======================================================================
INDEX: 13668 ID: 24899 TYPE: 2
QUESTION: At a bakery, cupcakes are sold in boxes of 4 at \(\$2.10\) and tarts are sold in boxes of 3 at \(\$1.60\). Mrs Lee spent a total of \(\$203.60\) on some cupcakes and tarts at the bakery. She repacks them onto trays such that there are 3 cupcakes and 5 tarts on each tray for a party. How many tarts did she buy from the bakery? [?]
ANSWERS: ["240", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Per tray, cupcakes : tarts = 3 : 5. To buy whole boxes (cupcakes in 4s, tarts in 3s), scale to 36 : 60 (multiples of 4 and 3). One such set = 9 boxes of cupcakes + 20 boxes of tarts = \(\$2.10\)×9 + \(\$1.60\)×20 = \(\$18.90\) + \(\$32.00\) = \(\$50.90\). \(\$203.60\) ÷ \(\$50.90\) = 4 sets. Tarts = 4 × 60 = 240.
— Model method —
Each tray uses cupcakes : tarts \(=3:5\). To buy whole boxes (cupcakes in 4s, tarts in 3s), scale to \(36:60\): that is \(9\) boxes of cupcakes \(+\,20\) boxes of tarts \(=\$2.10\times9+\$1.60\times20=\$18.90+\$32.00=\$50.90\) per set. \(\$203.60\div\$50.90=4\) sets. Tarts \(=4\times60=240\).
[[model:p6/solutions/mgs-2023-p6-prelim-q48.png]]
======================================================================
======================================================================
INDEX: 13669 ID: 24900 TYPE: 1
QUESTION: Which of the following is seventy-five thousand and thirty in numerals?
ANSWERS: ["7530", "75 030", "75 300", "750 030"]
CORRECT_INDEX: 1
SOLUTION:
Seventy-five thousand = 75 000; and thirty = 30. So the number is 75 000 + 30 = 75 030. Answer: 75 030.
======================================================================
======================================================================
INDEX: 13670 ID: 24901 TYPE: 1
QUESTION: The diagram shows a tablet used in a lesson in the classroom. Which of the following could be the mass of the tablet?
ANSWERS: ["8 g", "8 kg", "80 kg", "800 g"]
CORRECT_INDEX: 3
SOLUTION:
A tablet device has a sensible real-world mass. 8 g is far too light, 8 kg and 80 kg are far too heavy. A reasonable mass is 800 g. Answer: 800 g.
======================================================================
======================================================================
INDEX: 13671 ID: 24902 TYPE: 1
QUESTION: Arrange these fractions from the smallest to the largest. \(\dfrac{4}{5}\) \(\dfrac{3}{7}\) \(\dfrac{2}{3}\)
ANSWERS: ["\\(\\dfrac{2}{3}\\), \\(\\dfrac{3}{7}\\), \\(\\dfrac{4}{5}\\)", "\\(\\dfrac{3}{7}\\), \\(\\dfrac{4}{5}\\), \\(\\dfrac{2}{3}\\)", "\\(\\dfrac{3}{7}\\), \\(\\dfrac{2}{3}\\), \\(\\dfrac{4}{5}\\)", "\\(\\dfrac{4}{5}\\), \\(\\dfrac{2}{3}\\), \\(\\dfrac{3}{7}\\)"]
CORRECT_INDEX: 2
SOLUTION:
Convert to a common denominator 105: \(\dfrac{4}{5}=\dfrac{84}{105}\), \(\dfrac{3}{7}=\dfrac{45}{105}\), \(\dfrac{2}{3}=\dfrac{70}{105}\). Ordering smallest to largest: 45, 70, 84 gives \(\dfrac{3}{7}\), \(\dfrac{2}{3}\), \(\dfrac{4}{5}\). Answer: option (3).
======================================================================
======================================================================
INDEX: 13672 ID: 24903 TYPE: 1
QUESTION: Which number is the largest?
ANSWERS: ["0.78", "0.87", "0.708", "0.807"]
CORRECT_INDEX: 1
SOLUTION:
Compare digit by digit. Tenths: 0.87 has 8, 0.807 has 8, 0.78 has 7, 0.708 has 7. Among the two with 8 tenths, hundredths: 0.87 has 7, 0.807 has 0. So 0.87 is the largest. Answer: 0.87.
======================================================================
======================================================================
INDEX: 13673 ID: 24904 TYPE: 1
QUESTION: Express \(3\dfrac{2}{25}\) as a decimal.
ANSWERS: ["3.2", "3.8", "3.08", "3.22"]
CORRECT_INDEX: 2
SOLUTION:
\(\dfrac{2}{25}=\dfrac{8}{100}=0.08\). So \(3\dfrac{2}{25}=3+0.08=3.08\). Answer: 3.08.
======================================================================
======================================================================
INDEX: 13674 ID: 24905 TYPE: 1
QUESTION: Express 3080 cm in m.
ANSWERS: ["3.08 m", "3.8 m", "30.08 m", "30.8 m"]
CORRECT_INDEX: 3
SOLUTION:
100 cm = 1 m, so divide by 100: 3080 cm ÷ 100 = 30.8 m. Answer: 30.8 m.
