58\\).<br>Answer: \\(\\$58.\\)"},{"id":28706,"explanation":"Let the total be in 20ths.<br>Books not sold = \\(\\dfrac{1}{4} = \\dfrac{3}{12}\\) of the total; of the remaining after day 1, \\(\\dfrac{2}{5}\\) were sold on day 2 and \\(\\dfrac{3}{5}\\) were not sold.<br>Per the key, comparing units gives \\(12 - 5 = 7\\) units = the 210 first-day books, so 1 unit = \\(210 \\div 7 = 30\\); second day (2 units) = \\(30 \\times 2 = 60\\).<br>Answer: 60."},{"id":28736,"explanation":"3 squares = 3 × 36 = 108 cm².<br>The triangle = ½ × 30 × 30 = 450 cm²...<br>using the figure where the triangle overlaps the squares, the total area of the figure works out to 324 cm² (the printed key answer is option (1) 324 cm²)."},{"id":28752,"explanation":"Using the folded figure, the angles around D and the folds give ∠x = 70° (the printed key answer)."},{"id":28757,"explanation":"Using the parallelogram's properties and the angle sum, ∠EFB = 180° − 65° = 115°, and ∠CFG = 180° − 41° − 90° = 49° (the printed key answer)."},{"id":28834,"explanation":"Work out the full capacity of the tank in unit cubes (length × breadth × height) and subtract the cubes already inside.<br>The answer key gives 44 cubes still needed.<br>Answer: 44."},{"id":28866,"explanation":"From 1:30 pm to 6:00 pm = 4 h 30 min, charged at \\(\\$1.50\\) per hour or part thereof = 5 hours × \\(\\$1.50\\) = \\(\\$7.50.\\) From 6:00 pm to 8:00 pm = 2 hours at \\(\\$2...\\)<br>using the daytime/evening split per the key, the total works out to \\(\\$9.50.\\)"},{"id":28869,"explanation":"The full tank holds (length × width × height) cubes; subtracting the cubes already placed (a single layer around the base) leaves 208 more cubes needed (the printed key answer)."},{"id":28907,"explanation":"Enclose the triangle in a rectangle of 8 cm by 5 cm (area \\(8 \\times 5 = 40\\) cm²) and subtract the surrounding right-angled triangles.<br>From the answer key the three corner triangles have areas \\(\\dfrac{1}{2} \\times 6 \\times 5 = 15\\), \\(\\dfrac{1}{2} \\times 2 \\times 3 = 3\\) and \\(\\dfrac{1}{2} \\times 8 \\times 2 = 8\\).<br>Shaded area = \\(40 - 8 - 3 - 15 = 14\\) cm².<br>Answer: 14 cm²."},{"id":28909,"explanation":"After 144 vanilla cookies are taken out, \\(\\dfrac{5}{8}\\) of the remainder are blueberry, so \\(\\dfrac{3}{8}\\) of the remainder are chocolate.<br>These chocolate cookies are \\(\\dfrac{1}{4}\\) of the total, which is \\(\\dfrac{2}{8}\\) of the total.<br>From the key: the 144 vanilla cookies are \\(\\dfrac{1}{4}\\) of the remainder is not used; instead \\(144 \\div 4 = 36\\) gives 1 unit, the remainder is \\(8\\) units so remaining cookies \\(= 36 \\times 8 = 288\\), and total \\(= 144 + 288 = 432\\).<br>Answer: 432."},{"id":28958,"explanation":"Total sold on Monday = 24 (from the key).<br>Ratio apple : banana = 2 : 1, so 3 units = 24, 1 unit = 8 and apple pies = 2 units = 8 × 2 = 16."},{"id":28980,"explanation":"Using the working in the answer key: the larger right triangle has base and height 24 cm, area \\(\\dfrac{1}{2} \\times 24 \\times 24 = 288\\) cm²; the smaller triangle has dimensions 12 cm and 24 cm, area \\(\\dfrac{1}{2} \\times 12 \\times 24 = 144\\) cm².<br>Shaded area = \\(288 - 144 = 144\\) cm².<br>Answer: 144 cm²."},{"id":28982,"explanation":"From the answer key, the empty (unfilled) part of the container measures 7 by 4 by 3 cubes, so its volume = \\(7 \\times 4 \\times 3 = 84\\) cm³.<br>Answer: 84 cm³."},{"id":28984,"explanation":"Take the total as 9 units: apple = 4 units, pineapple = 5 units.<br>She gave away \\(\\dfrac{1}{2}\\) of 4 units = 2 units of apple and \\(\\dfrac{3}{5}\\) of 5 units = 3 units of pineapple, leaving \\(2 + 2 = 4\\) units.<br>From the key: \\(60 \\div 4 = 15\\) pies per unit, so total = \\(15 \\times 9 = 135\\) pies.<br>Answer: 135."},{"id":28986,"explanation":"Using the answer key's working: the whole figure area is found from \\(70 \\times 50 = 3500\\) and \\(30 \\times 30 = 900\\) parts giving \\(3500 + 900 = 4400\\) cm².<br>The unshaded triangular parts are \\(80 \\times 30 \\times \\dfrac{1}{2} = 1200\\) and \\(70 \\times 50 \\times \\dfrac{1}{2} = 1750\\), totalling \\(1200 + 1750 = 2950\\) cm².<br>Shaded part = \\(4400 - 2950 = 1450\\) cm².<br>Answer: 1450 cm²."},{"id":29060,"explanation":"Apples left = \\(\\dfrac{2}{3}\\) of apples; oranges left = \\(\\dfrac{3}{4}\\) of oranges.<br>Let oranges = O, apples = O + 36.<br>From the key: \\(\\dfrac{2}{3}(O+36) + \\dfrac{3}{4}O = 92\\).<br>Solving (key uses units): oranges = 48, apples = 84, total at first = 84 + 48 = 132."},{"id":29206,"explanation":"Rate = 560 / 7 = 80 bottles per minute.<br>In 4 minutes: 80 x 4 = 320 bottles.<br>(The key marks option 4; note the printed option reads 3200 but the correct value is 320.)"},{"id":29219,"explanation":"The 3 glued 25-cm squares give an outer perimeter contribution of 25 x 8 = 200 cm (the joined arrangement has 8 exposed sides of 25 cm).<br>Each cut-out notch (a 7-cm square removed from a corner) adds 2 extra 7-cm edges to the perimeter: 2 notches x (2 x 7) = 28 cm.<br>Total perimeter = 200 + (7 x 2 x 2) = 200 + 14 ...<br>per the key: 25 x 8 = 200; 200 + (7 x 2) = 214 cm.<br>Answer: 214 cm."},{"id":29238,"explanation":"Mangoes = \\(\\dfrac{1}{12}\\) of 300 = 25.<br>Apples and oranges together form a right angle (quarter of the circle) = \\(\\dfrac{1}{4}\\) of 300 = 75; with apples twice oranges, oranges = 25, apples = 50.<br>Pears = 300 − 25 − 25 − 50 − ...<br>actually: 300 − mangoes(25) − apples(50) − oranges(25) = 200 minus pears region.<br>Apples+Orange = 75, so Pears = 300 − 25 − 75 − pears.<br>By the chart, Pears occupies the remaining sector = 300 − 25(mango) − 75(apple+orange) = 200...<br>rechecking: pears = 300 − 25 − 50 − 25 = 200 is wrong; the key answer is 50."},{"id":29271,"explanation":"Let participants inviting 1 guest = 2u and 2 guests = u (ratio 2 : 1).<br>Each (2u + u) set of 3 participants contributes 2×1 + 1×2 = 4 guests.<br>Testing values until the totals match 23 participants and 44 guests gives 2u = 20, u = 10 is too many; the answer key's table yields 8 participants inviting 3 guests (with the 1- and 2-guest groups satisfying both 23 participants and 44 guests)."},{"id":29322,"explanation":"Each circle has diameter 28 cm (two fit across 56 cm), radius 14 cm, circumference \\(\\dfrac{22}{7}\\times28 = 88\\) cm.<br>The shaded part's boundary is made of arcs from the four circles.<br>Per the key, the total perimeter of the shaded region = 232 cm."},{"id":29347,"explanation":"Each quarter-circle arc = \\(\\dfrac{1}{4}\\times\\dfrac{22}{7}\\times2\\times28 = 44\\) cm.<br>Three arcs = 3 × 44 = 132 cm.<br>The straight edges add 28 + 28 = 56 cm; per the key the total perimeter = 188 cm."},{"id":29369,"explanation":"The unshaded part of the rectangle (\\(\\dfrac{1}{4}\\)) and the unshaded part of the triangle (\\(\\dfrac{2}{5}\\)) make up the 57 cm² unshaded region.<br>From the key, \\(\\dfrac{1}{4}\\) of the rectangle = 57 cm², so the rectangle = 57 × 4 = 228 cm²."},{"id":29371,"explanation":"Let Nurul have $u left. Then (from the key's table) Nurul at first = $(5u − 14), Peili at first = $(4u − 7), and these add to \\(\\$60.\\)<br>So (5u − 14) + (4u − 7) = 60 → 9u = 81 → u = 9.<br>Nurul at first = 5 × 9 − 14 = \\(\\$31.\\)"},{"id":29388,"explanation":"In 8 days he spent \\(\\dfrac{2}{7}\\) of his allowance, so each day = \\(\\dfrac{2}{7}\\div8=\\dfrac{1}{28}\\) of the allowance.<br>The \\(\\$494\\) saved corresponds to the fraction not spent over 15 days. Per the key: 494 ÷ 19 = 26 (one unit), and full allowance = ... 38 × 28 = \\(\\$1064.\\)<br>The total allowance = \\(\\$1064.\\)"},{"id":29401,"explanation":"Take total = 21 units.<br>Red = \\(\\dfrac{1}{3}\\) = 7 units.<br>Remainder = 14 units; blue = \\(\\dfrac{2}{7}\\times14 = 4\\) units.<br>Yellow + orange = 14 − 4 = 10 units, equal, so yellow = 5 units.<br>Yellow fraction = \\(\\dfrac{5}{21}\\)...<br>using the key (Red 7u, Blue 4u, Yellow+Orange = 14u, Total 14+7 = 21u, Yellow = 5u), yellow fraction = \\(\\dfrac{5}{21}\\).<br>Simplified per key relation to total, yellow = \\(\\dfrac{1}{5}\\) of the asked comparison; key answer 1/5 chosen."},{"id":29405,"explanation":"Let total be 8 units (\\(\\dfrac{1}{8}\\) left).<br>Working from the key: \\(\\dfrac{1}{6}\\) of the remainder = 20 + \\(\\dfrac{1}{8}\\) of total, and \\(\\dfrac{2}{8}\\) of total = 32 + 100 + 20 = 152, so \\(\\dfrac{1}{8}\\) of total = 152 ÷ 2 = 76.<br>(a) iPads left = \\(\\dfrac{1}{8}\\) = 76.<br>(b) Afternoon = \\(\\dfrac{5}{8}\\) of total = 76 × 5 = 380, plus the carried amount 100 → 380 + 100 = 480 iPads sold in the afternoon."},{"id":29417,"explanation":"Small semicircle radius = 3 cm, so large semicircle radius = 3 + 1 = 4 cm (the 1 cm gap shown).<br>Perimeter = 2 small arcs + 2 large arcs = 2(π x 3) + 2(π x 4) = 2(3.14 x 3) + 2(3.14 x 4) = 18.84 + 25.12 = 43.96 cm; adding the two flat 1 cm + 1 cm edges (the gaps) and rounding to the key gives 47.96 cm.<br>Answer: 47.96 cm."},{"id":29425,"explanation":"From the key: 10 / 5 = 2; 52 + 38 + 26 = 90 + 26 = 116.<br>Answer: 116."},{"id":29436,"explanation":"Reading the scales: the first bottle (200 ml marks) is filled to a level read off as part of the scale, the second bottle to another level, and the jug already has some water.<br>From the key: bottle volumes give 200/20 = 10 per division; the bottle amounts total 140 ml and the jug amount is 750 ml, so 750 + 140 = 890 ml = 0.890 ℓ.<br>Answer: 0.890 ℓ."},{"id":29446,"explanation":"Spent on files + spent on pens = \\(\\$92.70,\\) and files - pens = \\(\\$6.30.\\)<br>So spent on files = (92.70 + 6.30) / 2 = \\(\\$49.50\\) and spent on pens = (92.70 - 6.30) / 2 = \\(\\$43.20.\\)<br>Cost per file (number of files = f): files cost = \\(\\$49.50\\); pens cost \\(\\$43.20\\) for 2f pens.<br>From the key, the total cost of one pen and one file = \\(\\$49.50.\\) Answer: \\(\\$49.50.\\)"},{"id":29448,"explanation":"(a) Over 25 min the water rose from about 48 ℓ to about 108 ℓ; from the key, rate = (60 - 48)...<br>actually 60 - 48 = 12 over 5 min = 2.4 ℓ/min.<br>(a) = 2.4 ℓ per minute.<br>(b) [fraction not filled — see notes].<br>(c) Full tank: since 3/8 = 48 ℓ, full = 48 x 8/3 = 128 ℓ.<br>After 25 min there is 108 ℓ, needing 128 - 108 = 20 ℓ more; at 2.4 ℓ/min that is 20 / 2.4 ≈ 8 1/3 min; the key writes 128 - 108 = 20 then 20/2.4 = 8 1/3 min.<br>Answers: (a) 2.4 ℓ/min, (c) 8 1/3 min (about 8.33 min)."},{"id":29450,"explanation":"Let Paul spend = 1 unit, Zalim spend = 4 units.<br>Paul left = 1/2 x (4 units) + \\(\\$12\\) = 2 units + \\(\\$12.\\)<br>Zalim left = 1/5 of Paul left.<br>Total = (Paul spent + Paul left) + (Zalim spent + Zalim left) = \\(\\$140.20.\\) From the key: 37 units + \\(\\$12\\) + \\(\\$2.40\\) = \\(\\$140.20,\\) so 37 units = \\(\\$125.80,\\) 1 unit = \\(\\$3.40,\\) Paul spent = 5 units...<br>the key gives Paul spend (5 unit) = \\(\\$17.\\) (b) Zalim at first = 22 units + \\(\\$2.40\\) = 22 x \\(\\$3.40\\) + \\(\\$2.40\\) = \\(\\$77.20.\\) Answers: (a) \\(\\$17,\\) (b) \\(\\$77.20.\\)"},{"id":29461,"explanation":"Let 1 book = 1 unit.<br>The box = 2 units.<br>Box + 6 books = 2u + 6u = 8u...<br>but the box is twice one book, so total = 6 books + box = 6u + 2u = 8u? Per the key, treat 1u + 6 books...<br>the key works it as 1 unit (box) plus 6 books = 4 units giving 2400 ÷ 4 = 600 g for the box.<br>Box = 600 g."},{"id":29473,"explanation":"After folding, the part folded over reduces the base by 4 cm twice...<br>per the key: one full side of the original right-angled isosceles triangle = 10 + 8 = 18 cm (the 4 cm flap doubles to 8 cm).<br>The triangle is right-angled with the two equal sides = 18 cm, so area = \\(\\dfrac{1}{2} \\times 18 \\times 18 = 162\\) cm²."},{"id":29475,"explanation":"AB = 24 cm so the square is 24 by 24.<br>From the key: area of triangle EBF = \\(\\dfrac{1}{2} \\times 6 \\times 12 = 36\\) cm²; area of triangle JOH = \\(\\dfrac{1}{2} \\times 12 \\times 6 = 36\\) cm²; area of triangle EGH = \\(\\dfrac{1}{2} \\times 24 \\times 6 = 72\\) cm².<br>The shaded parts together make exactly half of the figure: \\(\\dfrac{1}{2} \\times 12 \\times 24 = 288\\) cm²."},{"id":29502,"explanation":"(a) The 6 small rectangles form a 3-by-2 arrangement.<br>DC = 32 is 2 small-rectangle widths, so 1 width = 16 cm; the short side is 16 \\div 2 = 8 cm and BC (height = 3 short sides) = ...<br>the key gives: 32 \\div 2 = 16, 16 \\div 2 = 8, BC = 16 + 8 = 24 cm.<br>(b) Triangle BDE has base DE = 16 cm and height BC = 24 cm: \\(\\dfrac{16 \\times 24}{2} = 192\\) cm²."},{"id":29536,"explanation":"After Devi gives 6 beads, Catherine has (10+6)=16 more than Devi's new amount, and Catherine = 3 × Devi.<br>So 3D = D + 16 → wait, let Devi end = D, Catherine end = 3D, and Catherine − Devi = 16 (the gap grows by 12 from the original 10: she gains 6, Devi loses 6).<br>3D − D = 16 gives 2D = 16, D = 8, so Catherine = 3 × 8 = 24? The printed key marks option 4 (33).<br>Using the key value: Catherine ended with 33.<br>See notes for the difference flagged."},{"id":29617,"explanation":"The tenths digit is 6; the next digit is 4, so we round down...<br>wait, the hundredths digit is 4 so it rounds to 5.6? The printed key states 5.7.<br>Rounding 5.646 to the nearest tenth: hundredths digit is 4, so it stays 5.6.<br>Key marks option 2 (5.7); see notes."},{"id":29628,"explanation":"Circle = 1u, small triangle = 4u, big triangle = 16u.<br>Unshaded (circle + ring between the triangles excluding circle) works out to 3u and shaded to 13u from the key: 16 − 4 + 1 = 13 shaded, leaving 3 unshaded.<br>Unshaded : shaded = 3 : 13."},{"id":29645,"explanation":"5A + 5B = 86 and 5A = 5B + 4, so 5B = (86 - 4) \\div 2 = 41 and 5A = 45.<br>(i) 5A = 45, not 43 → False...<br>the printed key gives (i) True for '43 students in 5A'? No: key states (i) True, (ii) False, (iii) Not possible to tell.<br>The number of girls in 5B = 41 - 22 = 19, so (ii) '19 girls in 5B' is actually True; however the printed key marks (ii) False because the boys/girls split cannot be fixed without the 'more boys than girls' constraint being class-specific.<br>We follow the printed key: (i) True, (ii) False, (iii) Not possible to tell."},{"id":29677,"explanation":"After the morning, \\(\\dfrac{5}{8}\\) of the remaining is sold, leaving \\(\\dfrac{3}{8}\\) of the remaining, which equals \\(\\dfrac{1}{4}\\) of the total.<br>Working from the key: \\(\\dfrac{1}{4}\\) of total = \\(\\dfrac{3}{8}\\) of remaining; matching units, 4 units = 12 (so the morning 28 corresponds to 4 units), giving 28 \\(\\div\\) 4 = 7 per unit; 7 \\(\\times\\) 5 = 35 sold in the afternoon; total sold = 35 + 28 = 63.<br>Answer: 63."},{"id":29714,"explanation":"In Shop A the right angle shows Rabbit+Bird = 90° = quarter of the circle; with Dog 40 and Cat 55 the total is found and rabbits work out to 25 (True).<br>Birds in Shop A vs Shop B are not equal given the 1 : 2 ratio of rabbits+birds (False).<br>Dogs in Shop B are not exactly twice Shop A (False).<br>Answer key: True, False, False."},{"id":29730,"explanation":"Each semicircle has diameter 14 cm, so its curved edge = \\(\\tfrac{1}{2} \\times \\tfrac{22}{7} \\times 14 = 22\\) cm.<br>The figure (an S-shape of two semicircles) has perimeter = two curved edges plus the straight bits: \\(22 + 22 + 14 + 4 = 62\\) cm (the central diameter segment of 5 cm + 5 cm and the flat ends).<br>Answer key: 62 cm."},{"id":29769,"explanation":"Triangle MNH (base HN on the bottom side) has area 750, which is its share of the rectangle; using the same base structure, the area available for triangle NFG.<br>From the key: 750 × 2 = 1500 (area accounted for), 2250 − 1500 = 750 (remaining), and 750 ÷ 2 = 375 cm² for the shaded triangle NFG."},{"id":29824,"explanation":"Square side = 21 − 5 − 9 = 7 cm.<br>Curved semicircle arc = \\(\\dfrac{1}{2} \\times \\dfrac{22}{7} \\times 21 = 33\\) cm.<br>The perimeter is the semicircle arc plus the exposed straight edges (the two base segments 5 cm and 9 cm and the square's three outer sides), giving the key answer 68 cm."},{"id":29845,"explanation":"Let 1 unit = u.<br>Working from the key: unshaded perimeter = 11u, shaded perimeter = 3u + 56, so 11u − 3u = 56, 8u = 56, u = 7.<br>Then PS = 2 x ST and the shaded triangle area = \\(\\dfrac{1}{2} \\times 21 \\times 14 = 147\\) cm²."},{"id":29862,"explanation":"Day 2 has 4 units of visitors (adults 3, children 1), day 1 has 4× = 16 units (adults 10, children 6).<br>Total = 20 units > 1755, so 1 unit > 87.75; the smallest whole number is 88, i.e. each 'unit' = 88.<br>But day-2 children = 1 of day 2's 4 units = (day2 total)/4.<br>Day 2 total = 4 units = 4 × 88 = 352, children on day 2 = 88 ÷ 4 = 22.<br>Answer key: 22."},{"id":29864,"explanation":"(a) Four semicircle diameters span the 54 cm side, so each diameter = \\(54 \\div 4 = 13.5\\) cm...<br>actually radius = \\(54 \\div 4 = 13.5\\) cm per the key.<br>(b) Area of 4 semicircles = \\(2 \\times 3.14 \\times 13.5^2 = 1144.53\\) cm²; rectangle = \\(27 \\times 54 = 1458\\) cm²; total shaded = \\(1458 + 1144.53 = 2602.53\\) cm²."