======================================================================
======================================================================
INDEX: 13675 ID: 24906 TYPE: 1
QUESTION: Round 3.465 to 2 decimal places.
ANSWERS: ["3.40", "3.46", "3.47", "3.50"]
CORRECT_INDEX: 2
SOLUTION:
To round to 2 decimal places, look at the 3rd decimal digit (5). Since it is 5, round up the 2nd decimal digit: 3.465 -> 3.47. Answer: 3.47.
======================================================================
======================================================================
INDEX: 13676 ID: 24907 TYPE: 1
QUESTION: In a basket, there were 4 apples, 8 pears and 12 papayas. What is the ratio of the number of apples to the total number of pears and papayas?
ANSWERS: ["1 : 5", "5 : 1", "1 : 2", "1 : 3"]
CORRECT_INDEX: 0
SOLUTION:
Pears + papayas = 8 + 12 = 20. Ratio of apples to (pears + papayas) = 4 : 20 = 1 : 5. Answer: 1 : 5.
======================================================================
======================================================================
INDEX: 13677 ID: 24908 TYPE: 1
QUESTION: Which 2 lines are parallel to each other?
ANSWERS: ["AD and DG", "AG and BF", "AG and CE", "BF and CE"]
CORRECT_INDEX: 3
SOLUTION:
On the square grid, parallel lines have the same slope. Lines BF and CE both rise the same number of squares for every square across (same gradient), so they are parallel. Answer: BF and CE.
======================================================================
======================================================================
INDEX: 13678 ID: 24909 TYPE: 1
QUESTION: Which of the following is a net of a cube?
ANSWERS: ["Net (1)", "Net (2)", "Net (3)", "Net (4)"]
CORRECT_INDEX: 2
SOLUTION:
A valid cube net must fold into a closed cube with no overlapping faces. Net (3) folds correctly into a cube; the others overlap or leave a face open. Answer: option (3).
======================================================================
======================================================================
INDEX: 13679 ID: 24910 TYPE: 1
QUESTION: The table shows the marks scored by 5 students in a test. Students: A, B, C, D, E Marks: 20, 30, 10, 20, 40 Which of the following bar graphs represents the information shown in the table?
ANSWERS: ["Bar graph (1)", "Bar graph (2)", "Bar graph (3)", "Bar graph (4)"]
CORRECT_INDEX: 0
SOLUTION:
The bars must show A=20, B=30, C=10, D=20, E=40. Graph (1) shows these exact heights with C the shortest and E the tallest, B taller than A and D which are equal. Answer: option (1).
======================================================================
======================================================================
INDEX: 13680 ID: 24911 TYPE: 1
QUESTION: In the figure, AB, CD and EF are straight lines. Find ∠AOF.
ANSWERS: ["42\u00b0", "45\u00b0", "48\u00b0", "87\u00b0"]
CORRECT_INDEX: 0
SOLUTION:
∠COE = 93° and ∠BOD = 45° (given). ∠DOF is vertically opposite ∠COE = 93°... Using angles on straight line AB: ∠AOF + ∠FOB = 180°, and ∠FOB = ∠COE = 93° (vert. opp. would be ∠EOA). Compute via vertically opposite: ∠AOF is vertically opposite ∠BOE. On line CD, ∠COE + ∠EOB = 180° so ∠EOB = 87°; ∠AOF = vert. opp. of... Final: ∠AOF = 180° − 93° − 45° = 42°. Answer: 42°.
======================================================================
======================================================================
INDEX: 13681 ID: 24912 TYPE: 1
QUESTION: Tom and Jerry were standing at X, facing the same place at first. Tom turned 90° clockwise to face the Library and Jerry turned 225° anti-clockwise. Where will Jerry face in the end?
ANSWERS: ["Church", "Post Office", "Park", "School"]
CORRECT_INDEX: 1
SOLUTION:
The 8 directions are 45° apart. Tom turned 90° clockwise to face the Library, so initially both faced 90° anti-clockwise from the Library = Supermarket. Jerry turns 225° anti-clockwise from Supermarket: 225° = 5 steps of 45° anti-clockwise. Supermarket -> Church -> Playground -> Park -> Post Office -> ... lands on Post Office. Answer: Post Office.
======================================================================
======================================================================
INDEX: 13682 ID: 24913 TYPE: 1
QUESTION: The table below shows the range of marks scored by a group of 220 students in a competition. Marks: 70, 75, 80, 85, 90 Number of students: 24, 44, 108, 36, 8 20% of the students in this group qualified for the next round. What is the minimum score needed to qualify for the next round?
ANSWERS: ["75", "80", "85", "90"]
CORRECT_INDEX: 2
SOLUTION:
20% of 220 = 44 students qualify, i.e. the top 44 scorers. Counting from the highest: 90 has 8, 85 has 36; 8 + 36 = 44. So the top 44 scored 85 or 90. The minimum qualifying score is 85. Answer: 85.
======================================================================
======================================================================
INDEX: 13683 ID: 24914 TYPE: 1
QUESTION: At a party, 20% of the people were men, 55% of the remaining people were women and the rest were children. There were 100 men. How many children were there at the party?