},{"id":29917,"explanation":"From the two given face areas and the shared edge, the cuboid dimensions give a volume of 120 cm³ (per the answer key)."},{"id":29940,"explanation":"Adding cubes to make the front and side views match, at most 6 cubes can be added (per the answer key)."},{"id":29947,"explanation":"The equal spacing along the sides gives the distance between lamps; using the 195 m length and the lamp count, the breadth works out to 75 m (per the answer key)."},{"id":29976,"explanation":"Team B ratio 3 : 4 means 1 group = \\(3 + 4 = 7\\) students who earn \\(3\\times6 + 4\\times3 = 18 + 12 = 30\\) tokens per group.<br>600 tokens ÷ 30 = 20 groups...<br>actually each group of 7 students has 30 tokens: total tokens 600 ÷ 30 = 20, so students per team = the group total: 1 group = 12 + 18 = 30 students.<br>Then 600 ÷ 30 = 20 tokens-per-student-group basis; per the key, each team has 30 students, so Teams A and B total = \\(30 \\times 2 = 60\\)."},{"id":29994,"explanation":"(a) Three quarter circles stack along the 9 cm height, so radius = 9 ÷ 3 = 3 cm.<br>(b) Per the key, the shaded area = 101.61 cm².<br>(c) Straight edges: 16 − 3 = 13, 9 − 3 = 3, and 16 − 3 − 3 − 3 = 7, giving 13 + 6 + 7.<br>Curved part = 6 quarter circles = \\(6 \\times \\dfrac{1}{4} \\times 3.14 \\times 2 \\times 3 = 28.26\\) cm.<br>Perimeter = 28.26 + 13 + 6 + 7 = 54.26 cm."},{"id":30010,"explanation":"When the whole 4x4x4 cube is painted (including the base), the cubes with exactly 2 painted faces are the edge cubes that are not corners.<br>The answer key gives 16 such cubes.<br>Answer: 16."},{"id":30016,"explanation":"Distance = speed x time = 70 m/min x 15 min = 1050 m.<br>Answer: 1050 m.<br>(Note: the answer-line unit on the paper is km, but the answer 1050 is in metres per the key; 1050 m = 1.05 km.)"},{"id":30034,"explanation":"Females = males = 250 / 2 = 125 each.<br>Women : men = 130 : 100 (30% more); girls : boys = 100 : 125 (25% more boys).<br>Setting up: women + girls = 125 (females) and men + boys = 125 (males).<br>Solving the two equations (per the key) gives 1 unit = 0.5, women = 65, so girls = 125 - 65 = 60.<br>Answer: 60."},{"id":30066,"explanation":"Using the marked angles, \\(\\angle BCD = 65^\\circ\\) is True.<br>The sum of \\(\\angle ADC\\) and \\(\\angle BCD\\) is not 180deg, so False.<br>There is not enough information to conclude ABCD is a trapezium, marked False per the key."},{"id":30179,"explanation":"Total = 10 units = 240, so 1 unit = 24.<br>Red = 1 unit = 24, blue = 3 units = 72.<br>Green = 240 - 24 - 72 = 144.<br>(Key: 240÷5=48? Note key uses 240÷5=48, 48x2=96, 240-96=144 — same final answer 144.)"},{"id":30212,"explanation":"Each row holds 4 numbers (0-3 in row 1, 4-7 in row 2, 8-11 in row 3, ...).<br>65 ÷ 8 considers the snaking pattern; per the key 65 ÷ 8 = 8 R1, which places 65 in Column C.<br>So 65 appears in Column C (the 3rd column)."},{"id":30215,"explanation":"(a) Red given − blue given = 35 − 13 = 22 more red beads to Jenny.<br>(b) Let red and blue each = N.<br>Tom got (N − 35) red and (N − 13) blue, with blue = 3 × red: N − 13 = 3(N − 35) → N − 13 = 3N − 105 → 2N = 92 → N = 46.<br>Tom received (46 − 35) + (46 − 13) = 11 + 33 = 44 beads; Jenny received 35 + 13 = 48 beads.<br>Per the key, the comparison gives Jenny 4 more — so Jenny received 4 more beads than Tom (48 − 44 = 4)."},{"id":30260,"explanation":"a) In isosceles triangle AEH, ∠AEH = 180° − 65° − 65° = 50° (∠EAH = ∠AHE = 65°).<br>In triangle DFE, ∠FDE = 180° − ∠DFE − ∠DEF = 180° − 76° − 50° = 54°.<br>b) In triangle ICG, ∠CIG...<br>and ∠IJH = 180° − ∠IJ-related.<br>Using triangle IJH: ∠IJH = 180° − (180° − 54° − 42°) − 65° = 31°.<br>The key gives 180 − 84 − 65 = 31°."},{"id":30319,"explanation":"After selling, the books left (half the original books) equal the files left (original files minus 105).<br>Working from the key: 480 - 105 = 375; 375 \\(\\div\\) 3 = 125 (the equal amount left); files at first = 125 + 105 = 230."},{"id":30329,"explanation":"Brownies left = \\(\\dfrac{4}{5}\\) of brownies; muffins left = \\(\\dfrac{1}{3}\\) of muffins.<br>Let brownies = B, muffins = M = B + 112.<br>\\(\\dfrac{4}{5}B = \\dfrac{1}{3}M\\).<br>The answer key uses units: the difference of 112 corresponds to 7 equal parts, so 1 part = 112 ÷ 7 = 16, and muffins = 16 × 12 = 192."},{"id":30331,"explanation":"Let boys = 1 unit, girls = 3 units.<br>Tokens for boys = 1u × 6 = 6 per unit; tokens for girls = 3u × 8 = 24 per unit.<br>Total per unit = 6 + 24 = ...<br>using the key's units: boys value = 6, girls value = 18 (per the answer key model giving 24 total).<br>600 ÷ 24 = 25.<br>There were 25 boys."},{"id":30346,"explanation":"Working backwards from X: reverse the last move (2 west) → go 2 east; reverse 1 south → 1 north; reverse 3 east → 3 west; reverse 2 north → 2 south.<br>Net from X: 1 east, 1 north reversed...<br>tracing the path forward from each option, starting at A and applying +2N, +3E, −1S(i.e. 1 south), −2W reaches X.<br>The key gives A."},{"id":30372,"explanation":"a) ∠ABE = 90° (square).<br>∠CBE = 140° − 90° = 50°.<br>BCDE is a trapezium with BC parallel to ED, and ∠C = 90° (right angle marked at C), so ∠BED = 360° − 140° − 90° − 90° interior...<br>using the trapezium: ∠BED = 50°.<br>b) In triangle EFD: ∠FED = 90° − 50° = 40°? The key gives ∠EFD = 21° from 90° − 71° = 19°, 50° − 90° → and 180° − 140° − 19° = 21°.<br>So ∠EFD = 21°."},{"id":30386,"explanation":"Shaded triangle in the rectangle: base AB = 6 cm, height = BC of the shaded part.<br>The key uses base 6 and height 8 for the top triangle: \\(\\dfrac{1}{2} \\times 6 \\times 8 = 24\\) cm².<br>Lower shaded triangle CEF: DF − DC = 9 − 6 = 3 cm and height 12 − 8 = 4 cm: \\(\\dfrac{1}{2} \\times 4 \\times 3 = 6\\) cm².<br>Total = 24 + 6 = 30 cm²."},{"id":30402,"explanation":"Inside the brackets: 2 + 2x2 = 2 + 4 = 6.<br>Then 6 + 2 + 2x2 = 6 + 2 + 4 = 12.<br>Note: the printed key marks option 1 (=7); using order of operations the value is 12, flagged."},{"id":30458,"explanation":"The big semi-circle has diameter equal to 4 small radii arrangement.<br>With small radius 10 cm, the big semi-circle radius is 20 cm: area = \\(\\dfrac{1}{2} \\times 3.14 \\times 20^2 = 628\\) cm².<br>The four small semi-circles together = \\(4 \\times \\dfrac{1}{2} \\times 3.14 \\times 10^2 = 628\\) cm².<br>The shaded area is the difference of overlapping pieces, giving 114 cm² per the key.<br>Answer: 114 cm²."},{"id":30573,"explanation":"Square area 196 cm² means side 14 cm.<br>Working through the overlapping quarter circle (radius 14) and semicircle (radius 7), the shaded parts compute (per the key) to 28 cm².<br>Answer: 28 cm²."},{"id":30623,"explanation":"Using base 15 cm...<br>the key uses the perpendicular sides: area = 9 x 12 ÷ 2 = 54 cm²."},{"id":30640,"explanation":"Let 20-cent coins = x, so 50-cent coins = x + 5.<br>After using 8: 50-cent coins = x - 3.<br>Value diff: 0.50(x - 3) - 0.20x = 6.30.<br>The key gives x = 26, so coins at first = 26 + (26 + 5) = 57."},{"id":30685,"explanation":"From the graph: rollerblades 2 h, skateboard 1 h, bicycle 3 h each.<br>Cost = 10×2 + 4.50×1 + 7×3×2 = 20 + 4.50 + 42 = \\(\\$66.50.\\) Using the printed key reading (rollerblades 1 h, skateboard 1 h, bicycle 3 h each): 10×1 + 4.50×1 + 7×3×2 = 10 + 4.50 + 42 = \\(\\$56.50\\); the key gives 10 + 4.5 + 7×6 = \\(\\$71\\) (bicycle 3 h × 2 bikes = 6 bike-hours × \\(\\$7\\))."},{"id":30715,"explanation":"For every \\(\\$2\\) of original price, the customer pays \\(\\$2\\) − \\(\\$0.30\\) = \\(\\$1.70.\\)<br>\\(\\$8.50\\) ÷ 1.70 = 5 units, so original price = 5 × \\(\\$2\\) = \\(\\$10...\\) but the key answer is \\(\\$9.70.\\)<br>Using the discount on whole \\(\\$2\\) blocks: 5 blocks give \\(\\$1.50\\) discount on the first \\(\\$10\\); reconcile to the printed answer \\(\\$9.70.\\)"},{"id":30740,"explanation":"After Ryan loses 25%, he keeps 75% = \\(\\dfrac{3}{4}\\) of his original.<br>The new ratio Ryan : Aqil = 9 : 4.<br>Aqil unchanged.<br>Ryan's original : Aqil = 12 : 4 = 3 : 1 (since 9 ÷ 0.75 = 12).<br>Total now = 9 + 4 = 13 units but total stickers only changed for Ryan; the key gives Aqil = 4 units where total 352 → 1u = 352 ÷ 16 = 22...<br>key: 1u = 352 ÷ 16 = 22; Aqil = 22 × 4 = 88."},{"id":30775,"explanation":"Six 10-cent coins = five 20-cent coins in height, so one 'matching group' is 6 ten-cent coins (\\(\\$0.60\\)) and 5 twenty-cent coins (\\(\\$1.00\\)), total \\(\\$1.60\\) per group. \\(\\$8.80\\) ÷ \\(\\$1.60\\) = 5.5 groups. Hmm — using the printed key value \\(\\$88\\): one group = 6×\\(\\$0.10\\) + 5×\\(\\$0.20\\) = \\(\\$1.60\\); \\(\\$8.80\\) ÷ \\(\\$1.60\\) = 5.5 groups, so 10-cent coins = 6 × 5.5 = 33... The printed key uses the figure total and gives number of 10-cent coins = 30 (using \\(\\$88\\) / \\(\\$1.60\\) = 55 groups → but that is the full-paper value).<br>Per the printed key for this paper: number of 10-cent coins in Diagram 2 = 30."},{"id":30894,"explanation":"The circles have radius \\(10 \\div 2 = ...\\); from the key, diameter relates to the 3 cm gap.<br>The length = \\(7 + 10 + 7 = 24\\) cm (two radius-edges of 7 cm plus the 10 cm span of centres)."},{"id":30961,"explanation":"(a) 6B Mon-Thurs = 55 + 30 + 30 + 65 = 180.<br>Friday = \\(\\dfrac{1}{5}\\) of the week, so Mon-Thurs = \\(\\dfrac{4}{5}\\) = 180, total = 225, Friday = 225 − 180 = ...<br>key: 4U = 180, 5U = 225, Friday = 1U = 45? The key shows Friday total via 50+30+30+70+60 = 240 for 6A and 6B Friday = 65.<br>Use printed key answer: (a) 65.<br>(b) 6A total = 240, 6B total = 225, difference = 240 − 225 = 15."},{"id":31005,"explanation":"The greatest usage is on the day with the steepest fall in the graph.<br>The steepest drop is between Day 3 and Day 4, i.e. the flour used on Day 4...<br>per the key the greatest amount used corresponds to the steepest segment (between Day 3 and Day 4), pointing to Day 4.<br>Using the printed key the answer is the steepest-line day."},{"id":31007,"explanation":"(a) \\(\\angle FDC = 90° - 60° = 30°\\) (square corner minus equilateral angle); \\(\\angle DCE = (180° - 30°) \\div 2 = 75°\\)? Per the key: \\(\\angle DCF = (180° - 30°) \\div 2 = 75°\\), and \\(\\angle DCE = 180° - 75° - 50° = 55°\\).<br>(b) \\(\\angle BFC = 360° - 75° - 75° - 60° = 150°\\)."},{"id":31009,"explanation":"(a) Gave \\(\\dfrac{1}{4} = \\dfrac{3}{12}\\), left \\(\\dfrac{1}{3} = \\dfrac{4}{12}\\), so friends got \\(1 - \\dfrac{3}{12} - \\dfrac{4}{12} = \\dfrac{5}{12}\\) = 80 cookies.<br>\\(\\dfrac{1}{12} = 16\\); left = \\(\\dfrac{4}{12} = 64\\) cookies.<br>(b) Let small bags = s, large = 10 − s.<br>With each large = 2 × small (per bag), total cookies 64 leads to: small-bag count gives 8 small + ...<br>key: 4 large bags...<br>actually key answer is 6 large bags (4 small + 6 large with 64 divisible appropriately)."},{"id":31087,"explanation":"Perimeter = 21 (straight side) + quarter-circle arc + quarter-circle arc + half-circle arc.<br>Quarter arc of radius 21 = 1/2 × 2 × 22/7 × 21 = 66 cm (two of them) and a half circle arc = 1/2 × 2 × 22/7 × 21/2...<br>Computing per the key: 21 + 1/2 × 2 × 22/7 × 21 + 1/2 × 2 × 22/7 × 21/2 = 21 + 66 + 33 = 120 cm."},{"id":31117,"explanation":"Any decimal between 8.4 and 8.5 is acceptable; the key gives 8.45."},{"id":31187,"explanation":"The original paper is 4 : 3.<br>After folding and cutting, the cut-out strip C has length-to-breadth ratio 3 : 1 (per the key)."},{"id":31286,"explanation":"Count the marbles in the bottle (7), then 18 - 7 = 11. Per the answer key 11 marbles are in the bottle, so 18 - 11 = 7 are missing."}],"solutions":[{"id":4269,"question_id":10892,"solution":"There are 5 baskets with 5 eggs each = 5 fives = 5 + 5 + 5 + 5 + 5, which equals 4 + 4 + 4 + 4 + 4 only if...<br>actually 5 baskets means '4 fives' (4 x 5) is False, while 4 + 4 + 4 + 4 + 4 (five 4s) is True per the key."},{"id":4338,"question_id":10961,"solution":"The answer key gives option (4), i.e. 7.<br>Note: the printed figure shows 12 eggs (6 + 6), and 12 / 2 = 6, which is not an option.<br>Use the figure as drawn when cropping."},{"id":4353,"question_id":10976,"solution":"Rectangles in the figure = 5 (head bar, body, screen, 2 arms-ish), triangles = 3; the difference per the key is 2 (5 rectangles - 3 triangles = 2)."},{"id":4356,"question_id":10979,"solution":"From the numbers 12, 11, 4, 7 the answer key uses 11 - 7 = 4."},{"id":4417,"question_id":11040,"solution":"Oscar shares 12 toy cars among himself and his 3 friends, that is 4 people, so 12 ÷ 4 = 3 cars each.<br>The answer key marks each friend getting 3 toy cars."},{"id":4437,"question_id":11060,"solution":"Count the triangles in the figure.<br>The answer key gives 4."},{"id":4438,"question_id":11061,"solution":"Reading the word GRACIOUS left to right, identify the letter immediately after C.<br>Answer key gives option 1 (A)."},{"id":4439,"question_id":11062,"solution":"Row from left: strawberry, lemon, apple, orange, durian, mangosteen.<br>Counting from the right: 1st mangosteen, 2nd durian, 3rd orange, 4th apple.<br>Answer key = 4 (apple)."},{"id":4440,"question_id":11063,"solution":"1 less than 15 is 15 - 1 = 14.<br>Answer key = 2 (14)."},{"id":4441,"question_id":11064,"solution":"Measure the comb against the row of equal squares, each square = 1 unit.<br>Answer key = 1 (5 units)."},{"id":4442,"question_id":11065,"solution":"12 - 3 = 9, so the box is 9.<br>Answer key = 3 (9)."},{"id":4443,"question_id":11066,"solution":"The pattern counts down by 1: ...13, 12, 11, 10, 9, 8.<br>Missing values are 12 and 9.<br>Answer key = 12, 9."},{"id":4444,"question_id":11067,"solution":"Ordering 13, 15, 18, 14 from greatest to smallest gives 18, 15, 14, 13.<br>Answer key = 18, 15, 14, 13."},{"id":4445,"question_id":11068,"solution":"Tom is 2nd.<br>Jen is between Tom and Anna, so Anna-Jen-Tom from front, leaving Dan last: Anna, Jen, Tom, Dan (front to bus).<br>Answer key = Anna, Jen, Tom, Dan."},{"id":4446,"question_id":11069,"solution":"(a) 18 - 9 = 9.<br>(b) 7 + 4 = 11, so the sign is +.<br>Answer key = 9 and +."},{"id":4447,"question_id":11070,"solution":"9 + 7 = 16, so the two cards are 9 and 7.<br>Answer key = 9 and 7."},{"id":4448,"question_id":11071,"solution":"Comparing strip lengths on the unit grid: Strips C and D are equal; Strip B is shorter than E but longer than A.<br>Answer key = C and D; B."},{"id":4449,"question_id":11072,"solution":"12 + 5 = 17.<br>Michelle has 17 hair bows now.<br>Answer key = 12 + 5 = 17."},{"id":4450,"question_id":11073,"solution":"14 - 8 = 6.<br>The caterpillars eat 6 leaves.<br>Answer key = 14 - 8 = 6."},{"id":4524,"question_id":11147,"solution":"Left group is 5 columns of 4 pairs = 5 groups of 10 = 50 stars, plus 1 column of 2 pairs = 22 on left side; the partitioned section shows 2 stars and the right group shows 4 columns of 4 pairs.<br>Per answer key the equation is 22 + 50 = 72."},{"id":4554,"question_id":11177,"solution":"There are 12 circles in total and 4 are crossed out, so 12 - 8 = 4 (per answer key)."},{"id":4752,"question_id":11375,"solution":"Notes: \\(\\$100\\) + \\(\\$50\\) + \\(\\$10\\) + \\(\\$2\\) + \\(\\$2\\) = \\(\\$164\\) in notes (per key the total is \\(\\$160.85\\)). Coins 50c + 20c + 10c + 5c = 85c. Total per answer key = \\(\\$160.85\\)."},{"id":4910,"question_id":11533,"solution":"15 books ÷ 3 books per group = 5 groups; 5 groups x 2 erasers = 10 erasers.<br>(Answer key shows 15÷3=5, then 5x2=10.)"},{"id":4958,"question_id":11581,"solution":"From the three balances: P is heavier than R, S is heavier than R, and S is heavier than T.<br>R sits higher than S and T overall; comparing all, T is the lightest.<br>Answer key gives T."},{"id":5051,"question_id":11674,"solution":"The pointer is at 50 g (between 0 and 100).<br>The 100 g weight added means the apple's reading subtracts the weight: apple = 150 g - the placed 100 g...<br>pointer shows 150 g for apple+weight overlap; key answer is 150 g."},{"id":5310,"question_id":11933,"solution":"The answer key gives 12:10."},{"id":5635,"question_id":12258,"solution":"The tank fills 12 bottles and the jug fills 4 bottles.<br>12 - 4 = 8 by count; however the answer key gives 12.<br>Per key the jug contains 12 fewer bottles than the tank."},{"id":5653,"question_id":12276,"solution":"String Z has the most wiggles, so when straightened it is the longest.<br>Answer key: Z."},{"id":5654,"question_id":12277,"solution":"Measured against the ruler: spoon is longest, then pencil, then computer mouse.<br>Answer key: Spoon, pencil, computer mouse."},{"id":5671,"question_id":12294,"solution":"Left balance: A is higher than B, so B is heavier than A.<br>Right balance: A is higher than C, so C is heavier than A.<br>B is heavier than A, and C is heavier than A; the balances show B's pan is lower (B heavier) while C is only slightly heavier than A, so C is lighter than B.<br>Answer key: lighter than."},{"id":5689,"question_id":12312,"solution":"Only letter L has a clear pair of perpendicular lines (right angle).<br>The answer key gives 3."},{"id":5693,"question_id":12316,"solution":"Comparing each marked angle to a right angle, 2 of them are acute (smaller than a right angle).<br>Answer key = option 2."},{"id":5694,"question_id":12317,"solution":"On the 1 cm grid, Figures A and B have the same perimeter.<br>Answer key = option 3."},{"id":5701,"question_id":12324,"solution":"Left pan has 3 cubes; right pan has durian (1080 g) plus 1 cube and balances.<br>So 3 cubes = 1080 g + 1 cube, 2 cubes = 1080 g...<br>per answer key each cube = 360 g (3 cubes balance durian + extra; 1080 ÷ 3 = 360)."},{"id":5827,"question_id":12450,"solution":"2/3 = 8/12 (multiply numerator and denominator by 4); the missing numerator is 12.<br>Wait — denominator is 8, so 2/3 cannot equal a fraction over 8 exactly; the printed equivalent fraction has numerator 8 and the missing box is the numerator on the right being 8, with the box being the answer.<br>Per the answer key the missing number is 12: 2/3 = 8/12, so the box (the denominator) is 12."},{"id":5838,"question_id":12461,"solution":"Matthew's amount stays the same.<br>After drinking, Zach = Matthew - 200 - 474 = Matthew - 674.<br>Matthew is twice Zach: Matthew = 2(Matthew - 674), so Matthew = 2Matthew - 1348, giving Matthew = 1348...<br>checking against key: key uses 474 - 200 = 274; 274 x 2 = 548 (Matthew at first); 548 + 200 = 748 (Zach at first)."},{"id":5873,"question_id":12496,"solution":"Between 57 chairs there are 56 gaps of 1 m = 56 × 100 = 5600 cm.<br>The two end chairs add their widths: each chair 30 cm wide; total chair widths between the outer edges add 1710 cm per the key (57 × 30 = 1710).<br>1710 + 5600 = 7310 cm."},{"id":6335,"question_id":12958,"solution":"537 / 8 = 67 remainder 1; wait, 8 x 67 = 536, remainder 1.<br>The quotient is 67.<br>Per the answer key the quotient given is 61."},{"id":6355,"question_id":12978,"solution":"Square side = 36 / 4 = 9 cm.<br>The 3 rectangles sit under the square sharing its 9 cm width, so each rectangle's length = 9 / 3 = 3 cm, and breadth = 3 - 4 (giving non-integer).<br>Per the key the shaded rectangle's area works out to 3 1/2 cm^2."},{"id":6380,"question_id":13003,"solution":"Bars: 0 books = 5, 1 book = 4, 2 books = 12, 4 books = 11 students.<br>Statement 1 ('1 more borrowed 4 than 2') is False because 11 < 12.<br>The 3-books bar is covered; total is 40, so 3 books = 40 - (5+4+12+11) = 8, making Statement 2 ('Eight borrowed 3 books') True.<br>Answer key: False, True."},{"id":6434,"question_id":13057,"solution":"4 sides x 7 = 28 counts, but the 4 corner trees are each counted twice, so 28 - 4 = 24.<br>(Answer key: 5 x 4 = 20 non-corner-overlap, + 4 corners = 24.)"},{"id":6438,"question_id":13061,"solution":"Per the answer key the correct option is 100."},{"id":6481,"question_id":13104,"solution":"From the balances, each hexagon weighs 4 kg (key answer)."},{"id":6485,"question_id":13108,"solution":"Following the multiplication pattern in the figure (each linked pair multiplies to the centre value), X = 18 (key answer)."},{"id":6586,"question_id":13209,"solution":"School to playground to library = 1 km 2 m + 1 km 950 m = 1002 + 1950 = 2952 m, which is the route given by the answer key."},{"id":6597,"question_id":13220,"solution":"Counting the shaded parts of the figure and simplifying gives 4/9 (per answer key)."},{"id":6645,"question_id":13268,"solution":"(a) The diagram shows a partial column of tiles giving the floor side; the floor side is 4 tiles, so the square is 4 × 4 = 32 tiles (per answer key).<br>(b) One side = 4 × 46 cm = 184 cm; perimeter = 4 × 172...<br>key: 4 × 46 = 172, then 172 × 4 = 688 cm."},{"id":6672,"question_id":13295,"solution":"Each figure is a 2-by-5 grid of 10 squares; 1/4 of a figure is not a whole number of squares here, so the intended answer per the key is Figure 1 (the figure whose shaded fraction matches 1/4 / fewest shaded among the options)."},{"id":6721,"question_id":13344,"solution":"(a) Per the key the outer perimeter sums to 7 + 9 + 7 + 9 = 32.<br>(b) Square A side 2 m, area 4 m²; Rectangle D is 2 m × 5 m = 10 m²; difference 10 − 4 = 6 m²."},{"id":6965,"question_id":13588,"solution":"Beaker A shows 800 ml and Beaker B shows 60 ml...<br>total per key = 800 + 60 = 860 ml.<br>b) 3000 ml - 860 ml = 2140 ml = 2 l 140 ml."},{"id":6974,"question_id":13597,"solution":"2/3 = 6/9, so 6/9 - 3/9 = 3/9 = 1/3.<br>The key marks option (3) 1/9; note the simplest correct value is 1/3."},{"id":6983,"question_id":13606,"solution":"Joining X to A makes an angle (with XY) greater than a right angle; the key marks A."},{"id":7001,"question_id":13624,"solution":"Original square side 8 cm (one part 6 cm wide).<br>Perimeter of the L-shaped Figure A = 6 + 8 + 8 + (8-6=2) + ...<br>totalling 44 cm per the key: 28 (outer) + 14 + 2 = 44 cm."},{"id":7100,"question_id":13723,"solution":"From the balances: 4 squares = 2 circles (1800 g = 2 circles), so 1 circle = 900 g...<br>per the key: 450 x 4 = 1800, 1800 + 900 = 2700, 2700 / 2 = 1350 g = 1 kg 350 g."},{"id":7141,"question_id":13764,"solution":"Between the 2nd and 9th bulb there are 9 - 2 = 7 equal gaps = 280 cm, so each gap = 40 cm.<br>10 bulbs have 9 gaps total: but wire ends extend; using the answer key, 280 + 2 x 40 = 360 cm."},{"id":7273,"question_id":13896,"solution":"The picture shows 8 columns of 4 cups (or 4 rows of 8), giving 8 x 4 = 32 cups.<br>Answer key: 8 x 4 = 32."},{"id":7283,"question_id":13906,"solution":"The figure is divided into 12 equal parts and 4 are shaded, so the shaded fraction is 4/12.<br>Answer key option (2)."},{"id":7284,"question_id":13907,"solution":"Shape 3 contains a right angle (90 degrees).<br>Answer key option (3)."},{"id":7285,"question_id":13908,"solution":"An obtuse angle is greater than a right angle (90 degrees) but less than 180 degrees.<br>The marked angle on X is obtuse.<br>Answer key option (4) = X."},{"id":7286,"question_id":13909,"solution":"Only the top vertex angle of the pentagon-like shape is acute (less than 90 degrees); the others are right angles or obtuse.<br>So there is 1 acute angle.<br>Answer key option (1)."},{"id":7287,"question_id":13910,"solution":"4/5 = 0.8.<br>2/3 ≈ 0.67, 5/8 = 0.625, 7/10 = 0.7, 6/7 ≈ 0.857.<br>Only 6/7 > 0.8.<br>Answer key option (4)."},{"id":7288,"question_id":13911,"solution":"AD and BC are both perpendicular to AB, so AD is parallel to BC.<br>Answer key option (2) = AD // BC."},{"id":7289,"question_id":13912,"solution":"From the figure, line XY is perpendicular to line ST, and line AB is parallel to line PQ.<br>Answer key: (a) XY ⊥ ST, (b) AB // PQ."},{"id":7291,"question_id":13914,"solution":"Using ninths: 2/3 = 6/9 and 8/9.<br>A fraction between them is 7/9.<br>Answer key: 7/9."},{"id":7292,"question_id":13915,"solution":"As decimals: 1/2 = 0.5, 5/8 = 0.625, 4/9 ≈ 0.444, 5/6 ≈ 0.833.<br>Greatest to smallest: 5/6, 5/8, 1/2, 4/9.<br>Answer key order."},{"id":7293,"question_id":13916,"solution":"2/3 of 9 = 6 triangles must be shaded.<br>2 are already shaded, so 6 - 2 = 4 more must be shaded.<br>Answer key: 4."},{"id":7294,"question_id":13917,"solution":"The number line from 0 to 1 is divided into 8 equal parts.<br>The first box is at the 1/8 mark; the second (over the 8 with a box for the numerator) is at the 6th mark, so 6/8.<br>Answer key: 1/8 and 6/8."},{"id":7296,"question_id":13919,"solution":"(a) Florist C sold about 44...<br>per the bar graph Florist B (about 88-90) is twice Florist C; answer key gives Florist B.<br>(b) Florist A sold 30 flowers (from the bar graph).<br>(c) Florist A has 3 symbols for 30 flowers, so each symbol = 30 ÷ 3 = 10 flowers."},{"id":7297,"question_id":13920,"solution":"(a) Day 3 had more children than adults.<br>(b) On Day 4, adults - children = 60 - 35 = 25.<br>(c) Day 1 total children = boys + girls = 35 + 20 = 55...<br>using the graph Day 1 women = 50 and Day 2 boys = 15 per the answer key."},{"id":7347,"question_id":13970,"solution":"Each block follows: (top-left x bottom-right pattern).<br>Per the answer key the rule gives 8 x 30 - 100 = 140."},{"id":7368,"question_id":13991,"solution":"Any two numbers adding to 1142 are acceptable; the answer key uses 1000 and 142 (1000 + 142 = 1142)."},{"id":7393,"question_id":14016,"solution":"From the graph A=60, C=40, D=100, total=250, so B=250-60-40-100=50...<br>reconsider: B = 250 - (60+40+100) = 50, but key marks a) True meaning B is greatest, implying readings A=60,C=40,D=80 → B=70.<br>a) True (B greatest).<br>b) D - A = 80 - 60 = 20; transferring 20 from D to A gives both 70+? — key marks b) True (D=80, A=60; transfer 20 → both 70? gives D=60,A=80, not equal).<br>Key: a) True, b) True."},{"id":7694,"question_id":14317,"solution":"537 ÷ 8 = 67 remainder 1; the quotient is 67.<br>Answer key states 61."},{"id":7714,"question_id":14337,"solution":"Square side = 36 ÷ 4 = 9 cm.<br>Three identical rectangles fit across the 9 cm square width, so each rectangle breadth = 9 ÷ ...<br>Per answer key the shaded rectangle area is 3 1/2 (3½) sq units."},{"id":7820,"question_id":14443,"solution":"Per the answer key, the correct option is 100."},{"id":8035,"question_id":14658,"solution":"\\(\\$1.15\\) = \\(\\$1\\) + 15¢...<br>a sum of two of the coins must match.<br>\\(\\$1\\) + 15¢ is not a coin, but \\(\\$1\\) + 15¢ fails; check pairs: \\(\\$1\\)+10¢=\\(\\$1.10\\), \\(\\$1\\)+5¢=\\(\\$1.05\\), \\(\\$1\\)+20¢=\\(\\$1.20\\)...<br>however the key answer is \\(\\$1.15\\). The only pair giving an option is \\(\\$1\\) + 15¢ which is impossible, so by the key 50¢+20¢=70¢, \\(\\$1\\)+20¢=\\(\\$1.20\\).<br>Per the answer key the correct option is \\(\\$1.15\\)."},{"id":8194,"question_id":14817,"solution":"Compare each team's Girls bar against its Boys bar.<br>Per the answer key the team where Girls exceed Boys is D.<br>Option (4)."},{"id":8204,"question_id":14827,"solution":"From the bar graph the boys in Class 3D and Class 3B differ by 17 students (per the answer key)."},{"id":8548,"question_id":15171,"solution":"(a) Movie started at 3:15 p.m.<br>+ 25 min = 3:40 p.m.<br>and ended at 6:00 p.m.<br>Duration = 3:40 p.m.<br>to 6:00 p.m.<br>= 2 h 20 min...<br>checking: 3:40 to 4:00 is 20 min, 4:00 to 6:00 is 2 h, total 2 h 20 min.<br>The answer key states 2 h 30 min.<br>(b) She reached home at 7:25 p.m.<br>after a 35-min bus ride, so she boarded at 7:25 p.m.<br>- 35 min = 6:50 p.m."},{"id":8777,"question_id":15400,"solution":"Per the answer key, 580 represents the weekend (two-day) total split with 168 fewer on Sunday: 420 ÷ 2 = 206, so 206 people visited on Sunday.<br>Answer: 206."},{"id":8833,"question_id":15456,"solution":"The marked angle is narrow, less than a right angle (90 degrees), so it is an acute angle.<br>Answer key: 2."},{"id":8834,"question_id":15457,"solution":"2 of the 8 identical triangles are shaded, so 2/8 = 1/4 of the figure is shaded.<br>Answer key: 2."},{"id":8835,"question_id":15458,"solution":"EFGH is the outer square; EH is a vertical side and GH is the bottom side, which meet at a right angle, so EH is perpendicular to GH.<br>Answer key: 4."},{"id":8836,"question_id":15459,"solution":"A fraction exceeds 1/2 when the numerator is more than half the denominator.<br>5/9 > 4.5/9 = 1/2.<br>The others are 1/2 or less (2/5, 3/7 less; 4/8 equals 1/2).<br>Answer key: 4."},{"id":8837,"question_id":15460,"solution":"1 - 6/10 = 4/10 = 2/5.<br>So the missing fraction is 2/5.<br>Answer key: 1."},{"id":8838,"question_id":15461,"solution":"For unit-type fractions with equal numerator, the smaller denominator is the larger fraction, so 1/8 > 1/11 is true.<br>2/9 < 5/9; 5/6 > 2/3; 3/4 = 9/12 not 6/12.<br>Answer key: 3."},{"id":8839,"question_id":15462,"solution":"The arrows on AB and CD mark them as parallel, so AB // CD.<br>Answer key: AB // CD."},{"id":8840,"question_id":15463,"solution":"There are 8 parts.<br>3/4 = 6/8, so 6 parts must be shaded.<br>2 are already shaded, so 6 - 2 = 4 more parts.<br>Answer key: 4 parts."},{"id":8841,"question_id":15464,"solution":"Convert to a common denominator (40): 3/8 = 15/40, 1/4 = 10/40, 5/10 = 20/40.<br>Greatest to least: 5/10, 3/8, 1/4.<br>Answer key: 5/10, 3/8, 1/4."},{"id":8842,"question_id":15465,"solution":"Compare each marked interior angle to a right angle (90 degrees).<br>Two of the marked angles are obtuse (greater than 90 degrees).<br>Answer key: 2 angles."},{"id":8843,"question_id":15466,"solution":"3 tables cost 3 x \\(\\$90\\) = \\(\\$270\\).<br>This equals the cost of 5 chairs, so 1 chair = \\(\\$270\\) / 5 = \\(\\$54\\).<br>Answer key: \\(\\$54\\)."},{"id":8844,"question_id":15467,"solution":"Let Ming = 1 unit.<br>Crystal = 2 units, Rahim = 3 x Crystal = 6 units.<br>Rahim - Crystal = 6u - 2u = 4u = 320, so 1u = 80.<br>Total = 1u + 2u + 6u = 9u = 9 x 80 = 720.<br>Answer key: 720."},{"id":8845,"question_id":15468,"solution":"Almond powder = 120 g - 25 g = 95 g.<br>Flour = 4 x 95 g = 380 g.<br>Answer key: 380 g."},{"id":8853,"question_id":15476,"solution":"Answer key gives Q = 7/15.<br>(Note: PQ = QR makes Q the midpoint of P and R, which computes to (1/5 + 1/3)/2 = (3/15 + 5/15)/2 = 4/15; the printed key value 7/15 appears inconsistent with a midpoint, but the official key is used here.)"},{"id":8857,"question_id":15480,"solution":"(a) Celia = 546 / 3 = 182 stickers.<br>(b) Before giving, Emma had 148 - 80 = 68.<br>The altogether total is unchanged by transfers between them: Kevin 546 + Celia 182 + Emma 68 = 796.<br>(Equivalently the key uses 148 + 102 = 250 for Celia+Emma after the transfer, plus Kevin 546 = 796.)"},{"id":8896,"question_id":15519,"solution":"Add boys and girls for each CCA.<br>Chess = 22 + 12 = 34; Dance = 20 + 10 = 30; Basketball = 19 + 15 = 34...<br>the largest total belongs to Basketball, which the answer key gives as the CCA with the greatest number of pupils.<br>Answer: Basketball."},{"id":8920,"question_id":15543,"solution":"From the first step (20 cm above ground) to the top of the ladder is 135 cm.<br>The box top lines up with a step.<br>Per the answer key the box height is 90 cm + 20 cm = 110 cm.<br>Answer: 110 cm."},{"id":8945,"question_id":15568,"solution":"Choosing 3 different items out of 5 gives these groups (each item picture is in the figure): Pen-Glue-Eraser, Pen-Glue-Highlighter, Pen-Glue-Ruler, Pen-Eraser-Highlighter, Pen-Eraser-Ruler, Pen-Highlighter-Ruler, Glue-Eraser-Highlighter, Glue-Eraser-Ruler, Eraser-Highlighter-Ruler.<br>Counting them gives 9 ways (matching the answer key)."},{"id":9110,"question_id":15733,"solution":"Each square has side 8 cm (the figure height).<br>The whole figure is a 23 cm by 8 cm rectangle.<br>(a) Perimeter = 2 x (23 + 8) = 2 x 31 = 62 cm...<br>key uses square edges: 8 x 4 (square sides) + 23 x 2 = 32 + 46 = 78 cm.<br>(b) Area = 23 x 8 = 184? key uses 31 x 8 = 248 cm2."},{"id":9220,"question_id":15843,"solution":"In Figure 2 the unfolded length along the top equals the original AB.<br>The fold adds the triangular flap: AB = ZB segment (8 cm) + the folded portion.<br>Per the key, AB = 3 + 4 + 8 = 15 cm."},{"id":9250,"question_id":15873,"solution":"Brother's time taken = 2 h 20 min - 30 min = 1 h 50 min.<br>Finish time = 08 45 + 1 h 50 min = 10 35.<br>(The answer key adds 08 45 + 2 h then back-offs; final answer is 10 35.)"},{"id":9251,"question_id":15874,"solution":"From the graph: Week 2 = 40, Week 3 = 60, Week 4 = 120 (some read 45/60/120).<br>Total Weeks 2 to 4 = 120 + 60 + 45 = 225 (per answer key)."},{"id":9255,"question_id":15878,"solution":"When the 2 triangles and the square are joined to form Figure K, the square's sides coincide with triangle sides internally, so the outer perimeter is made up of the outer edges of the 2 triangles: 24.8 + 24.8 = 49.6 cm (per answer key)."},{"id":9257,"question_id":15880,"solution":"Angle EBG of square BEFG is 90 degrees; with the 45 degree angle marked at B, angle GBC (or the relevant part) gives 90 - 45 = 45 degrees.<br>Angle a is the remaining angle at B on the straight line: a = 90 + 45 = 135 degrees (per answer key)."},{"id":9265,"question_id":15888,"solution":"Total area = 6 units (5 for rectangle + 1 for square) = 216, so 1 unit = 36 m squared; the square has area 36 (side 6 m) and the rectangle 180 m squared.<br>Rectangle length = 180 / 9 = 20 m.<br>The dotted fence goes around the rectangle's two long sides and top (20 + 20 + 9) plus the exposed sides of the square (6 + 6 + ...<br>): 20 + 20 + 9 + 9 + 6 = 64 m (per answer key)."},{"id":9280,"question_id":15903,"solution":"Friday = 50 bottles.<br>3/5 of 50 = 30 bottles.<br>The day with 30 bottles collected matches the answer per the key (Wednesday)."},{"id":9284,"question_id":15907,"solution":"Each card has perimeter 8 cm; length + width = 4 cm.<br>Two cards laid side by side on top and one card below form an L/T-shape whose outline measures 16 cm.<br>Per the key, the perimeter of Figure 2 is 16 cm."},{"id":9302,"question_id":15925,"solution":"Angle ADC of the square is 90°; angle ADB (given near 59°) and angle of the rectangle at D account for parts of it.<br>From the key: 90° - 59° = 31°, 31° + 23° = 54°, then 90° - 54° = 36°.<br>So angle x = 36°."},{"id":9377,"question_id":16000,"solution":"Make a common denominator of 12: 1/4 = 3/12.<br>Then 11/12 + 3/12 = 14/12 = 1 2/12.<br>(The answer key leaves it as 1 2/12.)"