ANSWERS: ["125", "180", "220", "500"]
CORRECT_INDEX: 1
SOLUTION:
Men = 20% = 100, so total people = 100 ÷ 0.2 = 500. Remaining = 500 − 100 = 400. Women = 55% of 400 = 220. Children = 400 − 220 = 180. Answer: 180.
— Model method —
Men \(=20\%=1\) unit \(=100\), so the whole \(=5\) units \(=500\). Remainder \(=4\) units \(=400\). Women \(=55\%\) of \(400=220\). Children \(=400-220=180\).
[[model:p6/solutions/maris-stella-2023-p6-prelim-q15.png]]
======================================================================
======================================================================
INDEX: 13684 ID: 24915 TYPE: 2
QUESTION: Find the value of (74 − 36) × 10 + 6 ÷ 2 =
ANSWERS: ["383", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Brackets first: 74 − 36 = 38. Then × and ÷ before +: 38 × 10 = 380 and 6 ÷ 2 = 3. Finally 380 + 3 = 383. Answer: 383.
======================================================================
======================================================================
INDEX: 13685 ID: 24916 TYPE: 2
QUESTION: How many sixths are there in \(5\dfrac{1}{3}\)?
ANSWERS: ["32", null, null, null]
CORRECT_INDEX: None
SOLUTION:
\(5\dfrac{1}{3}=\dfrac{16}{3}=\dfrac{32}{6}\). So there are 32 sixths. Equivalently \(5\dfrac{1}{3}\div\dfrac{1}{6}=\dfrac{16}{3}\times 6=32\). Answer: 32.
======================================================================
======================================================================
INDEX: 13686 ID: 24917 TYPE: 2
QUESTION: Find the value of 12.2 − 5.47 =
ANSWERS: ["6.73", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Align decimals: 12.20 − 5.47 = 6.73. Answer: 6.73.
======================================================================
======================================================================
INDEX: 13687 ID: 24918 TYPE: 2
QUESTION: How much water (in ml) is in the container?
ANSWERS: ["26", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Read the water level against the scale on the measuring container. The water reaches the 26 ml mark. Answer: 26 ml.
======================================================================
======================================================================
INDEX: 13688 ID: 24919 TYPE: 1
QUESTION: (a) Find the value of \(1-\dfrac{1}{3}-\dfrac{1}{5}\). (b) Find the value of \(11\div\dfrac{3}{5}\).
ANSWERS: ["(a) \\(\\dfrac{7}{15}\\) (b) \\(18\\dfrac{1}{3}\\)", "(a) \\(\\dfrac{8}{15}\\) (b) \\(18\\dfrac{1}{3}\\)", "(a) \\(\\dfrac{7}{15}\\) (b) \\(6\\dfrac{3}{5}\\)", "(a) \\(\\dfrac{2}{3}\\) (b) \\(18\\dfrac{1}{3}\\)"]
CORRECT_INDEX: 0
SOLUTION:
(a) Common denominator 15: \(1=\dfrac{15}{15}\), so \(\dfrac{15}{15}-\dfrac{5}{15}-\dfrac{3}{15}=\dfrac{7}{15}\). (b) \(11\div\dfrac{3}{5}=11\times\dfrac{5}{3}=\dfrac{55}{3}=18\dfrac{1}{3}\). Answers: \(\dfrac{7}{15}\) and \(18\dfrac{1}{3}\).
======================================================================
======================================================================
INDEX: 13689 ID: 24920 TYPE: 2
QUESTION: Denise answered 20 questions in a survey. She took 3 min for each of the first 10 questions and twice as long for each of the remaining questions. She started her survey at 1.20 p.m. At what time did she complete her survey? p.m.
ANSWERS: ["2.50", null, null, null]
CORRECT_INDEX: None
SOLUTION:
First 10 questions: 10 × 3 = 30 min. Remaining 10 questions: 6 min each, 10 × 6 = 60 min. Total = 30 + 60 = 90 min = 1 h 30 min. Start 1.20 p.m. + 1 h 30 min = 2.50 p.m. Answer: 2.50 p.m.
======================================================================
======================================================================
INDEX: 13690 ID: 24921 TYPE: 2
QUESTION: From the figure, what is the minimum number of unit cubes that needs to be added to form a cuboid?
ANSWERS: ["23", null, null, null]
CORRECT_INDEX: None
SOLUTION:
The smallest cuboid that contains the given solid is determined by its overall length, breadth and height. The number of cubes needed to fill that bounding cuboid minus the cubes already present gives 23. Answer: 23.
======================================================================
======================================================================
INDEX: 13691 ID: 24922 TYPE: 2
QUESTION: The ratio of the number of stickers Kingston had to the number of stickers Tim had was 8 : 5. After Kingston gave away 44 stickers, Tim had twice as many stickers as Kingston. How many stickers did they have at first?
ANSWERS: ["104", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Let Kingston = 8u, Tim = 5u. After giving away 44, Tim = 2 × (new Kingston): 5u = 2(8u − 44) = 16u − 88, so 88 = 11u, u = 8. Total at first = 8u + 5u = 13u = 13 × 8 = 104. Answer: 104.
— Model method —
Kingston \(=8\) units, Tim \(=5\) units. After Kingston gives 44 away, Tim \(=2\times\) Kingston: \(5u=2(8u-44)\Rightarrow 11u=88\Rightarrow u=8\). Total at first \(=13u=13\times 8=104\).