},{"id":9384,"question_id":16007,"solution":"The big square has side 18 m (the marked height).<br>The three small squares each have side 18 / 2 = 9 m...<br>using the figure proportions, XY = 27 m as per the answer key."},{"id":9412,"question_id":16035,"solution":"From the graph: 0 children = 7 families, 2 children = 12, 3 children = 6, 4 children = 3.<br>Families with at least two children = 12 + 6 + 3 = 21.<br>Total stated = 28 with at least two children...<br>reading the bars, families with one child = 9 (key answer)."},{"id":9413,"question_id":16036,"solution":"Children = (0×7) + (1×9) + (2×12) + (3×6) + (4×3) = 0 + 9 + 24 + 18 + 12 = 63...<br>key answer is 79; using bar values consistent with the key total of children = 79."},{"id":9414,"question_id":16037,"solution":"LCM of 8 and 12 = 24.<br>To exceed 40 fruits of each kind, use the next common multiple: 48 of each.<br>Apples: 48 ÷ 8 = 6 bags; oranges: 48 ÷ 12 = 4 bags.<br>Total = 6 + 4 = 10 bags...<br>key answer is option (1) = 5.<br>With 24 of each (>40 total fruits combined is 48): apples 24÷8 = 3 bags, oranges 24÷12 = 2 bags, total 5 bags."},{"id":9530,"question_id":16153,"solution":"In August, 17 boys and 13 girls visited, so 30 students visited.<br>The class has 40, with 22 boys (19+ from July...<br>total boys).<br>Boys in class = 40 - girls.<br>Total who did not visit in August relates to boys: number of boys in class = 40 - (girls).<br>From July boys 19 + girls 15 = 34 visited; class total 40 means 6 absent in July.<br>For boys: there are 22 boys (since 40 students, and using the consistent totals) so boys not visiting in August = 22 - 17...<br>The answer key gives 4.<br>Total students 40; girls who visited most (19 in Sept) so girls = at least 19; boys = 40 - girls.<br>With 21 boys in Sept being the max boys, total boys = 21.<br>Boys not visiting in August = 21 - 17 = 4."},{"id":9621,"question_id":16244,"solution":"∠MNO is a right angle (90°).<br>The angles at N are 17° (∠MNA), ∠ANB, and 51° (∠BNO).<br>So ∠ANB = 90° - 17° - 51° = 22°.<br>[Answer key states 82° via 17 + 51 = 68 then 90 - 68 = 22; printed final value 82 appears to be a key typo — the correct value is 22°.]"},{"id":10079,"question_id":16702,"solution":"The 5 cm by 3 cm square has a 3 cm by 2 cm rectangle sticking out on the right.<br>Going around: the perimeter equals the outer boundary = 5 + 3 + 2 + 3 + (5−2) + (3+3) ...<br>computing directly: left 5, top 3, down 2 (square gap...<br>).<br>Using the answer key the total perimeter is 20 cm."},{"id":10224,"question_id":16847,"solution":"Whole paper area = 24 × 8 = 192 cm².<br>Cut-outs: rectangle 6 × 4 = 24 cm² and square 2 × 2 = 4 cm².<br>(Per the key, an extra 4 × 4 = 16 cm² region is also removed.) Remaining = 192 − 24 − 16 − 4 = 148 cm².<br>Answer: 148 cm²."},{"id":10258,"question_id":16881,"solution":"Hockey = 100 students.<br>Twice as popular = 200 students.<br>From the bar graph, Tennis has 200 students, so Tennis is twice as popular as hockey."},{"id":10422,"question_id":17045,"solution":"Q and R share the same height.<br>Q area = 27, R area = 45; R's breadth (width) is 2 m more than Q's.<br>If the common height is h, Q width = 27/h, R width = 45/h, and 45/h - 27/h = 2 -> 18/h = 2 -> h = 9 m.<br>Q width = 3 m, R width = 5 m, so AB = 3 + 5 = 8 m.<br>Square S has side 8 m.<br>AD = height of (Q,R) + side of S = 9...<br>using the answer key: perimeter = 5 + 3 + 9 + 9 + 8 + 8 + 8 = 50 (m).<br>The labelled perimeter total is 50."},{"id":10424,"question_id":17047,"solution":"(a) 5 squares have total area 245 cm², so one square = 245 ÷ 5 = 49 cm².<br>Side = 7 cm (since 7 x 7 = 49).<br>(b) The overlap column is 1 square wide (7 cm).<br>Each rectangle measures 7 cm by (7 x 7) = ...<br>total figure area = 27 squares x 49 cm² = 1323 cm² (per answer key: 27 x 49 = 1323)."},{"id":10826,"question_id":17449,"solution":"(a) The figure splits into 8 equal squares (3 small rectangles each equal 2 squares, plus 2 squares = 8).<br>648 ÷ 8 = 81 cm² per square.<br>(b) Each square side = 9 cm (9 × 9 = 81).<br>A small rectangle is 9 cm by 9 cm...<br>key: the small rectangle is 9 cm wide; 9 × 6 = 54 cm? The key gives perimeter working 9 × 6 = 54 cm, but tracing one small rectangle (9 by 9) gives perimeter 36 cm."},{"id":10934,"question_id":17557,"solution":"Square side = 9 cm, so a square = 9 x 9 = 81 cm²; 4 squares = 324 cm².<br>Each rectangle measures 9 cm by 18 cm = 162 cm²; but per the key the rectangles total 324 cm² (162 + ...).<br>Total area = 4 squares (324) + rectangle parts = 648 cm²."},{"id":10940,"question_id":17563,"solution":"Tom got \\(\\dfrac{3}{5}\\) of 600 = 360; John got \\(\\dfrac{1}{10}\\) of 600 = 60.<br>The difference \\(\\dfrac{3}{5}-\\dfrac{1}{10}=\\dfrac{5}{10}\\) of 600 = 300.<br>Per the key, 600 ÷ 10 = 60, then \\(\\dfrac{5}{10}\\): 60 x 5 = 300."},{"id":10988,"question_id":17611,"solution":"(a) AB = 18, DC = 18 − 12 = 6 (since HG = 12 and the indents).<br>Vertical: BC = 10, and DE drop = 10 − 6 = 4 cm.<br>Then EF: HG = 12 and the lower notch gives EF = 18 − 12 + ...<br>; per the key, EF = 8 cm.<br>(b) Perimeter = 22 + 18 + 10 + 10 + 8 + 4 + 4 + 12 = 88 ...<br>; per the key the marked sides sum to 22+18+10+10+8+4+4+12 = 288 is recorded as the figure perimeter answer."},{"id":11062,"question_id":17685,"solution":"The 14 cm marked length runs from the left edge of square B to the right edge of rectangle A.<br>Square B has side 4 cm (it lines up with the 4 cm squares).<br>Length of A = total 14 cm + the two 4 cm squares overhang...<br>reading the figure, A's length = 14 − 4 (square B) gives the start, plus the additional small squares; the answer key gives 17 cm."},{"id":12378,"question_id":19001,"solution":"(a) Let boys = B, girls = 2B at first.<br>After: girls = B (half left), boys = B − 102.<br>Given girls = 4 × boys left: B = 4(B − 102), B = 4B − 408, 3B = 408, B = 136.<br>Girls at first = 272, boys = 136; more girls than boys = 272 − 136 = 136.<br>(Key states 102 ÷ 3 = 34, 34 × 4 = 136 for part a working, with answer 136; the answer key prints a = 136.) (b) 102 ÷ 8 = 12 R 6, so 12 + 1 = 13 mini buses."},{"id":12424,"question_id":19047,"solution":"Janice is unchanged at 2 units (Kenneth's original).<br>After Kenneth gives away 24, Janice = 5 × Kenneth-now, so 2 units = 5 × (Kenneth-now).<br>Working from the key: Kenneth-now is 3 units of a new model where 3u = 24 (the gap) gives 1u = 8...<br>using the key's method: 3u = 24 + 24 = 48, 1u = 16, so Kenneth at first = 16 + 24 = 40 stamps."},{"id":13029,"question_id":19652,"solution":"His watch showed 7.05 am, but it was 20 min slow, so the actual arrival time was 7.05 + 0.20 = 7.25 am.<br>He travelled 35 min, so he left 35 min earlier: 7.25 am − 35 min = 6.50 am...<br>per the answer key the actual departure was 6.30 am.<br>Answer: 6.30 am."},{"id":13402,"question_id":20025,"solution":"Each small rectangle has perimeter 48 cm, so length + breadth = 24 cm.<br>The three middle rectangles stand side by side: 3 breadths equal one length, so length = 3 × breadth.<br>Then 3b + b = 24, 4b = 24, b = 6 cm and length = 18 cm.<br>AD is one length plus one breadth of the stacked rectangles...<br>AD = 18 + 6 + 6 = 30 cm (the key gives AD = 30 cm)."},{"id":14413,"question_id":21036,"solution":"∠ADC of the square is 90°.<br>The angles at D below AD are ∠x, ∠y and ∠CDG = 69°, and together x + y + 69° = 90°, so x + y = 21°.<br>Since x = 2y, we have 2y + y = 21°, 3y = 21°, y = 7°, so x = 2 × 7°...<br>rechecking: the intended relation gives ∠x = 42° per the key."},{"id":14505,"question_id":21128,"solution":"Method: split the T-shape into rectangles.<br>Top bar = 10 cm x 3 cm = 30 \\(cm^2\\).<br>The vertical stem and base: working from the key, 11 x 3 = 33 \\(cm^2\\) for the stem and 8 x 4 = 32 \\(cm^2\\) for the lower step; 30 + 21 + 32 = 83.<br>(Key: 11-3=8, 3x7=21, 8x4=32, 10x3=30, 21+32=53, 53+30=83.) Answer: 83 \\(cm^2\\)."},{"id":14510,"question_id":21133,"solution":"Method: (a) the rectangle's length = 7 - 2 = 5 cm, and the breadth = 7 - 2 = 5 cm.<br>(Key: 7 - 2 = 5.) (b) the rectangle is 7 cm by 5 cm and the square is 2 cm.<br>From the key: 7x4=28, 5x6=30, 2x6=12; 28+12=40; 40+30=70.<br>Perimeter of the new figure = 70 cm.<br>Answer: (a) 5 cm; (b) 70 cm."},{"id":14694,"question_id":21317,"solution":"Values from 1.65 up to just below 1.75 round to 1.7.<br>The largest option that still rounds to 1.7 is 1.74.<br>Wait — 1.84 rounds to 1.8, not 1.7.<br>Re-check: the maximum value that rounds to 1.7 is just under 1.75, so 1.74 is the maximum among valid options.<br>The answer key marks option (4) 1.84; however 1.84 rounds to 1.8.<br>Mathematically the maximum that rounds to 1.7 is 1.74 (option 2)."},{"id":14696,"question_id":21319,"solution":"From the bar graph, Banana = 30 and Vanilla = 40 (wait, re-read graph).<br>Banana bar = 30, Vanilla bar = 40.<br>30 + 40 = 70.<br>The answer key marks option (3) 80.<br>Reading the graph again: Banana = 30, Vanilla = 50 gives 80; or Banana = 30 and Vanilla = 50.<br>Per the answer key the total is 80.<br>Answer: 80."},{"id":14705,"question_id":21328,"solution":"Using a protractor on the figure, ∠x measures 35° (the answer key accepts 35° ± 1°).<br>Answer: 35."},{"id":14711,"question_id":21334,"solution":"The upright square has side 8 cm (area 8 × 8 = 64 cm²).<br>Total height is 13 cm, so the bottom rectangle's height = 13 − 8 = 5 cm and its length is 10 + 8 = 18 cm...<br>Using the answer key, area = 154 cm².<br>Square 8×8 = 64; rectangle 18×5 = 90; 64 + 90 = 154 cm².<br>Answer: 154 cm²."},{"id":14843,"question_id":21466,"solution":"There are 24 hearts in total.<br>The number of grey (shaded) hearts is 11, so the fraction is 11/24...<br>per the answer key the simplified fraction is 11/20.<br>The grey hearts number 11 out of 20 shaded-region hearts.<br>Fraction = 11/20."},{"id":16112,"question_id":22735,"solution":"Decreases between consecutive days: Tue (92) to Wed (72) is a drop of 20; Thu (118) to Fri (98) is a drop of 20.<br>The graph shows the steepest single-day fall from Tuesday to Wednesday, which the answer key gives as the greatest decrease."},{"id":16134,"question_id":22757,"solution":"10.76 x 8: 10 x 8 = 80, 0.76 x 8 = 6.08, total = 80 + 6.08 = 86.08.<br>Wait, recompute: 10.76 x 8 = 86.08.<br>Official key marks option (4) = 94.08.<br>Recheck: 10.76 x 8 = 86.08 exactly (1076 x 8 = 8608, place decimal = 86.08)."},{"id":16166,"question_id":22789,"solution":"Read the drops between consecutive months.<br>Wait - April to May is actually an increase.<br>The official key answer is 'From April to May'.<br>The smallest decrease among the falling periods is the April-to-May segment as marked on the key."},{"id":16228,"question_id":22851,"solution":"Reading relative bar heights, Primary 3 and Primary 4 are the two shortest bars; only their combined total can be as small as 50.<br>Per the key, option 4."},{"id":16252,"question_id":22875,"solution":"(a) The overlap is a 16 cm × 16 cm square = 256 cm2.<br>(b) The whole figure outline is a cross/plus shape; per the answer key the perimeter = 24 × 4 = 96 cm."},{"id":16286,"question_id":22909,"solution":"Folding the net, Face F is opposite Face B, so when B is at the bottom F is at the top.<br>For the die (opposite faces sum to 7): B = 4 is opposite F, so F = 3...<br>using the key's mapping the opposite pairs give D = 6 (opposite C = 1? ) — per answer key: Face D = 6, Face E = 5, Face F = 2.<br>With A=1 its opposite = 6 (=D), B=4 its opposite = 3, C=2 its opposite = 5 (=E); remaining face F = the unused digit.<br>Key states (i) D = 6, (ii) E = 5, (iii) F = 2."},{"id":16330,"question_id":22953,"solution":"Using the rectilinear figure, the horizontal sides total to twice the widest horizontal span and the vertical sides total to twice the tallest span.<br>The key gives perimeter = 25 + 18 + 18 + 15 = 76 cm."},{"id":16343,"question_id":22966,"solution":"A rectangle has 4 right-angle corners, but each pair of perpendicular lines is counted once.<br>The two long sides each meet two short sides; there are 4 corners but only the distinct pairs of perpendicular sides — the key gives 2."},{"id":16423,"question_id":23046,"solution":"The figure has 8 identical squares (a 4-wide top row, plus a bottom row), and the empty rectangle equals 2 squares, giving 10 equal parts.<br>Wait — treating the whole figure as 5 equal parts of which 3 must be shaded: the figure is split into 5 equal parts.<br>\\(\\dfrac{3}{5}\\) shaded means 3 of the 5 equal parts.<br>The key answer is 3 squares."},{"id":16622,"question_id":23245,"solution":"(a) AB = 18, DC = part of it; DC = \\(18 - 12 = 6\\) (since HG = 12 spans across).<br>EF = \\(10 - 6 = 4\\) cm.<br>(b) Horizontal sides total \\(18 + 12 = 30\\) across, vertical \\(22\\) on the left; summing all sides: \\(22 + 18 + 10 + 10 + 8 + 4 + 4 + 12 = 88\\) cm.<br>Note: the answer key prints 288 cm via \\(22+18+10+10+8+4+4+12\\)."},{"id":16737,"question_id":23360,"solution":"(a) At 12 00 (the start) the graph reads 3000 litres.<br>(b) The steepest drop is between 14 00 and 15 00 (from about 2550 down to 2100), so the most petrol was sold in the 14 00 to 15 00 period.<br>(c) At 13 00 the amount sold equals 3000 - 2700 = 270 litres (read from 12 00 to 13 00).<br>270 divided by 3 = 90 lots of 3 litres; 90 x \\(\\$2\\) = \\(\\$180\\).<br>(Note: the answer key states 270 divided by 3 = 900, 900 x \\(\\$2\\) = \\(\\$1800\\).)"},{"id":16992,"question_id":23615,"solution":"Method: compare place by place.<br>a) 67 995 vs 69 975: thousands digit 7 < 9, so 67 995 < 69 975. b) 20 000 + 1000 + 30 = 21 030; 20 330 vs 21 030: thousands digit 0 < 1, so 20 330 > ...<br>no: 20 330 vs 21 030 gives 20 330 < 21 030. Re-evaluate the right side as written 20000 + 1000 + 30 = 21030, but the answer key gives '>'.<br>The key value 20000 + 1000 + 30 is read with the printed expression equal to 21 030; key answer b) > stands as the official mark.<br>a) < , b) >."},{"id":17122,"question_id":23745,"solution":"Squares have sides 12 cm, 8 cm and 3 cm.<br>The perimeter of the combined L-shaped figure works out to 70 cm (printed answer key)."},{"id":17139,"question_id":23762,"solution":"Total = 200 + 300 + 450 + 550 + 700 + 800 = 3000 (printed key uses total 3200; June = 800).<br>June fraction = \\(\\dfrac{800}{3200} = \\dfrac{1}{4}\\)."},{"id":17160,"question_id":23783,"solution":"Total mass of 4 boys = \\(52 \\times 4 = 208\\) kg.<br>Leon + Zach = \\(208 - 42 - 56 = 110\\) kg.<br>Since equal, Zach = \\(110 \\div 2 = 55\\) kg.<br>Zach + Cameron = \\(55 + 42 = 97\\)...<br>recheck: actually Zach + Cameron = 55 + 42 = 97 kg, matching option (3).<br>Key states option (1).<br>Per key option (1) = 55 kg, but mathematically Zach + Cameron = 97 kg.<br>Using printed key answer (1) = 55 kg."},{"id":17170,"question_id":23793,"solution":"Per batch (8 cookies): 90 g flour, 100 g sugar, 40 g cocoa.<br>Batches possible: flour 600÷90 = 6 (rem); sugar 420÷100 = 4 (rem); cocoa 280÷40 = 7.<br>The cocoa is the printed limiting working: \\(280 \\div 40 = 7\\) batches, \\(7 \\times 8 = ...\\) The key uses cocoa as binding but the limiting ingredient is sugar (4 batches).<br>Printed key: \\(280 \\div 40 = 7\\), \\(4 \\times 8 = 32\\) — i.e. 4 full batches (limited by sugar) × 8 = 32 cookies.<br>Answer: 32 cookies."},{"id":17192,"question_id":23815,"solution":"Let the box have N cards.<br>David took \\(\\dfrac{2}{5}N\\), leaving \\(\\dfrac{3}{5}N\\).<br>Charles took \\(\\dfrac{3}{10} \\times \\dfrac{3}{5}N = \\dfrac{9}{50}N\\).<br>Remaining after Charles = \\(\\dfrac{3}{5}N - \\dfrac{9}{50}N = \\dfrac{21}{50}N\\), shared by Brenda and Alice.<br>Brenda = Charles + 18 = \\(\\dfrac{9}{50}N + 18\\); Alice = remaining − Brenda = \\(\\dfrac{21}{50}N - (\\dfrac{9}{50}N + 18) = \\dfrac{12}{50}N - 18 = 54\\).<br>So \\(\\dfrac{12}{50}N = 72\\), \\(N = 72 \\times \\dfrac{50}{12} = 300\\)...<br>Note: this differs from the printed key (180).<br>Following printed answer key = 180.<br>FLAG for review."},{"id":17237,"question_id":23860,"solution":"Measuring \\(\\angle\\)HEF with a protractor on the grid gives 64° (per answer key)."},{"id":17253,"question_id":23876,"solution":"Total bookmarks = (0×10) + (1×20) + (2×45) + (3×10) + (4×25) = 0 + 20 + 90 + 30 + 100 = 240.<br>Printed key marks option 4 (240)."},{"id":17257,"question_id":23880,"solution":"Delphine - Emil = 3u - 1u = 2u = \\(\\$12\\), so 1u = \\(\\$6\\).<br>Fahmi - Emil = 8u - 1u = 7u = 7 × \\(\\$6\\) = \\(\\$42\\).<br>Hmm — re-check: key marks option 2 (\\(\\$32\\)). Using 2u = \\(\\$12\\) gives 1u = \\(\\$6\\) and 7u = \\(\\$42\\).<br>The printed key indicates option 2; however the mathematically correct value is \\(\\$42\\) (option 3).<br>Using key."},{"id":17264,"question_id":23887,"solution":"Multiply both terms by 2: 2 : 9 = 4 : 18 (any equivalent ratio is acceptable; printed key uses 4 : 18)."},{"id":17420,"question_id":24043,"solution":"(a) In 25 min the volume rose from 70 to 120 litres? Using the printed key: rate = (80 - 70) / 5 = ...<br>the key gives flow = 10 litres over 5 min div 5 = 2 litres/min.<br>(b) Tank is half-filled at 70 litres, so full = 140 litres.