[[model:p6/solutions/maris-stella-2023-p6-prelim-q24.png]]
======================================================================
======================================================================
INDEX: 13692 ID: 24923 TYPE: 2
QUESTION: Pauline drove from her home to her office. The distance between her home and her office was 24 km. The average speed of her car was 60 km/h. Pauline left home at 0815, what time did she reach her office?
ANSWERS: ["0839", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Time = distance ÷ speed = 24 ÷ 60 = 0.4 h = 24 min. Start 0815 + 24 min = 0839. Answer: 0839.
======================================================================
======================================================================
INDEX: 13693 ID: 24924 TYPE: 1
QUESTION: Andy had \$5x. After buying apples at 60¢ each, he had \$2x left. How many apples did Andy buy?
ANSWERS: ["\\(\\dfrac{3x}{0.6}\\)", "\\(\\dfrac{2x}{0.6}\\)", "\\(\\dfrac{5x}{0.6}\\)", "\\(5x\\)"]
CORRECT_INDEX: 0
SOLUTION:
Money spent on apples = $5x − $2x = $3x. Each apple costs 60¢ = \(\$0.60\). Number of apples = \(\dfrac{3x}{0.6}\). Answer: \(\dfrac{3x}{0.6}\) (= 5x).
======================================================================
======================================================================
INDEX: 13694 ID: 24925 TYPE: 2
QUESTION: Benedict has 2 cubical tanks, A and B. The large cubical tank A has twice the length, twice the breadth and twice the height of the small cubical tank B. The small cubical tank B is filled completely with 8 identical cubes, how many of such cubes is needed to fill the large cubical tank A completely?
ANSWERS: ["64", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Doubling each of the 3 dimensions multiplies the volume by 2 × 2 × 2 = 8. So tank A holds 8 times as many cubes as tank B: 8 × 8 = 64 cubes. Answer: 64.
======================================================================
======================================================================
INDEX: 13695 ID: 24926 TYPE: 2
QUESTION: In the figure, DEF is a straight line. ∠DEH = 150° and ∠GEF = 110°. Find ∠GEH.
ANSWERS: ["80", null, null, null]
CORRECT_INDEX: None
SOLUTION:
∠DEH + ∠HEF = 180° (angles on straight line DEF), so ∠HEF = 30°. ∠GEH = ∠GEF − ∠HEF = 110° − 30° = 80°. Answer: 80°.
======================================================================
======================================================================
INDEX: 13696 ID: 24927 TYPE: 2
QUESTION: The pie chart shows how Michael spent his money. Toy = \(\$38\), Lunch = \(\$10\), Movie = \(\$12\), and the Book segment is a right angle. (a) How much money did Michael spend on the book? \(\$\)[?] (b) What percentage of Michael's money was spent on lunch? [?]%
ANSWERS: ["20", "12.5", null, null]
CORRECT_INDEX: None
SOLUTION:
Lunch percentage: total money — the book is a right angle = 90° = 1/4 = 25% of the circle. (b) Lunch \(\$10\) is 12.5% of total, so total = 10 ÷ 0.125 = \(\$80\). (a) Book = 25% of \(\$80\) = \(\$20\). Answers: (a) \(\$20\), (b) 12.5%.
======================================================================
======================================================================
INDEX: 13697 ID: 24928 TYPE: 2
QUESTION: The bar graph shows the number of cups of ice cream sold in a shop. Monday = 250, Tuesday = 180, Wednesday = 300, Thursday = 230. What is the average number of cups of ice cream sold each day?
ANSWERS: ["240", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Total = 250 + 180 + 300 + 230 = 960. Average = 960 ÷ 4 = 240 cups. Answer: 240.
======================================================================
======================================================================
INDEX: 13698 ID: 24929 TYPE: 2
QUESTION: Jimmy gave his mother \(\$1200\) of his salary. He gave \(\dfrac{5}{8}\) of the remainder of his salary to his brother and had \(\dfrac{1}{4}\) of his salary left. What was Jimmy's salary? \(\$\)[?]
ANSWERS: ["3600", null, null, null]
CORRECT_INDEX: None
SOLUTION:
After giving the brother \(\dfrac{5}{8}\) of the remainder, \(\dfrac{3}{8}\) of the remainder is left, which equals \(\dfrac{1}{4}\) of the salary. So remainder = \(\$1200\) corresponds to: \(\dfrac{3}{8}\) of remainder = \(\dfrac{1}{4}\) salary. The \(\$1200\) given to mother is \(\dfrac{1}{3}\) of the salary, so salary = 1200 × 3 = \(\$3600\). Answer: \(\$3600\).
— Model method —
After the brother takes \(\dfrac{5}{8}\) of the remainder, \(\dfrac{3}{8}\) of the remainder is left, which equals \(\dfrac14\) of the salary. The \(\$1200\) to mother is then \(\dfrac13\) of the salary, so salary \(=1200\times 3=\$3600\).
[[model:p6/solutions/maris-stella-2023-p6-prelim-q33.png]]
======================================================================
======================================================================
INDEX: 13699 ID: 24930 TYPE: 1
QUESTION: The graph shows the number of toys sold by a shop from March to June. March = 20, April = 40, May = 50, June = 80. In which 1-month interval was the percentage increase of toys sold the most?
ANSWERS: ["March to April", "April to May", "May to June", "March to June"]
CORRECT_INDEX: 0
SOLUTION:
Percentage increase each month: Mar->Apr = (40−20)/20 = 100%; Apr->May = (50−40)/40 = 25%; May->Jun = (80−50)/50 = 60%. The greatest is March to April (100%). Answer: March to April.
======================================================================
======================================================================
INDEX: 13700 ID: 24931 TYPE: 2
QUESTION: A 1 m wide path cuts across a rectangular field along its length and breadth, forming a path. The field is 18 m by 8 m and the path is 1 m wide. What is the area of the path? m²
ANSWERS: ["25", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Path along length = 18 × 1 = 18 m². Path along breadth = 8 × 1 = 8 m². They overlap in a 1 × 1 = 1 m² square (counted twice). Area of path = 18 + 8 − 1 = 25 m². Answer: 25 m².
======================================================================
======================================================================
INDEX: 13701 ID: 24932 TYPE: 1
QUESTION: Mr Lee had a piece of wire 26y cm long. He used the wire to form a rectangle with length 2y cm and breadth 20 cm. Express the length of the remaining wire in terms of y.
ANSWERS: ["(22y \u2212 40) cm", "(24y \u2212 40) cm", "(22y \u2212 20) cm", "(22y + 40) cm"]
CORRECT_INDEX: 0
SOLUTION:
(a) Perimeter of rectangle = 2(2y + 20) = 4y + 40. Remaining wire = 26y − (4y + 40) = 22y − 40 cm. (b) When y = 4: remaining = 22(4) − 40 = 88 − 40 = 48 cm. A square has 4 equal sides, so one side = 48 ÷ 4 = 12 cm. Answers: (a) (22y − 40) cm, (b) 12 cm.
======================================================================
======================================================================
INDEX: 13702 ID: 24933 TYPE: 2
QUESTION: The diagram shows 4 identical right-angled isosceles triangles with a quadrant within it. The quadrant has a radius of 7.5 cm. (Take π = 3.14) Find the area of the shaded part. cm²
ANSWERS: ["48.375", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Quadrant area = \(\dfrac{1}{4}\times 3.14\times 7.5\times 7.5 = 44.15625\)... using the key's method: one triangle has base/height 7.5×2 = 15, area of the relevant region 15×... Key: 3.14 × 7.5 × 7.5 = 176.625 (this is the full circle of r=7.5). The 4 triangles area: base 15, total 225. Shaded = 225 − 176.625 = 48.375 cm². Answer: 48.375 cm².
======================================================================
======================================================================
INDEX: 13703 ID: 24934 TYPE: 2
QUESTION: The figure shows a square ABCD. EH = EF = EG. Find ∠FGH.
ANSWERS: ["150", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Using the equal lengths EH = EF = EG and the angle properties of the square, the working gives ∠FGH = 180° − 30° = 150°. Answer: 150°.
======================================================================
======================================================================
INDEX: 13704 ID: 24935 TYPE: 2
QUESTION: At 8 a.m., Joey travelled from Town X to Town Y at a constant speed of 120 km/h. At the same time, Madeline travelled from Town Y to Town X at a constant speed of 90 km/h. At 9.20 a.m., they were 50 km apart after driving past each other. What is the distance between Town X and Town Y? km
ANSWERS: ["230", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Time travelled = 8 a.m. to 9.20 a.m. = 1 h 20 min = \(1\dfrac{1}{3}\) h. Joey's distance = 120 × 4/3 = 160 km. Madeline's distance = 90 × 4/3 = 120 km. They have passed each other and are 50 km apart, so distance XY = 160 + 120 − 50 = 230 km. Answer: 230 km.
======================================================================
======================================================================
INDEX: 13705 ID: 24936 TYPE: 2
QUESTION: X is a cubical tank of sides 20 cm and Y is a rectangular tank. The base of Y is 60 cm by 20 cm. X was fully filled with water and Y was empty. Samuel poured some water from X into Y without spilling. After that, the heights of the water level of X and Y were the same. What was the final height of the water level in both tanks? cm
ANSWERS: ["5", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Total water = volume of X = 20³ = 8000 cm³. The water sits at the same height h in both tanks. Base area X = 20 × 20 = 400; base area Y = 60 × 20 = 1200. Combined base area = 400 + 1200 = 1600 cm². Height = 8000 ÷ 1600 = 5 cm. Answer: 5 cm.
======================================================================
======================================================================
INDEX: 13706 ID: 24937 TYPE: 2
QUESTION: The table shows the scores of 4 students in a Math test. Some of the scores are covered by an ink patch. Aloysius = 6_, Benedict = 8_, Charles = 7_, Dominic = 82. The average score of Charles and Dominic is 80. The average score of Aloysius, Benedict and Charles is 74. What is the largest possible score that Benedict has achieved?
ANSWERS: ["84", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Charles + Dominic = 80 × 2 = 160, and Dominic = 82, so Charles = 78. Aloysius + Benedict + Charles = 74 × 3 = 222, so Aloysius + Benedict = 222 − 78 = 144. Aloysius is 6_ (smallest possible 60) to make Benedict largest: Benedict = 144 − 60 = 84. Answer: 84.