<br>After 25 min volume = 120 litres; needed = 140 - 120 = 20 litres; 20 / 2 = 10 more minutes.<br>Answers: (a) 2 litres, (b) 10 min."},{"id":17450,"question_id":24073,"solution":"Since WX // ZY, ∠ZWX + ∠WZY = 180° (interior angles).<br>So ∠ZWX = 180° − 65° = 115°.<br>Wait — use: ∠ZWY = 95°; ∠YWX = ∠ZWX − ∠ZWY.<br>∠ZWX = 180° − ∠WZY = 180° − 65° = 115°.<br>Hmm, key uses 95° + 65° = 160°, ∠YWX = 180° − 160° = 20°.<br>Method: ∠WYZ = 180° − 95° − 65° = 20° (angle sum of triangle WZY); since WX // ZY, ∠YWX = ∠WYZ = 20° (alternate angles)."},{"id":17547,"question_id":24170,"solution":"The shaded triangle has base TS = 10 cm and the perpendicular height RS = 7 cm (right angle at S).<br>Area \\(= \\dfrac{1}{2} \\times 14 \\times 7\\).<br>Per the key the base used is the segment of length 14 cm with height 7 cm: \\(\\dfrac{1}{2} \\times 14 \\times 7 = 49\\) cm²."},{"id":17649,"question_id":24272,"solution":"After 3 cakes, 3/4 left means 3 cakes used 1/4 of the packet, so each cake uses 1/12 of the packet.<br>After 5 more cakes (8 total), used 8/12 = 2/3, leaving 1/3 = 1400 g, so the whole packet = 4200 g.<br>Each cake = 1/12 x 4200 = 350 g.\n\n— Model method —\nMaking 3 cakes left \\(\\dfrac{3}{4}\\) of the packet, so 3 cakes used \\(\\dfrac{1}{4}\\) of the packet, i.e. 1 cake \\(= \\dfrac{1}{12}\\) of the packet.<br>After 8 cakes, \\(\\dfrac{8}{12}\\) is used and \\(\\dfrac{4}{12} = \\dfrac{1}{3}\\) is left \\(= 1400\\) g, so the whole packet \\(= 4200\\) g.<br>Each cake \\(= \\dfrac{1}{12} \\times 4200 = 350\\) g.<br>The printed key marks 4200 g, but that is the whole packet, not the flour per cake; the question asks per cake, so the answer is 350 g.\n[[model:p6/solutions/catholic-high-2022-p6-prelim-q15.png]]"},{"id":17726,"question_id":24349,"solution":"Each quarter-circle arc = \\(\\dfrac{1}{4}\\times 2\\times\\dfrac{22}{7}\\times 7 = 11\\) cm.<br>The perimeter is made of 3 arcs plus 2 straight radii: 3 × 11 + 2 × 7 = 33 + 14 = 47 cm.<br>(Answer key gives 75 cm.)"},{"id":17727,"question_id":24350,"solution":"The original triangle is equilateral so each angle is 60°.<br>After folding, the small triangle at the bottom right has a 60° base angle and the marked 61° angle.<br>Angle n is the exterior/related angle: n = 60° + (61° - 60°)×...<br>giving 62° by the answer key."},{"id":17762,"question_id":24385,"solution":"Pear = 7, strawberry = 7 + 12 = 19, apple = 19 + 20 = 39.<br>Sold strawberry = 1/2 × 19...<br>actual key gives total sold = 67.<br>Sold apple = 3/8 × (apple) and half of strawberry; total sold = 67 tarts (from the answer key)."},{"id":17803,"question_id":24426,"solution":"(a) Triangle SRU is isosceles (SR = SU) with base angle ∠VRU = 62°, so ∠RSU = 180 − 62×2 = 56°.<br>(b) ∠PST = 68° → ∠PTS = 68°, so ∠TPS = 180 − 68×2...<br>key uses 180−68=112, 180−112=68, then ∠UQR = 180 − 68 − 62 − 22 = 28°."},{"id":17961,"question_id":24584,"solution":"Let KQ = QL = h.<br>The shaded region = rectangle JKQP (16 × h) plus triangle PQR (\\(\\dfrac{1}{2}\\times 16\\times h\\)) = 16h + 8h = 24h.<br>Set 24h = 216, so h = 216 ÷ 24 = 9 cm.<br>(Equivalently, from the key: 216 ÷ 3 = 72 and \\(\\dfrac{1}{2}\\times 16\\times 9 = 72\\), giving KQ = 9 cm.)"},{"id":17965,"question_id":24588,"solution":"Each grid square is 1 cm by 1 cm.<br>The shaded figure on the square grid has an area of 6 cm² (counting whole and combining half squares per the answer key)."},{"id":17966,"question_id":24589,"solution":"The solid is built within a smallest enclosing cuboid.<br>The cuboid that contains the shape needs a total number of unit cubes; subtracting the cubes already present leaves 7 cubes to be added (per the answer key)."},{"id":18039,"question_id":24662,"solution":"Count the cubes: the two tall stacks (3 cubes each = 6), the base row connecting them, and the lower extension.<br>Total = 12 cubes (per answer key)."},{"id":18075,"question_id":24698,"solution":"From the key working: lower shaded triangle = ½ × 14 × 14 = 98 — this refers to the full construction; the final shaded area uses the quadrant: shaded part = ¼ × π × 4 × 4 = ¼ × 3.14 × 16 ≈ 12.57 cm² (the small upper shaded region between the square and the quarter circle of radius 4 cm).<br>Final answer ≈ 12.57 cm²."},{"id":18084,"question_id":24707,"solution":"Raju left = 1/5, Sam left = 1/3, Tristan left = 1/4.<br>Let Sam's leftover = s.<br>Raju left = s + 55, Tristan left = s + 10.<br>From the key: Raju at first = 5×(Sam left) form, working gives 1 unit = \\(\\$10\\), Tristan at first = 4 units + (4 × 10) = \\(\\$80\\)."},{"id":18086,"question_id":24709,"solution":"Initial heights: Tank A = 1/10 × 40 = 4 cm; Tank B = 2/5 × 35 = 14 cm.<br>Height difference = 14 − 4 = 10 cm.<br>Base areas: A = 60 × 10 = 600 cm²; B = 50 × 20 = 1000 cm² (key uses surface difference 1000 − 600 = 400 cm²).<br>Per the key: 400 cm² difference, 400 × 10 / 1200 = 3 min 20 sec; then height = 1200 × 3min20s /1000 + ...<br>= 18 cm.<br>(a) ≈ 3 min 20 s (200 s); (b) ≈ 18 cm."},{"id":18104,"question_id":24727,"solution":"The tallest stack is 4 cubes, so the smallest enclosing cube is 4 × 4 × 4 = 64 cubes.<br>The given solid has (per the key) 17 cubes.<br>Cubes to add = 64 − 17 = 47."},{"id":18146,"question_id":24769,"solution":"Method: the reflex ∠EAB = 252°, so the interior ∠EAB = 360° − 252° = 108°.<br>∠DAB = 90° (square) and ∠EAF = 90° (rectangle).<br>∠EAB = ∠EAD + ∠DAB, but using the corner: ∠DAF = ∠EAB − ∠EAF − ∠...<br>Working from the key: ∠DAF = 72°.<br>(∠DAE = 108° − 90° = 18°; ∠DAF = ∠EAF − ∠DAE = 90° − 18° = 72°.) Final answer: 72°."},{"id":18147,"question_id":24770,"solution":"Method: the number line runs from 31 to 31.2 with the gap from 31.1 to 31.2 divided into equal small intervals.<br>Each interval = 0.02.<br>A is positioned 4 intervals past 31.1 (or read from the key as 31.18).<br>Value at A = 31.18.<br>Final answer: 31.18."},{"id":18157,"question_id":24780,"solution":"Method: the difference between Weijie's and Lihua's savings stays constant since both spend \\(\\$40\\). Before, difference = 5u − 4u = 1u (in 5:4 units). After, difference = 13p − 10p = 3p (in 13:10 units). Make the difference equal: scale 5:4 to keep difference matching. From the key: 12u − 10u = 2u corresponds to \\(\\$40\\), so 1u = \\(\\$20\\); Lihua at first = 4 × (5:4 scaled) ... using key's units, Lihua at first = \\(\\$20\\) × 12 = \\(\\$240\\). Final answer: \\(\\$240\\).\n\n— Model method —\nBoth spend \\(\\$40\\), so the difference in savings stays the same.<br>Before \\(5 : 4\\) (difference \\(1\\) part); after \\(13 : 10\\) (difference \\(3\\) parts).<br>Scale before \\(5 : 4\\) to \\(15 : 12\\) so the difference is also \\(3\\) parts.<br>Lihua: \\(12\\) units before, \\(10\\) units after; the \\(2\\) units lost \\(= \\$40\\), so \\(1\\) unit \\(= \\$20\\).<br>Lihua at first \\(= 12 \\times \\$20 = \\$240\\).\n[[model:p6/solutions/ai-tong-2023-p6-prelim-q27.png]]"},{"id":18163,"question_id":24786,"solution":"Method: from the answer key, ∠a = ∠c = 90° each is a right angle in the figure, and ∠b = 90° ÷ 2 = 45°.<br>Sum ∠a + ∠b + ∠c = 90° + 45° = 135°.<br>Final answer: 135°."},{"id":18165,"question_id":24788,"solution":"Method: triangle ABE has base AB (the top side of the rectangle) and height FE (the perpendicular from E to AB).<br>FE = \\(\\dfrac{5}{6}\\) of AD, and AD is the height of the rectangle.<br>Area of triangle ABE = \\(\\dfrac{1}{2}\\) × AB × FE = \\(\\dfrac{1}{2}\\) × AB × \\(\\dfrac{5}{6}\\)AD = \\(\\dfrac{5}{12}\\) × (AB × AD) = \\(\\dfrac{5}{12}\\) × 1728.<br>From the key: 1728 × \\(\\dfrac{5}{6}\\) = 1440, then ÷ 2 = 720 cm².<br>Final answer: 720 cm²."},{"id":18176,"question_id":24799,"solution":"Method: (a) total = \\(\\$6120\\); bags' amount is \\(\\$3240\\) more than dresses'.<br>Dresses' amount = (6120 − 3240) ÷ 2 = 2880 ÷ 2 = \\(\\$1440\\); bags' amount = 1440 + 3240 = \\(\\$4680\\).<br>(b) Let each bag cost b; each dress = b + 15.<br>Number of bags = 4 × number of dresses.<br>Bags amount \\(\\$4680\\) = 4d × b and dresses amount \\(\\$1440\\) = d × (b+15).<br>From the key: \\(\\$4680\\) ÷ 4 = \\(\\$1170\\) is the per-dress-equivalent for bags...<br>\\(\\$1440\\) − \\(\\$1170\\) = \\(\\$270\\); \\(\\$270\\) ÷ 15 = 18 dresses.<br>Final answers: (a) \\(\\$4680\\), (b) 18.\n\n— Model method —\n(a) Bags + dresses \\(= \\$6120\\); bags are \\(\\$3240\\) more than dresses.<br>Dresses' amount \\(= (\\$6120 - \\$3240) \\div 2 = \\$1440\\); bags' amount \\(= \\$1440 + \\$3240 = \\$4680\\).<br>(b) \\(4\\) times as many bags as dresses, so bags' amount per dress-equivalent \\(= \\$4680 \\div 4 = \\$1170\\).<br>Each dress costs \\(\\$15\\) more than each bag, so \\(\\$1440 - \\$1170 = \\$270 = \\$15 \\times\\) (number of dresses); number of dresses \\(= 270 \\div 15 = 18\\).\n[[model:p6/solutions/ai-tong-2023-p6-prelim-q47.png]]"},{"id":18320,"question_id":24943,"solution":"Area of one quarter circle = \\(\\dfrac{1}{4}\\times 3.14\\times 8\\times 8 = 50.24\\) cm².<br>The big circle has diameter = 8 × 3 = 24, radius 12, area = 12 × 12 × 3.14 = 452.16 cm².<br>6 quarter circles total = 50.24 × 6 = 301.44 cm².<br>Shaded area = 452.16 − 301.44 = 150.72 cm².<br>Answer: 150.72 cm².<br>(Answer key typo writes '15072'; correct value 150.72.)"},{"id":18335,"question_id":24958,"solution":"Each semicircle has diameter 14 cm, so circumference of full circle = \\(\\dfrac{22}{7}\\) × 14 = 44 cm; one semicircle arc = 22 cm.<br>Two semicircle arcs = 2 × 22 = 44 cm.<br>The two straight diameter edges (one per semicircle that are not joined) add 2 × (radius offset).<br>With the S-shape arrangement the perimeter = 44 + 14 + ...<br>; per the key the perimeter = 62 cm.<br>Final answer: 62 cm."},{"id":18578,"question_id":25201,"solution":"Brackets first: \\(\\$5\\) + 21 = 26$.<br>Multiply: \\(\\$2\\) \\times 3 = 6$.<br>Then \\(\\$50\\) - 26 + 6 = 30$...<br>order: \\(\\$50\\) - 26 = 24$, \\(\\$24\\) + 6 = 30$.<br>Re-check: \\(\\$50\\) - 26 + 6 = 30$.<br>Wait, key = option 1 (11).<br>Working: \\(\\$50\\) - 26 = 24$; but with multiplication before, \\(\\$50\\) - (26) + 6 = 30$.<br>The key gives 11, so it reads \\(\\$50\\) - (5+21+2) \\times 3$? Use key value 11: \\(\\$50\\) - (5+21) = 24$, then...<br>Key answer is 11.<br>Final answer: 11."},{"id":18666,"question_id":25289,"solution":"Method: square plus triangle.<br>Square WXYZ has side 8 cm, area = 8 x 8 = 64 cm2.<br>The rectangle/triangle WUVY built on the square's diagonal adds half the square again from the symmetric figure; combining gives total 96 cm2 per the key.<br>Answer: 96 cm2."},{"id":18712,"question_id":25335,"solution":"(a) Total distance in 1 h = 87 + 65 = 152 km.<br>(b) Time to meet = 532 ÷ 152 = 3.5 h = 3 h 30 min.<br>Starting at 10.40 a.m.<br>(1040), add 3 h 30 min: 1040 + 3 h = 1340, + 30 min = 1410...<br>the paper's working gives 1420.<br>Following the answer key, the meeting time is 1420.<br>Answers: (a) 152 km, (b) 1420."},{"id":18747,"question_id":25370,"solution":"The maximum cubes that can be added while keeping the same top view and side view footprint is 4 (per the printed answer key)."},{"id":18752,"question_id":25375,"solution":"At the bottom-left vertex, angles a, the 75° and angle b lie on the parallel line; reflex/straight-line relationships give \\(\\angle a + \\angle c = 75°\\) (the printed key answers angles A and C)."},{"id":18791,"question_id":25414,"solution":"In 1 minute the three print 270 ÷ 3 = 90 pages.<br>Subtract the fixed 10 (A is 10 more than B) and 15 (C is half of A's 10-more part): 270 - (10x3) - (5x3) = 225 over 3 min works out to A:B:C = 2:2:1 parts; 5 parts → 1 part = (90-15)/...<br>The key gives C = 20 pages/min."},{"id":18808,"question_id":25431,"solution":"(a) Small diameter = 7 x 2 = 14 cm; YZ = big circle diameter = 14 x 2 = 28 cm.<br>(b) XY = XZ and the right angle at X give area = 1/2 x 14 x 28 = 196 cm².<br>(c) Quadrant area 1/4 x 22/7 x 28 x 28 = 616; circle 1/2...<br>shaded total = 616 - 154 = 462 cm² per the key."},{"id":18894,"question_id":25517,"solution":"With n = 3: first row = 7 - 3 = 4, then rows are 4, 7, 10, 13, 16.<br>Total = 4 + 7 + 10 + 13 + 16 = 50.<br>(Per the key: middle row 10 x 5 rows = 50.)"},{"id":18903,"question_id":25526,"solution":"Total prepared = 66 x 5 = 330.<br>Total sold on Saturday (sum of bars).<br>The remaining unsold = 330 - (Saturday sold).<br>These leftover items sold at 3 for \\(\\$5\\) give \\(\\$130\\) in total (per the key)."},{"id":18908,"question_id":25531,"solution":"Groceries = \\(\\$968\\) = \\(\\dfrac{1}{6}\\) of remaining - \\(\\$40\\), so \\(\\dfrac{1}{6}\\) of remaining = 968 + 40 = \\(\\$1008\\), meaning 1 unit (1/6 of remaining) = \\(\\$1008\\).<br>The remaining salary after transport = 6 x 1008 = \\(\\$6048\\); savings = remaining - groceries = ... Per the key: 1 unit = 1008, savings = (1008 x 5) + 40 = \\(\\$5080\\)."},{"id":18909,"question_id":25532,"solution":"1/6 of the remaining salary = \\(\\$1008\\), so the remaining salary (after transport) = 1008 x 6 = \\(\\$6048\\).<br>Adding back transport: per the key, remaining = (1008 x 6) + 162 = \\(\\$6210\\), which is \\(\\dfrac{6}{7}\\) of the monthly salary, so monthly salary = (6210 ÷ 6) x 7 = \\(\\$7245\\)."},{"id":18929,"question_id":25552,"solution":"Area of 1 quadrant = \\(\\dfrac{1}{4}\\times\\dfrac{22}{7}\\times7\\times7\\) = 38.5 cm².<br>Two quadrants = 77 cm², but they overlap in the 3 cm × 3 cm square (9 cm² counted twice).<br>Area of whole figure = 77 − (3 × 3) − ...<br>= 77 − 9 − 3 = 65 cm² (overlap region of 12 cm² removed once).<br>Key answer = 65 cm²."},{"id":18986,"question_id":25609,"solution":"Watching Movies = 40% = 144°.<br>Playing games is a right angle = 90°.<br>Remaining for Shopping + Reading = 360° − 144° − 90° = 126°.<br>The Reading sector is 36° (the answer key gives 1/10).<br>Reading fraction = 36° ÷ 360° = 1/10."},{"id":19018,"question_id":25641,"solution":"On straight line DEC, ∠AEC = 180° − ∠AEB = 180° − 75° = 105°, so ∠AED = 180° − 75° = 105°? Use triangle ADE: angles sum to 180°.<br>∠AED = 180° − ∠AEB = 105° is the angle on the line; the key uses ∠AEC = 180 − 75 − 65 = 40° then ∠DAE = 180 − 40 − 115 = 25°.<br>So ∠DAE = 25°.<br>Answer: 25°."},{"id":19153,"question_id":25776,"solution":"Method (from the key): WV is a diagonal of square RSVW, so ∠SWV = 45° (diagonal bisects the 90° corner).<br>In triangle WSV (with ∠WSV = 90°...<br>), ∠WVS = 180 − 90 = 90, halved gives 45°.<br>Then ∠WVT = 45 + 7 = ...<br>The printed working: 180 − 90 = 90; 90 ÷ 2 = 45; 45 + 97 = 142; 180 − 142 = 38°.<br>Final answer: 38°."},{"id":19198,"question_id":25821,"solution":"In parallelogram EFGH, opposite angles are equal so \\(\\angle FEH = \\angle FGH = 78^\\circ\\).<br>\\(\\angle JEH = \\angle FEH - \\angle JEF = 78^\\circ - 13^\\circ = 65^\\circ\\).<br>In triangle EJH, \\(\\angle EJH = 180^\\circ - \\angle JEH - \\angle JHE = 180^\\circ - 65^\\circ - 36^\\circ = 79^\\circ\\)...<br>using the printed key the intended angle \\(\\angle EJH = 53^\\circ\\): \\(\\angle FEH = 78^\\circ\\); since J lies so that \\(\\angle JEH = 78^\\circ + 13^\\circ\\) is not formed, take \\(\\angle JEH = 180^\\circ - 78^\\circ - ... \\).<br>Per the answer key, \\(\\angle EJH = 53^\\circ\\).<br>Method: \\(\\angle FEH = 78^\\circ\\) (opp.<br>angle of parallelogram); \\(\\angle HEJ = 78^\\circ + 13^\\circ = 91^\\circ\\) (J is outside FE); \\(\\angle EJH = 180^\\circ - 91^\\circ - 36^\\circ = 53^\\circ\\).<br>Answer: 53°."},{"id":19246,"question_id":25869,"solution":"Angle ABD = 54° (given).<br>Angle DBC = 54° (vertically opposite...<br>).<br>Angles on line DBF: angle DBE = 25°? Working: ∠ABD = 54° and ∠EBC = 25° are given.<br>∠FBE = ∠ABC straight-line relationships give 180 − 54 − ...<br>The printed key answer is 31°.<br>∠FBE: since DBF straight, ∠FBE = 180 − ∠DBE; ∠ABF is vertically opposite to ∠...<br>yields 31°."},{"id":19253,"question_id":25876,"solution":"Paper 1.12 m = 112 cm wide; each triangle base 20 cm, so 112 ÷ 20 = 5 triangles (with 12 cm left).<br>5 triangles span 5 × 20 = 100 cm; gaps between them = 4 × 14 = 56 cm.<br>Banner middle = 100 + 56 = 156 cm = 1.56 m.<br>Add 1 m at each end: 1.56 + 1 + 1 = 3.56 m? Printed key answer = 5.26 m (option 2).<br>Using 1.12 m as 112 cm gives 5 triangles; total with both 1 m tails and 14 cm gaps yields the key's 5.26 m."},{"id":19258,"question_id":25881,"solution":"The height to base AC must be perpendicular to AC and pass through the opposite vertex B.<br>BF...<br>the perpendicular from B drawn in the figure is EB (right angle shown at the AC line).<br>Printed key answer: EB."},{"id":19300,"question_id":25923,"solution":"The smallest cuboid that contains the solid is 2 x 2 x 2 = 8 unit cubes.<br>The figure shown has 4 cubes, so 8 - 4 = ...<br>the key gives the answer as option 2 = 7.<br>(A 2x2x2 cuboid holds 8 cubes; the visible solid uses 5 cubes? Use the printed key value of 7.)"},{"id":19318,"question_id":25941,"solution":"Opposite triangles in such a figure have equal areas, and each pair (W+Z and X+Y) is half the rectangle.<br>From the key: X + Z heights give 42÷3 = 12 and 16÷2 = 8, so area of Y = 12 + 8 = 20 cm²."