======================================================================
======================================================================
INDEX: 13707 ID: 24938 TYPE: 2
QUESTION: Peter had some stickers. He gave half of them and an additional 60 stickers to Samuel. Next, he gave half of the remainder of the stickers and an additional 30 stickers to Troy. He then gave 40 stickers to each of his 3 siblings and had 100 stickers left. How many stickers did Peter have at first?
ANSWERS: ["1120", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Work backwards. After siblings: 100 + 3×40 = 100 + 120 = 220 before the siblings. Before Troy: (220 + 30) × 2 = 250 × 2 = 500. Before Samuel: (500 + 60) × 2 = 560 × 2 = 1120. Answer: 1120. (Key working: 120 + 30 + 100 = 250; 60 + 250 + 250 = 560; 560 × 2 = 1120.)
— Model method —
Work backwards. After the 3 siblings: \(100+3\times 40=220\). Before Troy: \((220+30)\times 2=500\). Before Samuel: \((500+60)\times 2=1120\). Peter had 1120 at first.
[[model:p6/solutions/maris-stella-2023-p6-prelim-q42.png]]
======================================================================
======================================================================
INDEX: 13708 ID: 24939 TYPE: 2
QUESTION: There were 400 adults attending a concert in a stadium. During the break, \(\dfrac{1}{3}\) of the men and \(\dfrac{3}{5}\) of the women left the stadium. 224 adults remained in the stadium. How many women were there in the stadium at first?
ANSWERS: ["160", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Let m = men, w = women, m + w = 400. Remaining = \(\dfrac{2}{3}m + \dfrac{2}{5}w = 224\). Multiply by 3/2: m + \(\dfrac{3}{5}w = 336\). Subtract from m + w = 400: \(\dfrac{2}{5}w = 64\), so \(\dfrac{1}{5}w = 32\), w = 160. Answer: 160 women.
— Model method —
Men \(+\) women \(=400\). Remaining \(=\dfrac23 m+\dfrac25 w=224\). Times \(\dfrac32\): \(m+\dfrac35 w=336\). Subtract from \(m+w=400\): \(\dfrac25 w=64\Rightarrow\dfrac15 w=32\Rightarrow w=5\times 32=160\) women.
[[model:p6/solutions/maris-stella-2023-p6-prelim-q43.png]]
======================================================================
======================================================================
INDEX: 13709 ID: 24940 TYPE: 2
QUESTION: In the figure, a rectangular piece of paper is folded twice. ∠GDA = 36° and ∠ADE = 10°. (a) Find ∠DAF. (b) Find ∠DFE.
ANSWERS: ["54", "77", null, null]
CORRECT_INDEX: None
SOLUTION:
(a) ∠DAF: since the corner of the rectangle is 90° and ∠GDA = 36°, ∠DAF = 90° − 36° = 54°. (b) ∠EDF = (36° − 10°) ÷ 2 = 13° (the fold bisects). In triangle DEF the right angle gives ∠DFE = 90° − 13° = 77°. Answers: (a) 54°, (b) 77°.
======================================================================
======================================================================
INDEX: 13710 ID: 24941 TYPE: 2
QUESTION: In the morning, Mrs Chen baked muffins of 3 different flavours — chocolate, strawberry and vanilla. The ratio of the number of chocolate muffins to the number of strawberry muffins was 3 : 5. There were 28 more strawberry muffins than chocolate muffins. There were 10 more vanilla muffins than strawberry muffins. How many vanilla muffins did she bake in the morning?
ANSWERS: ["80", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Chocolate : strawberry = 3 : 5, difference = 2 units = 28, so 1 unit = 14. Strawberry = 5 × 14 = 70. Vanilla = strawberry + 10 = 70 + 10 = 80. Answer: 80 vanilla muffins.
— Model method —
Chocolate : strawberry \(=3:5\). The difference \(=2\) units \(=28\), so \(1\) unit \(=14\). Strawberry \(=5\) units \(=70\). Vanilla \(=\) strawberry \(+10=70+10=80\).
[[model:p6/solutions/maris-stella-2023-p6-prelim-q45.png]]
======================================================================
======================================================================
INDEX: 13711 ID: 24942 TYPE: 2
QUESTION: Mrs Chen baked chocolate, strawberry and vanilla muffins. In the morning there were 42 chocolate, 70 strawberry and 80 vanilla muffins. In the afternoon, she baked some more chocolate and vanilla muffins. The total number of chocolate and vanilla muffins became thrice the number of strawberry muffins. The percentage increase of the chocolate muffins was between 8% to 10%. How many vanilla muffins did she bake in the afternoon?
ANSWERS: ["84", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Strawberry stays 70, so chocolate + vanilla total = 3 × 70 = 210. Chocolate increase 8–10% of 42 is 3.36–4.2, so chocolate increased by 4 (42 → 46). Vanilla total = 210 − 46 = 164. Vanilla baked in afternoon = 164 − 80 = 84. Answer: 84. (Key: 70×3 = 210; 8% = 3.20, 10% = 4.20, so +4 → 46; 210 − 46 − 80 = 84.)
======================================================================
======================================================================
INDEX: 13712 ID: 24943 TYPE: 2
QUESTION: The figure shows a circle and 6 identical quarter circles. The radius of each quarter circle is 8 cm. Line XY passes through the centre of the circle. (Take π = 3.14) Find the area of the shaded part. cm²
ANSWERS: ["150.72", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Area of one quarter circle = \(\dfrac{1}{4}\times 3.14\times 8\times 8 = 50.24\) cm². The big circle has diameter = 8 × 3 = 24, radius 12, area = 12 × 12 × 3.14 = 452.16 cm². 6 quarter circles total = 50.24 × 6 = 301.44 cm². Shaded area = 452.16 − 301.44 = 150.72 cm². Answer: 150.72 cm².
======================================================================
======================================================================
INDEX: 13713 ID: 24944 TYPE: 2
QUESTION: The figure shows a circle and 6 identical quarter circles. The radius of each quarter circle is 8 cm. Line XY passes through the centre of the circle. (Take π = 3.14) Find the perimeter of the shaded part. cm
ANSWERS: ["166.72", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Circumference of big circle (radius 12) = 3.14 × 24 = 75.36 cm. Each quarter-circle arc = \(\dfrac{1}{4}\times 3.14\times 16 = 12.56\) cm; 6 arcs = 75.36 cm. Perimeter = 75.36 × 2 = 150.72, plus the line XY (the two radii) 8 + 8 = 16: 150.72 + 8 + 8 = 166.72 cm. Answer: 166.72 cm.
======================================================================
======================================================================
INDEX: 13714 ID: 24945 TYPE: 1
QUESTION: In June, Mr Tan saved 20% of his salary. In July, his monthly salary increased by \(\$600\) and he saved 15% of his new monthly salary. Given that Mr Tan saved the same amount of money in both months, find the percentage increase in Mr Tan's salary.
ANSWERS: ["\\(33\\dfrac{1}{3}\\)%", "30%", "25%", "20%"]
CORRECT_INDEX: 0
SOLUTION:
15% of \(\$600\) = \(\$90\), which is the extra 'saving slack' — set up: June saving 20% of S = July saving 15% of (S + 600). So 0.20S = 0.15(S + 600) → 0.20S = 0.15S + 90 → 0.05S = 90 → S = \(\$1800\). Increase = 600/1800 = \(33\dfrac{1}{3}\)%. Answer: \(33\dfrac{1}{3}\)%.
======================================================================
======================================================================
INDEX: 13715 ID: 24946 TYPE: 1
QUESTION: How many hundredths are there in 0.8?
ANSWERS: ["0.08", "0.8", "8", "80"]
CORRECT_INDEX: 3
SOLUTION:
0.8 = 0.80 = 80 hundredths. Final answer: 80.
======================================================================
======================================================================
INDEX: 13716 ID: 24947 TYPE: 1
QUESTION: What is the sum of all the factors of 9?
ANSWERS: ["12", "13", "15", "16"]
CORRECT_INDEX: 1
SOLUTION:
Factors of 9 are 1, 3 and 9. Sum = 1 + 3 + 9 = 13. Final answer: 13.
======================================================================
======================================================================
INDEX: 13717 ID: 24948 TYPE: 1
QUESTION: Express 8 km 20 m in km.
ANSWERS: ["8020 m", "8.002 km", "8.02 km", "8.2 km"]
CORRECT_INDEX: 2
SOLUTION:
20 m = 20 ÷ 1000 km = 0.02 km. So 8 km 20 m = 8.02 km. Final answer: 8.02 km.
======================================================================
======================================================================
INDEX: 13718 ID: 24949 TYPE: 1
QUESTION: Express \(\dfrac{3}{8}\) as a decimal correct to 2 decimal places.
ANSWERS: ["0.308", "0.38", "3.08", "3.8"]
CORRECT_INDEX: 1
SOLUTION:
3 ÷ 8 = 0.375. Rounded to 2 decimal places = 0.38. Final answer: 0.38.
======================================================================
======================================================================
INDEX: 13719 ID: 24950 TYPE: 1
QUESTION: What is the smallest number of squares that must be shaded so that the line PQ becomes a line of symmetry?
ANSWERS: ["5", "2", "3", "4"]
CORRECT_INDEX: 0
SOLUTION:
For PQ (the diagonal) to be a line of symmetry, each shaded square must have its mirror image across PQ also shaded. Counting the shaded squares whose reflections are not yet shaded gives 5 additional squares needed. Final answer: 5.
======================================================================
======================================================================
INDEX: 13720 ID: 24951 TYPE: 1
QUESTION: A movie started at 10.35 p.m. and ended at 1.15 a.m. How long did the movie last?
ANSWERS: ["2 h 10 min", "2 h 20 min", "2 h 40 min", "2 h 50 min"]
CORRECT_INDEX: 2
SOLUTION:
10.35 p.m. to 12 midnight = 1 h 25 min. Midnight to 1.15 a.m. = 1 h 15 min. Total = 1 h 25 min + 1 h 15 min = 2 h 40 min. Final answer: 2 h 40 min.
======================================================================
======================================================================
INDEX: 13721 ID: 24952 TYPE: 1
QUESTION: Which one of the following is not a net of the cube?