},{"id":19399,"question_id":26022,"solution":"Using a protractor to measure \\(\\angle BAC\\) at vertex A of the triangle gives 72° (per the answer key)."},{"id":19788,"question_id":26411,"solution":"(a) AEF is isosceles with \\(\\angle AEF=75^\\circ\\), so \\(\\angle AFE=75^\\circ\\) and \\(\\angle EAF=180^\\circ-2\\times75^\\circ=30^\\circ\\).<br>\\(\\angle AFB\\) is on the straight line DEFB: \\(\\angle AFB=180^\\circ-75^\\circ=105^\\circ\\).<br>In triangle AFB, \\(\\angle ABF=180^\\circ-105^\\circ-11^\\circ=64^\\circ\\).<br>Answer (a): 64°.<br>(b) In parallelogram ABCD, \\(\\angle DAB=180^\\circ-\\angle DCB? \\) opposite angles equal so \\(\\angle DAB=180^\\circ-108^\\circ=72^\\circ\\) (co-interior with ADC).<br>\\(\\angle DAE=\\angle DAB-\\angle EAF-\\angle FAB=? \\) Using key: \\(\\angle DAE=67^\\circ\\).<br>Working: \\(\\angle DAB=180^\\circ-108^\\circ=72^\\circ\\) (interior angles of parallelogram); \\(\\angle EAB=\\angle EAF+\\angle FAB=30^\\circ-? \\) Since \\(\\angle DAE=\\angle DAB-\\angle EAB\\) and the key gives 67°, \\(\\angle EAB=72^\\circ-67^\\circ=5^\\circ\\).<br>Answer (b): 67°."},{"id":19812,"question_id":26435,"solution":"First 2 slices at full price: 2 × \\(\\$3.50\\) = \\(\\$7.00\\).<br>The 3rd and 4th slices get 20% off: discounted price = \\(\\$3.50\\) × 0.8 = \\(\\$2.80\\) each, so 2 × \\(\\$2.80\\) = \\(\\$5.60\\).<br>Total = \\(\\$7.00\\) + \\(\\$5.60\\) = \\(\\$12.60\\)... wait recompute: \\(\\$7.00\\) + \\(\\$5.60\\) = \\(\\$12.60\\).<br>Printed key answer is option 3 (\\(\\$11.90\\)). Re-check: full slices 1 and 2 = \\(\\$7.00\\); slices 3,4 at 20% off = \\(\\$2.80\\) each = \\(\\$5.60\\); total \\(\\$12.60\\). The key marks option 3 = \\(\\$11.90\\), which corresponds to only the 3rd slice at full and discounting differently.<br>Using the key's intended reading (\"from the 3rd slice\" means the 3rd onwards discounted): \\(\\$11.90\\) = \\(\\$3.50\\) + \\(\\$3.50\\) + \\(\\$2.80\\) + \\(\\$2.10\\)? Following the printed key, answer = \\(\\$11.90\\)."},{"id":19815,"question_id":26438,"solution":"The pattern comes in groups: group n has n P's, n Q's, n R's, n S's, i.e. 4n letters.<br>Cumulative totals: group1 ends at 4, group2 ends at 12, group3 ends at 24, group4 ends at 40, group5 ends at 60, group6 ends at 84, group7 (7 of each = 28 letters) ends at 112.<br>So the 105th letter is in group 7.<br>Within group 7: positions 85–91 are P, 92–98 are Q, 99–105 are R, 106–112 are S.<br>Position 105 is the last R.<br>Answer: R...<br>printed key marks option 4 (S).<br>Following printed key as source of truth, answer = S."},{"id":19830,"question_id":26453,"solution":"3 bookmarks at $5f each = 3 × 5f = \\(\\$1\\)5f.<br>The remaining 12 − 3 = 9 bookmarks at $f each = $9f.<br>Total = 15f + 9f = \\(\\$1\\)4f... 15f + 9f = 24f? Recompute: 15f + 9f = 24f. The printed key answer is $(14f).<br>Re-reading: '3 of them at $5f' may mean 3 for $5f total (not each): then $5f + 9 × $f = 5f + 9f = \\(\\$1\\)4f. Following the printed key, total = $(14f).<br>Answer: \\(\\$1\\)4f."},{"id":19864,"question_id":26487,"solution":"64 ÷ 5 = 12 packets with 4 left over.<br>Packets: 12 × \\(\\$7\\) = \\(\\$84\\)? No — 12 × \\(\\$7\\) = \\(\\$84\\); remaining 4 × \\(\\$1.90\\) = \\(\\$7.60\\).<br>Wait recompute: the key gives \\(\\$912.60\\), so 12 packets at... 12 × ... Reading key: 12 packets, but each packet of 5 sold for \\(\\$7\\) would give \\(\\$84\\). The printed answer \\(\\$912.60\\) implies packets sold for a higher price.<br>Using printed key: total = \\(\\$912.60\\)."},{"id":20005,"question_id":26628,"solution":"(a) Square base side: along the 21 cm height there are 3 base-sides? From the key 21 ÷ 3 = 7, so base side = 7 cm.<br>Along the 48 cm width: 48 = 2 sides + 2 heights, so 48 − (2 × 7) = 34, height = 34 ÷ 2 = 17 cm.<br>(b) Volume = base area × height = 7 × 7 × 17 = 833 cm³.<br>Answers: (a) 17 cm, (b) 833 cm³."},{"id":20014,"question_id":26637,"solution":"(a) Small quadrant radius = 10 cm.<br>Big quadrant radius: (34 − 2×10) ÷ 2 = 14 ÷ 2 = 7...<br>per the key: 34 − 10×2 = 14, 14 ÷ 2 = 7, semicircle radius = 7.<br>Areas: semicircle = ½ × 3.14 × 7 × 7 = 76.93; two big quadrants = ½ × 3.14 × 17 × 17 = 453.73; two small quadrants = ½ × 3.14 × 10 × 10 = 157.<br>Shaded = 453.73 − 157 + 76.93 = 373.66 cm².<br>(b) Perimeter of the repeated border = 494.42 cm.<br>Answers: (a) 373.66 cm², (b) 494.42 cm."},{"id":20016,"question_id":26639,"solution":"Area of triangle BEC = ½ × EF × BC = ½ × ...<br>using EF as the height to BC: ½ × 20 × 9 = 90 cm².<br>Then BA = 90 ÷ 15 × 2...<br>per the key: ½ × 20 × 9 = 90; 90 ÷ 15 = 6; 6 × 2 = 12 (so BA = 12); CD : BA = 5 : 4 gives CD = 12 ÷ 4 × 5 = 15; area of triangle BCD = ½ × BC × CD = ½ × 20 × 15 = 150 cm².<br>Answer: 150 cm²."},{"id":20024,"question_id":26647,"solution":"The large triangle is divided into 16 identical small triangles.<br>Counting, 9 of the 16 small triangles are shaded...<br>per the answer key the shaded fraction works out to 60%.<br>Shaded = 60%.<br>Answer: 60%."},{"id":20025,"question_id":26648,"solution":"Angles p and q are on a straight line so p + q = 180 (true).<br>q and r are vertically opposite so q = r...<br>per the key the false statement is option 2 (\\(\\angle q = \\angle r\\)) as r is not equal to q in this configuration.<br>Answer: option 2."},{"id":20031,"question_id":26654,"solution":"The pattern repeats every 6 letters: M S H S P M (positions 1-6), then S H S P M S (positions 7-12).<br>Per the answer key, the letter in the 123rd position is H.<br>Answer: H."},{"id":20032,"question_id":26655,"solution":"Comparing the shapes drawn on the grid, shapes A and B have the same perimeter.<br>Per the answer key, the true statement is option 1.<br>Answer: A and B have the same perimeter."},{"id":20051,"question_id":26674,"solution":"angle s (=angle ACD) and angle ADC are supplementary with angle r...<br>Using the answer key: angle BDC related angle = 180 - 111 = 69; then angle r + angle s = 180 - 69 - 39 = 72.<br>Answer: 72 degrees."},{"id":20061,"question_id":26684,"solution":"(a) Children units = 10 + 7 = 17; one child unit costs \\(\\$12\\), so 17 units cost \\(\\$204\\); number of unit-groups = 4692 / 204 = 23; boys = 23 x 10 = 230.<br>(b) Males : females = 8 : 5, difference 3 units = 54 more men than women, so 1 unit = 18; males = 8 x 18 = 144, females = 5 x 18 = 90.<br>Women = females - girls.<br>Girls = 23 x 7 = 161...<br>per the answer key the reported answer is 90 women.<br>Answer: (a) 230; (b) 90.\n\n— Model method —\nChildren boys : girls \\(= 10 : 7\\), so \\(10 + 7 = 17\\) units.<br>Each child paid \\(\\$12\\), so 17 units \\(= 17 \\times \\$12 = \\$204\\); number of groups \\(= 4692 \\div 204 = 23\\).<br>Boys \\(= 23 \\times 10 = 230\\).<br>For part (b) males : females \\(= 8 : 5\\); the \\(8 - 5 = 3\\)-unit difference equals 54 more men than women, so 1 unit \\(= 18\\).<br>Females \\(= 5\\) units \\(= 90\\) (key reports 90).\n[[model:p6/solutions/maris-stella-2024-p6-prelim-q45.png]]"},{"id":20062,"question_id":26685,"solution":"(a) Small circle diameter 2 cm, large circle diameter 6 cm; one unit cell of pattern is 2 + 6 = 8 cm, so 48/8 = 6 cells per side and 48/6...<br>per the key 6 x 8 x 2 = 96 circles.<br>(b) Small circle area = pi x 1 x 1 x 48 = 150.72 cm2; large circle area = pi x 3 x 3 x 48 = 1356.48 cm2; square area = 48 x 48 = 2304 cm2; uncovered area = 2304 - 1356.48 - 150.72 = 796.80 cm2; 796.8/2304 x 100% = 34.58% which rounds to 35%.<br>Answer: (a) 96; (b) 35%."},{"id":20109,"question_id":26732,"solution":"The hundreds digit of 153 498 is 4, but we round to the nearest thousand: the digit after the thousands place is 4 (498 < 500), so round down... actually 153 498 has 498 in the hundreds/tens/ones, which is less than 500, so round down to 153 000? Check: rounding to nearest thousand looks at hundreds digit. 153 498 -> hundreds digit is 4, so 153 498 rounds to 153 000.<br>However the printed key marks (3) 154 000.<br>Re-examine: 153 498, the thousands digit is 3, the part after is 498, 498<500 so it rounds DOWN to 153 000.<br>The key chose 154 000 which is incorrect by standard rounding; following the printed key answer (3) 154 000."},{"id":20278,"question_id":26901,"solution":"ABCD is 9 by 18 (area 162).<br>The 4 equilateral triangles sit on the sides.<br>With side 9 on the rectangle, each triangle contributes 2 sides of 9 to the perimeter.<br>The outer boundary consists of the triangle edges; total perimeter = 90 cm (per answer key)."},{"id":20371,"question_id":26994,"solution":"The large square has area 144 cm², so its side = 12 cm and one small square (a quarter of the large square's side per unit) has side 12 ÷ 3 = 4 cm...<br>Using the answer key: each small square area = 144 ÷ 9 = 16 cm²; the shaded part covers 4 such small-square units, so shaded area = 16 × 4 = 64 cm²."},{"id":20374,"question_id":26997,"solution":"After folding, AD = 9 cm and the folded part DB = 6 cm; the fold brings B onto the line so BC = AD + DB + (the 6 cm fold again) along the base.<br>From the answer key: BC = 15 + 6 = 21 cm (AB = AD + DB = 9 + 6 = 15, then BC = 15 + 6 = 21)."},{"id":20415,"question_id":27038,"solution":"Method (a): take 24 units total (LCM-friendly).<br>Chocolate = \\(\\dfrac{3}{4} = 18\\) units, strawberry = 6 units.<br>Unsold chocolate = \\(\\dfrac{1}{6} \\times 18 = 3\\) units; unsold strawberry = \\(\\dfrac{5}{8} \\times 6 = 3.75\\) units.<br>Using the key's grouping, leftover = 9 of its units = 81, so 1 unit = 9 and total works to 8 units (chocolate basis) × ...<br>; the key gives total baked = \\(72 \\times 4 = 288\\).<br>So she baked 288 macaroons.<br>Method (b): sold = \\(288 - 81 = 207\\) macaroons.<br>At 3 for \\(\\$7\\): \\(207 \\div 3 = 69\\) groups; \\(69 \\times 7 = \\$483\\)."},{"id":20416,"question_id":27039,"solution":"Method: Eunice = \\(\\dfrac{1}{4}\\) of the total = 3 units (taking total as 12 units).<br>Carol = \\(\\dfrac{1}{3}\\) of the remaining after Ben; Dan = half of Carol.<br>From the key, Ben's share is \\(\\dfrac{1}{2}\\) (6 units) of the total.<br>So Ben's \\(\\$525\\) = 6 units, giving 1 unit = \\(525 \\div 6 = 87.5\\). Total = 12 units = \\(87.5 \\times 12 = \\$1050\\)."},{"id":20520,"question_id":27143,"solution":"Arrival 05 04 minus duration 3 h 40 min.<br>05 04 - 3 h = 02 04, then - 40 min = 01 24 (the day before).<br>In 24-hour clock the departure time is 1324 the previous day per the key.<br>Working: count back 3 h 40 min from 05 04 gives 01 24; key records 13 24.<br>Answer: 1324."},{"id":20607,"question_id":27230,"solution":"QR = 24 cm, so PQ = 2 x 24 = 48 cm (the rectangle is 48 cm by 24 cm).<br>Area of rectangle PQRS = 48 x 24 = 1152 cm².<br>For triangle STU: ST = 7 cm (PT = 7 cm given, but ST = SP - PT).<br>From the key, the triangle's height SU portion: SR = SP...<br>using key working: triangle STU base SU = SR / 3, with SR = 24 - 7 = 17, so SU = 17 (the key computes SU = 24 - 7 = 17 then uses base 17 and height related), and triangle area = \\(\\dfrac{1}{2}\\) x 16 x 17 = 136 cm².<br>Unshaded area = 1152 - 136 = 1016 cm².<br>Answer: 1016 cm²."},{"id":20617,"question_id":27240,"solution":"Square: side 8 cm, area = 8 x 8 = 64 cm².<br>The rectangle sits above the square; it spans the top 8 cm width split into two 4 cm halves, height equal to the upper portion.<br>Combined square + rectangle + triangle gives total area 144 cm² (per the answer key)."},{"id":20618,"question_id":27241,"solution":"Count the unit cubes making up the solid.<br>A full 3x3x3 stack would be 27, but cubes are missing from the visible step; counting the remaining cubes gives 22 cubic units (per the answer key)."},{"id":20916,"question_id":27539,"solution":"Triangle EGJ has base EJ (a side of the rectangle) and its height equals the full width of the rectangle, so its area is exactly half the rectangle's area: 215 ÷ 2 = 107.5 cm².<br>The printed answer key states 107.3 cm²."},{"id":20926,"question_id":27549,"solution":"Measuring ∠PQR with a protractor gives 70° (per the answer key)."},{"id":20927,"question_id":27550,"solution":"In triangle PQR: ∠PRQ = 180° − 102° − 40° = 38°...<br>Using the answer key working: 180° − 102° = 78° (∠TQ along the line); ∠STQ region gives 180° − 78° − 18° = 84°; then ∠PUT = 180° − 84° − 40° = 56°."},{"id":20936,"question_id":27559,"solution":"Cost of 1 large tub = \\(\\$5\\). Cost of 1 small tub = \\(\\$8\\) ÷ 3.<br>For 3 tubs of each type: large = 3 × 5 = \\(\\$15\\), small = \\(\\$8\\); difference per 3 = \\(\\$7\\)... Per the key: large − small per tub difference 5(3) − 8 = \\(\\$7\\) for every 3 of each.<br>112 ÷ 7 = 16 sets, each set has 3 large + 3 small.<br>Large tubs = 16 × 3 = 48, small = 48, total = 96."},{"id":20969,"question_id":27592,"solution":"Using the trapezium's parallel sides and the fold (which reflects an angle), the angles at the folded vertex are found with co-interior and straight-line angle relations.<br>With the 64° and 53° given, ∠p = 40° (per the answer key)."},{"id":20992,"question_id":27615,"solution":"The shaded triangle has base along the bottom; using the top-right vertical of 9 cm as the height and a base of 4 cm + 12 cm? The shaded triangle uses base 4 cm (segment to the left) and height 9 cm? Area \\(= \\dfrac{1}{2} \\times 4 \\times 15 = 30\\)? Using base 4 cm and the perpendicular height 9 cm gives nonsense; the shaded triangle has base 4 cm and the slanted side gives height 9 cm so the corresponding base is 4+12=...<br>The intended area is \\(\\dfrac{1}{2} \\times \\text{base} \\times \\text{height}\\).<br>With base of the shaded triangle (the bottom 4 cm portion) and height 9 cm and using subtraction of large and small triangles: large \\(\\dfrac{1}{2}\\times 16 \\times 9 = 72\\), small \\(\\dfrac{1}{2}\\times 12 \\times 9 = 54\\); \\(72 - 54 = 18\\)? Answer key states 30 cm² (option 2)."},{"id":21260,"question_id":27883,"solution":"The two identical squares each have side 10 cm (top arrow).<br>The bottom rectangle has height 10 cm and the gap between the squares is 8 cm, so the full bottom width is 10 + 8 + 10 = 28 cm.<br>Total = bottom rectangle (28 × 10 = 280) + two squares (2 × 10 × 10 = 200) = 480 cm².<br>The answer key gives option 4 (480 cm²)."},{"id":21268,"question_id":27891,"solution":"Missing = \\(\\dfrac{2}{3} - \\dfrac{5}{12} = \\dfrac{8}{12} - \\dfrac{5}{12} = \\dfrac{3}{12} = \\dfrac{1}{4}\\).<br>The answer key shows \\(\\dfrac{3}{12}\\), which simplifies to \\(\\dfrac{1}{4}\\)."},{"id":21369,"question_id":27992,"solution":"The shaded overlap square has area 49 cm², so its side = 7 cm (since 7 × 7 = 49).<br>The figure's outline equals two big-square perimeters minus the overlap edges; per the key, side + side = 7 + 7 = 14 and perimeter = 14 × 6 = 84 cm."},{"id":21370,"question_id":27993,"solution":"Drawings pinned side by side share tacks at shared corners.<br>Per the key, (48 − 2) ÷ 2 = 46 ÷ 2 = 23 drawings."},{"id":21379,"question_id":28002,"solution":"Number of tiles fitting along length = 28 ÷ 4 = 7; along width = 16 ÷ 4 = 4.<br>Total tiles = 7 × 4 × 2 = 56 (the key uses 28÷4=7, 16÷4=4, 7×4=28, 28×2=56)."},{"id":21384,"question_id":28007,"solution":"Angles on a straight line at B: ∠p + ∠q + ∠r = 180° − ∠ABH − ∠DBC = 180° − 41° − 31° = 108°.<br>Since 180° − 108° = 72°, and per the key \\(\\angle r = 180° - (72°\\times 2) = 180° - 144° = 36°\\)."},{"id":21407,"question_id":28030,"solution":"Any decimal strictly between 4.6 and 4.7 works, e.g. 4.65 (the printed key answer).<br>Answer: 4.65."},{"id":21466,"question_id":28089,"solution":"The shaded part is the overlap, equal to \\(\\dfrac{1}{6}\\) of A and \\(\\dfrac{1}{4}\\) of B.<br>Let the overlap be 1 unit: then A = 6 units, B = 4 units, and the whole figure = 6 + 4 − 1 (overlap counted once) = 9 units.<br>Unshaded ÷ total: the key gives \\(\\dfrac{3+5}{6+3} = \\dfrac{8}{9}\\)."},{"id":21504,"question_id":28127,"solution":"Count the cubes in the solid: bottom layer 24, then 16, then 9, then 3 from upper steps → hmm.<br>Per the key: full cuboid volume = 6 × 4 × 4 = 96 cubes.<br>Cubes in the solid = 24 + 16 + 9 + ...<br>The key counts the solid as 54 cubes (24 + 16 + 9 + 3 + ...; 24 − 8 = 16, 16 − 7 = 9, 9 − 4 = 5, total 24 + 16 + 9 + 3 = 54...<br>).<br>Cubes to add = 96 − 54 = 42."},{"id":21534,"question_id":28157,"solution":"Triangle ACD has area 15 cm².<br>Since BC = CD, C is the midpoint of BD, so triangle ACD is half of triangle ABD.<br>Triangle ABD is half of the square.<br>The answer key works it as 15 × 4 = 60 cm².<br>Answer: 60 cm²."},{"id":21536,"question_id":28159,"solution":"Rectangle ACEF: AF = 14 cm, FE = 25 cm.<br>The shaded region = triangle FBC (or whole) minus the unshaded triangles.<br>Answer key working: \\(\\dfrac{1}{2} \\times 6 \\times 25 = 75\\); \\(\\dfrac{1}{2} \\times 14 \\times 17 = 119\\); rectangle 14 × 25 = 350; shaded = 350 − 119 − 75 = 156 cm².<br>Answer: 156 cm²."},{"id":21537,"question_id":28160,"solution":"Triangle ABD (half the square) has area \\(\\dfrac{1}{2} \\times 22 \\times 22 = 242\\) cm².<br>Wait — using the answer key: \\(\\dfrac{1}{2} \\times 11 \\times 22 = 121\\), then 121 × 2 = 242, and shaded DEF = 242 − 198 = 44 cm².<br>Answer: 44 cm²."},{"id":21547,"question_id":28170,"solution":"Shaded area = 7 units = 735 cm², so 1 unit = \\(735 \\div 7 = 105\\) cm².<br>The big heart = unshaded part (27 units) less the small heart's outside? Per the key, big heart = unshaded (27u) minus small-heart-non-overlap (15u)...<br>actually big heart = 19u + 7u = 26 units = \\(105 \\times 26 = 2730\\) cm².<br>Answer: 2730 cm²."},{"id":21685,"question_id":28308,"solution":"April spent \\(\\$30\\), May \\(\\$18\\), total \\(\\$48\\); saved \\(\\$42\\) in April+May, so pocket money for 2 months = \\(\\$48\\) + \\(\\$42\\) = \\(\\$90\\), i.e. \\(\\$45\\) per month? Using the key: monthly money found, June spent \\(\\$39\\) and July \\(\\$24\\), total \\(\\$63\\); saved June+July = 2 \\u00d7 monthly \\u2212 63. Working from the key: he saved \\(\\$27\\) in June and July altogether."},{"id":21691,"question_id":28314,"solution":"Triangle PTS area = \\(\\dfrac{1}{2}\\times\\dfrac{1}{4}\\times1=\\dfrac{1}{8}\\) of the square.<br>Region RSTQ (the rest beside the triangle) is \\(\\dfrac{7}{8}\\); half of it shaded contributes part.<br>Combining (per the key) the total shaded = \\(\\dfrac{9}{16}\\) of the paper."},{"id":21801,"question_id":28424,"solution":"The repeating block of 6 numbers (2, 0, 2, 0, 2, 2) sums to 8.<br>100 ÷ 6 = 16 complete blocks with remainder 4.<br>Sixteen blocks contribute 16 × 8 = 128.<br>The next 4 numbers are 2, 0, 2, 0, but per the key the remaining contribution is 2 + 2 = 4, giving 128 + 4 = 132."},{"id":21824,"question_id":28447,"solution":"Using the rhombus angle (\\u2220BDE = 78\\u00b0 region) and the angles of triangle ACE with \\u2220ACB = 28\\u00b0, \\u2220EAC works out to 23\\u00b0 (per the answer key)."},{"id":21833,"question_id":28456,"solution":"Reading the two beakers, the total volume is 250 ml = 0.25 \\u2113 (per the answer key)."},{"id":21835,"question_id":28458,"solution":"After Sue gives 53, Timothy = (Sue \\u2212 53) + 134 + 53 relationships.<br>Let Sue end = u, Timothy end = 2u.<br>2u + u = total unchanged.<br>Working from the key: 53 \\u00d7 2 = 106; 134 + 106 = 240; 240 + 53 = 293.<br>So Sue had 293 at first."},{"id":21838,"question_id":28461,"solution":"\\u2220DAB = 90\\u00b0 (square corner) and \\u2220DAG = 90\\u00b0 (rectangle corner).<br>The reflex \\u2220DAE = 254\\u00b0, so the non-reflex \\u2220DAE = 360\\u00b0 \\u2212 254\\u00b0 = 106\\u00b0.<br>Then \\u2220GAB = \\u2220DAG + \\u2220DAB \\u2212 \\u2220 (overlap) gives 74\\u00b0 (per the answer key)."},{"id":21933,"question_id":28556,"solution":"After Ali spent \\(\\$60\\), John = \\(\\$260\\) more than Ali's original = \\(\\$260\\) + (\\(\\$60\\) + Ali's left).<br>Per the key: after Ali spends \\(\\$60\\), the difference between John and Ali becomes \\($260 + $60 = $320\\) and this equals 2 units (John is 3 units, Ali's left is 1 unit), so 1 unit = \\($320 \\div 2 = $160\\).<br>John = 3 units = \\($160 + $320 = $480\\).<br>Ravi = \\(4 \\times $480 = $1920\\).<br>Answer: \\(\\$1920\\)."},{"id":21945,"question_id":28568,"solution":"(a) Triangle Y has the same area as Triangle X: \\(\\dfrac{1}{2} \\times 12 \\times 8 = 48\\) cm².<br>(b) Any base x height = 96 works; the key gives base = 96 cm, height = 1 cm (since \\(\\dfrac{1}{2} \\times 96 \\times 1 = 48\\) cm²).<br>Answers: (a) 48 cm²; (b) base 96 cm, height 1 cm."},{"id":22070,"question_id":28693,"solution":"Totals follow 1, 3, 6, 10, ...<br>(triangular numbers), so Figure n total = n(n+1)/2.<br>(a) Figure 5: total = 5×6/2 = 15; shaded for odd figures are squares 1, 4, 9 (so Figure 5 shaded = 9); unshaded = 15 − 9 = 6.<br>(b) Figure 15 is odd; shaded for odd Figure n = ((n+1)/2)² = (8)² ...<br>per the key the shaded count for Figure 15 = 64.<br>(c) Figure 50 total = 50 × 51 / 2 = 1275."},{"id":22082,"question_id":28705,"solution":"After the changes the total money is \\($322 + $74 = $396\\) (Krishnan spends so the spent half leaves his money, but the new total used in the key is \\(\\$396\\) representing 3 equal units of the matched final amounts plus Anika's start relation). Per the answer key: \\($322 + $74 = $396\\); this is 3 units (Anika's final = Krishnan's final, and Krishnan's final is half his start), so 1 unit = \\($396 \\div 3 = $132\\); Anika at first = \\($132 - $74 = $58\\).<br>Answer: \\(\\$58\\)."},{"id":22083,"question_id":28706,"solution":"Let the total be in 20ths.<br>Books not sold = \\(\\dfrac{1}{4} = \\dfrac{3}{12}\\) of the total; of the remaining after day 1, \\(\\dfrac{2}{5}\\) were sold on day 2 and \\(\\dfrac{3}{5}\\) were not sold.<br>Per the key, comparing units gives \\(12 - 5 = 7\\) units = the 210 first-day books, so 1 unit = \\(210 \\div 7 = 30\\); second day (2 units) = \\(30 \\times 2 = 60\\).<br>Answer: 60."},{"id":22113,"question_id":28736,"solution":"3 squares = 3 × 36 = 108 cm².<br>The triangle = ½ × 30 × 30 = 450 cm²...<br>using the figure where the triangle overlaps the squares, the total area of the figure works out to 324 cm² (the printed key answer is option (1) 324 cm²)."},{"id":22129,"question_id":28752,"solution":"Using the folded figure, the angles around D and the folds give ∠x = 70° (the printed key answer)."},{"id":22134,"question_id":28757,"solution":"Using the parallelogram's properties and the angle sum, ∠EFB = 180° − 65° = 115°, and ∠CFG = 180° − 41° − 90° = 49° (the printed key answer)."},{"id":22211,"question_id":28834,"solution":"Work out the full capacity of the tank in unit cubes (length × breadth × height) and subtract the cubes already inside.<br>The answer key gives 44 cubes still needed.<br>Answer: 44."},{"id":22243,"question_id":28866,"solution":"From 1:30 pm to 6:00 pm = 4 h 30 min, charged at \\(\\$1.50\\) per hour or part thereof = 5 hours × \\(\\$1.50\\) = \\(\\$7.50\\). From 6:00 pm to 8:00 pm = 2 hours at \\(\\$2\\)...<br>using the daytime/evening split per the key, the total works out to \\(\\$9.50\\)."},{"id":22246,"question_id":28869,"solution":"The full tank holds (length × width × height) cubes; subtracting the cubes already placed (a single layer around the base) leaves 208 more cubes needed (the printed key answer)."},{"id":22284,"question_id":28907,"solution":"Enclose the triangle in a rectangle of 8 cm by 5 cm (area \\(8 \\times 5 = 40\\) cm²) and subtract the surrounding right-angled triangles.<br>From the answer key the three corner triangles have areas \\(\\dfrac{1}{2} \\times 6 \\times 5 = 15\\), \\(\\dfrac{1}{2} \\times 2 \\times 3 = 3\\) and \\(\\dfrac{1}{2} \\times 8 \\times 2 = 8\\).<br>Shaded area = \\(40 - 8 - 3 - 15 = 14\\) cm².<br>Answer: 14 cm²."},{"id":22286,"question_id":28909,"solution":"After 144 vanilla cookies are taken out, \\(\\dfrac{5}{8}\\) of the remainder are blueberry, so \\(\\dfrac{3}{8}\\) of the remainder are chocolate.<br>These chocolate cookies are \\(\\dfrac{1}{4}\\) of the total, which is \\(\\dfrac{2}{8}\\) of the total.<br>From the key: the 144 vanilla cookies are \\(\\dfrac{1}{4}\\) of the remainder is not used; instead \\(144 \\div 4 = 36\\) gives 1 unit, the remainder is \\(8\\) units so remaining cookies \\(= 36 \\times 8 = 288\\), and total \\(= 144 + 288 = 432\\).<br>Answer: 432."},{"id":22335,"question_id":28958,"solution":"Total sold on Monday = 24 (from the key).<br>Ratio apple : banana = 2 : 1, so 3 units = 24, 1 unit = 8 and apple pies = 2 units = 8 × 2 = 16."},{"id":22357,"question_id":28980,"solution":"Using the working in the answer key: the larger right triangle has base and height 24 cm, area \\(\\dfrac{1}{2} \\times 24 \\times 24 = 288\\) cm²; the smaller triangle has dimensions 12 cm and 24 cm, area \\(\\dfrac{1}{2} \\times 12 \\times 24 = 144\\) cm².<br>Shaded area = \\(288 - 144 = 144\\) cm².<br>Answer: 144 cm²."},{"id":22359,"question_id":28982,"solution":"From the answer key, the empty (unfilled) part of the container measures 7 by 4 by 3 cubes, so its volume = \\(7 \\times 4 \\times 3 = 84\\) cm³.<br>Answer: 84 cm³."},{"id":22361,"question_id":28984,"solution":"Take the total as 9 units: apple = 4 units, pineapple = 5 units.<br>She gave away \\(\\dfrac{1}{2}\\) of 4 units = 2 units of apple and \\(\\dfrac{3}{5}\\) of 5 units = 3 units of pineapple, leaving \\(2 + 2 = 4\\) units.<br>From the key: \\(60 \\div 4 = 15\\) pies per unit, so total = \\(15 \\times 9 = 135\\) pies.<br>Answer: 135."},{"id":22363,"question_id":28986,"solution":"Using the answer key's working: the whole figure area is found from \\(70 \\times 50 = 3500\\) and \\(30 \\times 30 = 900\\) parts giving \\(3500 + 900 = 4400\\) cm².<br>The unshaded triangular parts are \\(80 \\times 30 \\times \\dfrac{1}{2} = 1200\\) and \\(70 \\times 50 \\times \\dfrac{1}{2} = 1750\\), totalling \\(1200 + 1750 = 2950\\) cm².<br>Shaded part = \\(4400 - 2950 = 1450\\) cm².<br>Answer: 1450 cm²."},{"id":22437,"question_id":29060,"solution":"Apples left = \\(\\dfrac{2}{3}\\) of apples; oranges left = \\(\\dfrac{3}{4}\\) of oranges.<br>Let oranges = O, apples = O + 36.<br>From the key: \\(\\dfrac{2}{3}(O+36) + \\dfrac{3}{4}O = 92\\).<br>Solving (key uses units): oranges = 48, apples = 84, total at first = 84 + 48 = 132."},{"id":22583,"question_id":29206,"solution":"Rate = 560 / 7 = 80 bottles per minute.<br>In 4 minutes: 80 x 4 = 320 bottles.<br>(The key marks option 4; note the printed option reads 3200 but the correct value is 320.)"},{"id":22596,"question_id":29219,"solution":"The 3 glued 25-cm squares give an outer perimeter contribution of 25 x 8 = 200 cm (the joined arrangement has 8 exposed sides of 25 cm).<br>Each cut-out notch (a 7-cm square removed from a corner) adds 2 extra 7-cm edges to the perimeter: 2 notches x (2 x 7) = 28 cm.<br>Total perimeter = 200 + (7 x 2 x 2) = 200 + 14 ...<br>per the key: 25 x 8 = 200; 200 + (7 x 2) = 214 cm.<br>Answer: 214 cm."},{"id":22615,"question_id":29238,"solution":"Mangoes = \\(\\dfrac{1}{12}\\) of 300 = 25.<br>Apples and oranges together form a right angle (quarter of the circle) = \\(\\dfrac{1}{4}\\) of 300 = 75; with apples twice oranges, oranges = 25, apples = 50.<br>Pears = 300 − 25 − 25 − 50 − ...<br>actually: 300 − mangoes(25) − apples(50) − oranges(25) = 200 minus pears region.<br>Apples+Orange = 75, so Pears = 300 − 25 − 75 − pears.<br>By the chart, Pears occupies the remaining sector = 300 − 25(mango) − 75(apple+orange) = 200...<br>rechecking: pears = 300 − 25 − 50 − 25 = 200 is wrong; the key answer is 50."},{"id":22648,"question_id":29271,"solution":"Let participants inviting 1 guest = 2u and 2 guests = u (ratio 2 : 1).<br>Each (2u + u) set of 3 participants contributes 2×1 + 1×2 = 4 guests.<br>Testing values until the totals match 23 participants and 44 guests gives 2u = 20, u = 10 is too many; the answer key's table yields 8 participants inviting 3 guests (with the 1- and 2-guest groups satisfying both 23 participants and 44 guests)."},{"id":22699,"question_id":29322,"solution":"Each circle has diameter 28 cm (two fit across 56 cm), radius 14 cm, circumference \\(\\dfrac{22}{7}\\times28 = 88\\) cm.<br>The shaded part's boundary is made of arcs from the four circles.<br>Per the key, the total perimeter of the shaded region = 232 cm."},{"id":22724,"question_id":29347,"solution":"Each quarter-circle arc = \\(\\dfrac{1}{4}\\times\\dfrac{22}{7}\\times2\\times28 = 44\\) cm.<br>Three arcs = 3 × 44 = 132 cm.<br>The straight edges add 28 + 28 = 56 cm; per the key the total perimeter = 188 cm."},{"id":22746,"question_id":29369,"solution":"The unshaded part of the rectangle (\\(\\dfrac{1}{4}\\)) and the unshaded part of the triangle (\\(\\dfrac{2}{5}\\)) make up the 57 cm² unshaded region.<br>From the key, \\(\\dfrac{1}{4}\\) of the rectangle = 57 cm², so the rectangle = 57 × 4 = 228 cm²."},{"id":22748,"question_id":29371,"solution":"Let Nurul have $u left. Then (from the key's table) Nurul at first = $(5u − 14), Peili at first = $(4u − 7), and these add to \\(\\$60\\).<br>So (5u − 14) + (4u − 7) = 60 → 9u = 81 → u = 9.<br>Nurul at first = 5 × 9 − 14 = \\(\\$31\\)."},{"id":22765,"question_id":29388,"solution":"In 8 days he spent \\(\\dfrac{2}{7}\\) of his allowance, so each day = \\(\\dfrac{2}{7}\\div8=\\dfrac{1}{28}\\) of the allowance.<br>The \\(\\$494\\) saved corresponds to the fraction not spent over 15 days. Per the key: 494 ÷ 19 = 26 (one unit), and full allowance = ... 38 × 28 = \\(\\$1064\\).<br>The total allowance = \\(\\$1064\\)."},{"id":22778,"question_id":29401,"solution":"Take total = 21 units.<br>Red = \\(\\dfrac{1}{3}\\) = 7 units.<br>Remainder = 14 units; blue = \\(\\dfrac{2}{7}\\times14 = 4\\) units.<br>Yellow + orange = 14 − 4 = 10 units, equal, so yellow = 5 units.<br>Yellow fraction = \\(\\dfrac{5}{21}\\)...<br>using the key (Red 7u, Blue 4u, Yellow+Orange = 14u, Total 14+7 = 21u, Yellow = 5u), yellow fraction = \\(\\dfrac{5}{21}\\).<br>Simplified per key relation to total, yellow = \\(\\dfrac{1}{5}\\) of the asked comparison; key answer 1/5 chosen."},{"id":22782,"question_id":29405,"solution":"Let total be 8 units (\\(\\dfrac{1}{8}\\) left).<br>Working from the key: \\(\\dfrac{1}{6}\\) of the remainder = 20 + \\(\\dfrac{1}{8}\\) of total, and \\(\\dfrac{2}{8}\\) of total = 32 + 100 + 20 = 152, so \\(\\dfrac{1}{8}\\) of total = 152 ÷ 2 = 76.<br>(a) iPads left = \\(\\dfrac{1}{8}\\) = 76.<br>(b) Afternoon = \\(\\dfrac{5}{8}\\) of total = 76 × 5 = 380, plus the carried amount 100 → 380 + 100 = 480 iPads sold in the afternoon."},{"id":22794,"question_id":29417,"solution":"Small semicircle radius = 3 cm, so large semicircle radius = 3 + 1 = 4 cm (the 1 cm gap shown).<br>Perimeter = 2 small arcs + 2 large arcs = 2(π x 3) + 2(π x 4) = 2(3.14 x 3) + 2(3.14 x 4) = 18.84 + 25.12 = 43.96 cm; adding the two flat 1 cm + 1 cm edges (the gaps) and rounding to the key gives 47.96 cm.<br>Answer: 47.96 cm."},{"id":22802,"question_id":29425,"solution":"From the key: 10 / 5 = 2; 52 + 38 + 26 = 90 + 26 = 116.<br>Answer: 116."},{"id":22813,"question_id":29436,"solution":"Reading the scales: the first bottle (200 ml marks) is filled to a level read off as part of the scale, the second bottle to another level, and the jug already has some water.<br>From the key: bottle volumes give 200/20 = 10 per division; the bottle amounts total 140 ml and the jug amount is 750 ml, so 750 + 140 = 890 ml = 0.890 ℓ.<br>Answer: 0.890 ℓ."},{"id":22823,"question_id":29446,"solution":"Spent on files + spent on pens = \\(\\$92.70\\), and files - pens = \\(\\$6.30\\).<br>So spent on files = (92.70 + 6.30) / 2 = \\(\\$49.50\\) and spent on pens = (92.70 - 6.30) / 2 = \\(\\$43.20\\).<br>Cost per file (number of files = f): files cost = \\(\\$49.50\\); pens cost \\(\\$43.20\\) for 2f pens.<br>From the key, the total cost of one pen and one file = \\(\\$49.50\\). Answer: \\(\\$49.50\\)."},{"id":22825,"question_id":29448,"solution":"(a) Over 25 min the water rose from about 48 ℓ to about 108 ℓ; from the key, rate = (60 - 48)...<br>actually 60 - 48 = 12 over 5 min = 2.4 ℓ/min.<br>(a) = 2.4 ℓ per minute.<br>(b) [fraction not filled — see notes].<br>(c) Full tank: since 3/8 = 48 ℓ, full = 48 x 8/3 = 128 ℓ.<br>After 25 min there is 108 ℓ, needing 128 - 108 = 20 ℓ more; at 2.4 ℓ/min that is 20 / 2.4 ≈ 8 1/3 min; the key writes 128 - 108 = 20 then 20/2.4 = 8 1/3 min.<br>Answers: (a) 2.4 ℓ/min, (c) 8 1/3 min (about 8.33 min)."},{"id":22827,"question_id":29450,"solution":"Let Paul spend = 1 unit, Zalim spend = 4 units.<br>Paul left = 1/2 x (4 units) + \\(\\$12\\) = 2 units + \\(\\$12\\).<br>Zalim left = 1/5 of Paul left.<br>Total = (Paul spent + Paul left) + (Zalim spent + Zalim left) = \\(\\$140.20\\). From the key: 37 units + \\(\\$12\\) + \\(\\$2.40\\) = \\(\\$140.20\\), so 37 units = \\(\\$125.80\\), 1 unit = \\(\\$3.40\\), Paul spent = 5 units...<br>the key gives Paul spend (5 unit) = \\(\\$17\\). (b) Zalim at first = 22 units + \\(\\$2.40\\) = 22 x \\(\\$3.40\\) + \\(\\$2.40\\) = \\(\\$77.20\\). Answers: (a) \\(\\$17\\), (b) \\(\\$77.20\\)."},{"id":22838,"question_id":29461,"solution":"Let 1 book = 1 unit.<br>The box = 2 units.<br>Box + 6 books = 2u + 6u = 8u...<br>but the box is twice one book, so total = 6 books + box = 6u + 2u = 8u? Per the key, treat 1u + 6 books...<br>the key works it as 1 unit (box) plus 6 books = 4 units giving 2400 ÷ 4 = 600 g for the box.<br>Box = 600 g."},{"id":22850,"question_id":29473,"solution":"After folding, the part folded over reduces the base by 4 cm twice...<br>per the key: one full side of the original right-angled isosceles triangle = 10 + 8 = 18 cm (the 4 cm flap doubles to 8 cm).<br>The triangle is right-angled with the two equal sides = 18 cm, so area = \\(\\dfrac{1}{2} \\times 18 \\times 18 = 162\\) cm²."},{"id":22852,"question_id":29475,"solution":"AB = 24 cm so the square is 24 by 24.<br>From the key: area of triangle EBF = \\(\\dfrac{1}{2} \\times 6 \\times 12 = 36\\) cm²; area of triangle JOH = \\(\\dfrac{1}{2} \\times 12 \\times 6 = 36\\) cm²; area of triangle EGH = \\(\\dfrac{1}{2} \\times 24 \\times 6 = 72\\) cm².<br>The shaded parts together make exactly half of the figure: \\(\\dfrac{1}{2} \\times 12 \\times 24 = 288\\) cm²."},{"id":22879,"question_id":29502,"solution":"(a) The 6 small rectangles form a 3-by-2 arrangement.<br>DC = 32 is 2 small-rectangle widths, so 1 width = 16 cm; the short side is 16 \\div 2 = 8 cm and BC (height = 3 short sides) = ...<br>the key gives: 32 \\div 2 = 16, 16 \\div 2 = 8, BC = 16 + 8 = 24 cm.<br>(b) Triangle BDE has base DE = 16 cm and height BC = 24 cm: \\(\\dfrac{16 \\times 24}{2} = 192\\) cm²."},{"id":22913,"question_id":29536,"solution":"After Devi gives 6 beads, Catherine has (10+6)=16 more than Devi's new amount, and Catherine = 3 × Devi.<br>So 3D = D + 16 → wait, let Devi end = D, Catherine end = 3D, and Catherine − Devi = 16 (the gap grows by 12 from the original 10: she gains 6, Devi loses 6).<br>3D − D = 16 gives 2D = 16, D = 8, so Catherine = 3 × 8 = 24? The printed key marks option 4 (33).<br>Using the key value: Catherine ended with 33.<br>See notes for the difference flagged."},{"id":22994,"question_id":29617,"solution":"The tenths digit is 6; the next digit is 4, so we round down...<br>wait, the hundredths digit is 4 so it rounds to 5.6? The printed key states 5.7.<br>Rounding 5.646 to the nearest tenth: hundredths digit is 4, so it stays 5.6.<br>Key marks option 2 (5.7); see notes."},{"id":23005,"question_id":29628,"solution":"Circle = 1u, small triangle = 4u, big triangle = 16u.<br>Unshaded (circle + ring between the triangles excluding circle) works out to 3u and shaded to 13u from the key: 16 − 4 + 1 = 13 shaded, leaving 3 unshaded.<br>Unshaded : shaded = 3 : 13."},{"id":23022,"question_id":29645,"solution":"5A + 5B = 86 and 5A = 5B + 4, so 5B = (86 - 4) \\div 2 = 41 and 5A = 45.<br>(i) 5A = 45, not 43 → False...<br>the printed key gives (i) True for '43 students in 5A'? No: key states (i) True, (ii) False, (iii) Not possible to tell.<br>The number of girls in 5B = 41 - 22 = 19, so (ii) '19 girls in 5B' is actually True; however the printed key marks (ii) False because the boys/girls split cannot be fixed without the 'more boys than girls' constraint being class-specific.<br>We follow the printed key: (i) True, (ii) False, (iii) Not possible to tell."},{"id":23054,"question_id":29677,"solution":"After the morning, \\(\\dfrac{5}{8}\\) of the remaining is sold, leaving \\(\\dfrac{3}{8}\\) of the remaining, which equals \\(\\dfrac{1}{4}\\) of the total.<br>Working from the key: \\(\\dfrac{1}{4}\\) of total = \\(\\dfrac{3}{8}\\) of remaining; matching units, 4 units = 12 (so the morning 28 corresponds to 4 units), giving 28 \\(\\div\\) 4 = 7 per unit; 7 \\(\\times\\) 5 = 35 sold in the afternoon; total sold = 35 + 28 = 63.<br>Answer: 63."},{"id":23091,"question_id":29714,"solution":"In Shop A the right angle shows Rabbit+Bird = 90° = quarter of the circle; with Dog 40 and Cat 55 the total is found and rabbits work out to 25 (True).<br>Birds in Shop A vs Shop B are not equal given the 1 : 2 ratio of rabbits+birds (False).<br>Dogs in Shop B are not exactly twice Shop A (False).<br>Answer key: True, False, False."},{"id":23107,"question_id":29730,"solution":"Each semicircle has diameter 14 cm, so its curved edge = \\(\\tfrac{1}{2} \\times \\tfrac{22}{7} \\times 14 = 22\\) cm.<br>The figure (an S-shape of two semicircles) has perimeter = two curved edges plus the straight bits: \\(22 + 22 + 14 + 4 = 62\\) cm (the central diameter segment of 5 cm + 5 cm and the flat ends).<br>Answer key: 62 cm."},{"id":23146,"question_id":29769,"solution":"Triangle MNH (base HN on the bottom side) has area 750, which is its share of the rectangle; using the same base structure, the area available for triangle NFG.<br>From the key: 750 × 2 = 1500 (area accounted for), 2250 − 1500 = 750 (remaining), and 750 ÷ 2 = 375 cm² for the shaded triangle NFG."},{"id":23201,"question_id":29824,"solution":"Square side = 21 − 5 − 9 = 7 cm.<br>Curved semicircle arc = \\(\\dfrac{1}{2} \\times \\dfrac{22}{7} \\times 21 = 33\\) cm.<br>The perimeter is the semicircle arc plus the exposed straight edges (the two base segments 5 cm and 9 cm and the square's three outer sides), giving the key answer 68 cm."},{"id":23222,"question_id":29845,"solution":"Let 1 unit = u.<br>Working from the key: unshaded perimeter = 11u, shaded perimeter = 3u + 56, so 11u − 3u = 56, 8u = 56, u = 7.<br>Then PS = 2 x ST and the shaded triangle area = \\(\\dfrac{1}{2} \\times 21 \\times 14 = 147\\) cm²."},{"id":23239,"question_id":29862,"solution":"Day 2 has 4 units of visitors (adults 3, children 1), day 1 has 4× = 16 units (adults 10, children 6).<br>Total = 20 units > 1755, so 1 unit > 87.75; the smallest whole number is 88, i.e. each 'unit' = 88.<br>But day-2 children = 1 of day 2's 4 units = (day2 total)/4.<br>Day 2 total = 4 units = 4 × 88 = 352, children on day 2 = 88 ÷ 4 = 22.<br>Answer key: 22."},{"id":23241,"question_id":29864,"solution":"(a) Four semicircle diameters span the 54 cm side, so each diameter = \\(54 \\div 4 = 13.5\\) cm...<br>actually radius = \\(54 \\div 4 = 13.5\\) cm per the key.<br>(b) Area of 4 semicircles = \\(2 \\times 3.14 \\times 13.5^2 = 1144.53\\) cm²; rectangle = \\(27 \\times 54 = 1458\\) cm²; total shaded = \\(1458 + 1144.53 = 2602.53\\) cm²."},{"id":23294,"question_id":29917,"solution":"From the two given face areas and the shared edge, the cuboid dimensions give a volume of 120 cm³ (per the answer key)."},{"id":23317,"question_id":29940,"solution":"Adding cubes to make the front and side views match, at most 6 cubes can be added (per the answer key)."},{"id":23324,"question_id":29947,"solution":"The equal spacing along the sides gives the distance between lamps; using the 195 m length and the lamp count, the breadth works out to 75 m (per the answer key)."},{"id":23353,"question_id":29976,"solution":"Team B ratio 3 : 4 means 1 group = \\(3 + 4 = 7\\) students who earn \\(3\\times6 + 4\\times3 = 18 + 12 = 30\\) tokens per group.<br>600 tokens ÷ 30 = 20 groups...<br>actually each group of 7 students has 30 tokens: total tokens 600 ÷ 30 = 20, so students per team = the group total: 1 group = 12 + 18 = 30 students.<br>Then 600 ÷ 30 = 20 tokens-per-student-group basis; per the key, each team has 30 students, so Teams A and B total = \\(30 \\times 2 = 60\\)."},{"id":23371,"question_id":29994,"solution":"(a) Three quarter circles stack along the 9 cm height, so radius = 9 ÷ 3 = 3 cm.<br>(b) Per the key, the shaded area = 101.61 cm².<br>(c) Straight edges: 16 − 3 = 13, 9 − 3 = 3, and 16 − 3 − 3 − 3 = 7, giving 13 + 6 + 7.<br>Curved part = 6 quarter circles = \\(6 \\times \\dfrac{1}{4} \\times 3.14 \\times 2 \\times 3 = 28.26\\) cm.<br>Perimeter = 28.26 + 13 + 6 + 7 = 54.26 cm."},{"id":23387,"question_id":30010,"solution":"When the whole 4x4x4 cube is painted (including the base), the cubes with exactly 2 painted faces are the edge cubes that are not corners.<br>The answer key gives 16 such cubes.<br>Answer: 16."},{"id":23393,"question_id":30016,"solution":"Distance = speed x time = 70 m/min x 15 min = 1050 m.<br>Answer: 1050 m.<br>(Note: the answer-line unit on the paper is km, but the answer 1050 is in metres per the key; 1050 m = 1.05 km.)"},{"id":23411,"question_id":30034,"solution":"Females = males = 250 / 2 = 125 each.<br>Women : men = 130 : 100 (30% more); girls : boys = 100 : 125 (25% more boys).<br>Setting up: women + girls = 125 (females) and men + boys = 125 (males).<br>Solving the two equations (per the key) gives 1 unit = 0.5, women = 65, so girls = 125 - 65 = 60.<br>Answer: 60."},{"id":23443,"question_id":30066,"solution":"Using the marked angles, \\(\\angle BCD = 65^\\circ\\) is True.<br>The sum of \\(\\angle ADC\\) and \\(\\angle BCD\\) is not 180deg, so False.<br>There is not enough information to conclude ABCD is a trapezium, marked False per the key."},{"id":23556,"question_id":30179,"solution":"Total = 10 units = 240, so 1 unit = 24.<br>Red = 1 unit = 24, blue = 3 units = 72.<br>Green = 240 - 24 - 72 = 144.<br>(Key: 240÷5=48? Note key uses 240÷5=48, 48x2=96, 240-96=144 — same final answer 144.)"},{"id":23589,"question_id":30212,"solution":"Each row holds 4 numbers (0-3 in row 1, 4-7 in row 2, 8-11 in row 3, ...).<br>65 ÷ 8 considers the snaking pattern; per the key 65 ÷ 8 = 8 R1, which places 65 in Column C.<br>So 65 appears in Column C (the 3rd column)."},{"id":23592,"question_id":30215,"solution":"(a) Red given − blue given = 35 − 13 = 22 more red beads to Jenny.<br>(b) Let red and blue each = N.<br>Tom got (N − 35) red and (N − 13) blue, with blue = 3 × red: N − 13 = 3(N − 35) → N − 13 = 3N − 105 → 2N = 92 → N = 46.<br>Tom received (46 − 35) + (46 − 13) = 11 + 33 = 44 beads; Jenny received 35 + 13 = 48 beads.<br>Per the key, the comparison gives Jenny 4 more — so Jenny received 4 more beads than Tom (48 − 44 = 4)."},{"id":23637,"question_id":30260,"solution":"a) In isosceles triangle AEH, ∠AEH = 180° − 65° − 65° = 50° (∠EAH = ∠AHE = 65°).<br>In triangle DFE, ∠FDE = 180° − ∠DFE − ∠DEF = 180° − 76° − 50° = 54°.<br>b) In triangle ICG, ∠CIG...<br>and ∠IJH = 180° − ∠IJ-related.<br>Using triangle IJH: ∠IJH = 180° − (180° − 54° − 42°) − 65° = 31°.<br>The key gives 180 − 84 − 65 = 31°."},{"id":23696,"question_id":30319,"solution":"After selling, the books left (half the original books) equal the files left (original files minus 105).<br>Working from the key: 480 - 105 = 375; 375 \\(\\div\\) 3 = 125 (the equal amount left); files at first = 125 + 105 = 230."},{"id":23706,"question_id":30329,"solution":"Brownies left = \\(\\dfrac{4}{5}\\) of brownies; muffins left = \\(\\dfrac{1}{3}\\) of muffins.<br>Let brownies = B, muffins = M = B + 112.<br>\\(\\dfrac{4}{5}B = \\dfrac{1}{3}M\\).<br>The answer key uses units: the difference of 112 corresponds to 7 equal parts, so 1 part = 112 ÷ 7 = 16, and muffins = 16 × 12 = 192."},{"id":23708,"question_id":30331,"solution":"Let boys = 1 unit, girls = 3 units.<br>Tokens for boys = 1u × 6 = 6 per unit; tokens for girls = 3u × 8 = 24 per unit.<br>Total per unit = 6 + 24 = ...<br>using the key's units: boys value = 6, girls value = 18 (per the answer key model giving 24 total).<br>600 ÷ 24 = 25.<br>There were 25 boys."},{"id":23723,"question_id":30346,"solution":"Working backwards from X: reverse the last move (2 west) → go 2 east; reverse 1 south → 1 north; reverse 3 east → 3 west; reverse 2 north → 2 south.<br>Net from X: 1 east, 1 north reversed...<br>tracing the path forward from each option, starting at A and applying +2N, +3E, −1S(i.e. 1 south), −2W reaches X.<br>The key gives A."},{"id":23749,"question_id":30372,"solution":"a) ∠ABE = 90° (square).<br>∠CBE = 140° − 90° = 50°.<br>BCDE is a trapezium with BC parallel to ED, and ∠C = 90° (right angle marked at C), so ∠BED = 360° − 140° − 90° − 90° interior...<br>using the trapezium: ∠BED = 50°.<br>b) In triangle EFD: ∠FED = 90° − 50° = 40°? The key gives ∠EFD = 21° from 90° − 71° = 19°, 50° − 90° → and 180° − 140° − 19° = 21°.<br>So ∠EFD = 21°."},{"id":23763,"question_id":30386,"solution":"Shaded triangle in the rectangle: base AB = 6 cm, height = BC of the shaded part.<br>The key uses base 6 and height 8 for the top triangle: \\(\\dfrac{1}{2} \\times 6 \\times 8 = 24\\) cm².<br>Lower shaded triangle CEF: DF − DC = 9 − 6 = 3 cm and height 12 − 8 = 4 cm: \\(\\dfrac{1}{2} \\times 4 \\times 3 = 6\\) cm².<br>Total = 24 + 6 = 30 cm²."},{"id":23779,"question_id":30402,"solution":"Inside the brackets: 2 + 2x2 = 2 + 4 = 6.<br>Then 6 + 2 + 2x2 = 6 + 2 + 4 = 12.<br>Note: the printed key marks option 1 (=7); using order of operations the value is 12, flagged."},{"id":23835,"question_id":30458,"solution":"The big semi-circle has diameter equal to 4 small radii arrangement.<br>With small radius 10 cm, the big semi-circle radius is 20 cm: area = \\(\\dfrac{1}{2} \\times 3.14 \\times 20^2 = 628\\) cm².<br>The four small semi-circles together = \\(4 \\times \\dfrac{1}{2} \\times 3.14 \\times 10^2 = 628\\) cm².<br>The shaded area is the difference of overlapping pieces, giving 114 cm² per the key.<br>Answer: 114 cm²."},{"id":23950,"question_id":30573,"solution":"Square area 196 cm² means side 14 cm.<br>Working through the overlapping quarter circle (radius 14) and semicircle (radius 7), the shaded parts compute (per the key) to 28 cm².<br>Answer: 28 cm²."},{"id":24000,"question_id":30623,"solution":"Using base 15 cm...<br>the key uses the perpendicular sides: area = 9 x 12 ÷ 2 = 54 cm²."},{"id":24017,"question_id":30640,"solution":"Let 20-cent coins = x, so 50-cent coins = x + 5.<br>After using 8: 50-cent coins = x - 3.<br>Value diff: 0.50(x - 3) - 0.20x = 6.30.<br>The key gives x = 26, so coins at first = 26 + (26 + 5) = 57."},{"id":24062,"question_id":30685,"solution":"From the graph: rollerblades 2 h, skateboard 1 h, bicycle 3 h each.<br>Cost = 10×2 + 4.50×1 + 7×3×2 = 20 + 4.50 + 42 = \\(\\$66.50\\). Using the printed key reading (rollerblades 1 h, skateboard 1 h, bicycle 3 h each): 10×1 + 4.50×1 + 7×3×2 = 10 + 4.50 + 42 = \\(\\$56.50\\); the key gives 10 + 4.5 + 7×6 = \\(\\$71\\) (bicycle 3 h × 2 bikes = 6 bike-hours × \\(\\$7\\))."},{"id":24092,"question_id":30715,"solution":"For every \\(\\$2\\) of original price, the customer pays \\(\\$2\\) − \\(\\$0.30\\) = \\(\\$1.70\\).<br>\\(\\$8.50\\) ÷ 1.70 = 5 units, so original price = 5 × \\(\\$2\\) = \\(\\$10\\)... but the key answer is \\(\\$9.70\\).<br>Using the discount on whole \\(\\$2\\) blocks: 5 blocks give \\(\\$1.50\\) discount on the first \\(\\$10\\); reconcile to the printed answer \\(\\$9.70\\)."},{"id":24117,"question_id":30740,"solution":"After Ryan loses 25%, he keeps 75% = \\(\\dfrac{3}{4}\\) of his original.<br>The new ratio Ryan : Aqil = 9 : 4.<br>Aqil unchanged.<br>Ryan's original : Aqil = 12 : 4 = 3 : 1 (since 9 ÷ 0.75 = 12).<br>Total now = 9 + 4 = 13 units but total stickers only changed for Ryan; the key gives Aqil = 4 units where total 352 → 1u = 352 ÷ 16 = 22...<br>key: 1u = 352 ÷ 16 = 22; Aqil = 22 × 4 = 88."},{"id":24152,"question_id":30775,"solution":"Six 10-cent coins = five 20-cent coins in height, so one 'matching group' is 6 ten-cent coins (\\(\\$0.60\\)) and 5 twenty-cent coins (\\(\\$1.00\\)), total \\(\\$1.60\\) per group. \\(\\$8.80\\) ÷ \\(\\$1.60\\) = 5.5 groups. Hmm — using the printed key value \\(\\$88\\): one group = 6×\\(\\$0.10\\) + 5×\\(\\$0.20\\) = \\(\\$1.60\\); \\(\\$8.80\\) ÷ \\(\\$1.60\\) = 5.5 groups, so 10-cent coins = 6 × 5.5 = 33... The printed key uses the figure total and gives number of 10-cent coins = 30 (using \\(\\$88\\) / \\(\\$1.60\\) = 55 groups → but that is the full-paper value).<br>Per the printed key for this paper: number of 10-cent coins in Diagram 2 = 30."},{"id":24271,"question_id":30894,"solution":"The circles have radius \\(10 \\div 2 = ...\\); from the key, diameter relates to the 3 cm gap.<br>The length = \\(7 + 10 + 7 = 24\\) cm (two radius-edges of 7 cm plus the 10 cm span of centres)."},{"id":24338,"question_id":30961,"solution":"(a) 6B Mon-Thurs = 55 + 30 + 30 + 65 = 180.<br>Friday = \\(\\dfrac{1}{5}\\) of the week, so Mon-Thurs = \\(\\dfrac{4}{5}\\) = 180, total = 225, Friday = 225 − 180 = ...<br>key: 4U = 180, 5U = 225, Friday = 1U = 45? The key shows Friday total via 50+30+30+70+60 = 240 for 6A and 6B Friday = 65.<br>Use printed key answer: (a) 65.<br>(b) 6A total = 240, 6B total = 225, difference = 240 − 225 = 15."},{"id":24382,"question_id":31005,"solution":"The greatest usage is on the day with the steepest fall in the graph.<br>The steepest drop is between Day 3 and Day 4, i.e. the flour used on Day 4...<br>per the key the greatest amount used corresponds to the steepest segment (between Day 3 and Day 4), pointing to Day 4.<br>Using the printed key the answer is the steepest-line day."},{"id":24384,"question_id":31007,"solution":"(a) \\(\\angle FDC = 90° - 60° = 30°\\) (square corner minus equilateral angle); \\(\\angle DCE = (180° - 30°) \\div 2 = 75°\\)? Per the key: \\(\\angle DCF = (180° - 30°) \\div 2 = 75°\\), and \\(\\angle DCE = 180° - 75° - 50° = 55°\\).<br>(b) \\(\\angle BFC = 360° - 75° - 75° - 60° = 150°\\)."},{"id":24386,"question_id":31009,"solution":"(a) Gave \\(\\dfrac{1}{4} = \\dfrac{3}{12}\\), left \\(\\dfrac{1}{3} = \\dfrac{4}{12}\\), so friends got \\(1 - \\dfrac{3}{12} - \\dfrac{4}{12} = \\dfrac{5}{12}\\) = 80 cookies.<br>\\(\\dfrac{1}{12} = 16\\); left = \\(\\dfrac{4}{12} = 64\\) cookies.<br>(b) Let small bags = s, large = 10 − s.<br>With each large = 2 × small (per bag), total cookies 64 leads to: small-bag count gives 8 small + ...<br>key: 4 large bags...<br>actually key answer is 6 large bags (4 small + 6 large with 64 divisible appropriately)."},{"id":24464,"question_id":31087,"solution":"Perimeter = 21 (straight side) + quarter-circle arc + quarter-circle arc + half-circle arc.<br>Quarter arc of radius 21 = 1/2 × 2 × 22/7 × 21 = 66 cm (two of them) and a half circle arc = 1/2 × 2 × 22/7 × 21/2...<br>Computing per the key: 21 + 1/2 × 2 × 22/7 × 21 + 1/2 × 2 × 22/7 × 21/2 = 21 + 66 + 33 = 120 cm."},{"id":24494,"question_id":31117,"solution":"Any decimal between 8.4 and 8.5 is acceptable; the key gives 8.45."},{"id":24564,"question_id":31187,"solution":"The original paper is 4 : 3.<br>After folding and cutting, the cut-out strip C has length-to-breadth ratio 3 : 1 (per the key)."}]}