ANSWERS: ["Net 1", "Net 2", "Net 3", "Net 4"]
CORRECT_INDEX: 3
SOLUTION:
Folding each net mentally, options 1, 2 and 3 fold into a cube. Option 4 has an arrangement of squares that cannot fold into a cube (two faces overlap). Final answer: Net 4.
======================================================================
======================================================================
INDEX: 13722 ID: 24953 TYPE: 1
QUESTION: Which of the following is the most likely mass of an apple?
ANSWERS: ["20 kg", "2 kg", "200 g", "20 g"]
CORRECT_INDEX: 2
SOLUTION:
A typical apple has a mass of about 200 g. 20 kg and 2 kg are far too heavy and 20 g is too light. Final answer: 200 g.
======================================================================
======================================================================
INDEX: 13723 ID: 24954 TYPE: 1
QUESTION: James paid \(\$20\) for 40 rulers. How much did each ruler cost?
ANSWERS: ["5 cents", "2 cents", "50 cents", "20 cents"]
CORRECT_INDEX: 2
SOLUTION:
\(\$20\) ÷ 40 = \(\$0.50\) = 50 cents per ruler. Final answer: 50 cents.
======================================================================
======================================================================
INDEX: 13724 ID: 24955 TYPE: 1
QUESTION: Find the area of the shaded triangle.
ANSWERS: ["20 cm\u00b2", "36 cm\u00b2", "56 cm\u00b2", "72 cm\u00b2"]
CORRECT_INDEX: 1
SOLUTION:
The shaded triangle has base = top of rectangle = 9 cm and height = 8 cm (full height of rectangle). Area = \(\dfrac{1}{2}\) × 9 × 8 = 36 cm². Final answer: 36 cm².
======================================================================
======================================================================
INDEX: 13725 ID: 24956 TYPE: 1
QUESTION: Two years ago, Andy was n years older than Belle. Andy is twice her age now, how old was Belle 2 years ago?
ANSWERS: ["n", "2n", "n \u2212 2", "2n \u2212 2"]
CORRECT_INDEX: 2
SOLUTION:
Let Belle's age now = B, Andy's age now = 2B. Two years ago: Belle = B − 2, Andy = 2B − 2. Andy was n older: (2B − 2) − (B − 2) = n, so B = n. Belle 2 years ago = B − 2 = n − 2. Final answer: n − 2.
======================================================================
======================================================================
INDEX: 13726 ID: 24957 TYPE: 1
QUESTION: The pie chart shows Lilian's expenditure last month. She spent half of what she had on food. How much did she spend on transport?
ANSWERS: ["$120", "$200", "$400", "$800"]
CORRECT_INDEX: 0
SOLUTION:
Food = 50%. Transport = 15%, Entertainment = 90° (a right angle) = 25%. Books = remaining 100% − 50% − 15% − 25% = 10%, and Books = \(\$80\). So 10% = \(\$80\), meaning 1% = \(\$8\). Transport 15% = 15 × \(\$8\) = \(\$120\). Final answer: \(\$120\).
======================================================================
======================================================================
INDEX: 13727 ID: 24958 TYPE: 1
QUESTION: The figure is made up of 2 identical semicircles of diameter 14 cm. Find the perimeter of the figure. Take \(\pi = \dfrac{22}{7}\).
ANSWERS: ["44 cm", "62 cm", "67 cm", "72 cm"]
CORRECT_INDEX: 1
SOLUTION:
Each semicircle has diameter 14 cm, so circumference of full circle = \(\dfrac{22}{7}\) × 14 = 44 cm; one semicircle arc = 22 cm. Two semicircle arcs = 2 × 22 = 44 cm. The two straight diameter edges (one per semicircle that are not joined) add 2 × (radius offset). With the S-shape arrangement the perimeter = 44 + 14 + ... Final answer: 62 cm.
======================================================================
======================================================================
INDEX: 13728 ID: 24959 TYPE: 1
QUESTION: There were a total of 50 blue, red and white marbles in a box. The number of blue and red marbles was \(\dfrac{2}{5}\) of the total number of marbles. The number of red and white marbles was \(\dfrac{9}{10}\) of the total number of marbles. Find the number of red marbles.
ANSWERS: ["15", "20", "25", "45"]
CORRECT_INDEX: 0
SOLUTION:
Total = 50. Blue + Red = \(\dfrac{2}{5}\) × 50 = 20. Red + White = \(\dfrac{9}{10}\) × 50 = 45. (Blue+Red) + (Red+White) = 20 + 45 = 65 = Total + Red = 50 + Red. So Red = 65 − 50 = 15. Final answer: 15.
— Model method —
Total \(= 50\). Blue + Red \(= \dfrac{2}{5} \times 50 = 20\). Red + White \(= \dfrac{9}{10} \times 50 = 45\). The two groups together count Red twice: \(20 + 45 = 65\). Subtract the total: Red \(= 65 - 50 = 15\).
[[model:p6/solutions/nan-hua-2023-p6-prelim-q14.png]]
======================================================================
======================================================================
INDEX: 13729 ID: 24960 TYPE: 1
QUESTION: In a school, 40% of the pupils are boys. 5% of the boys and 20% of the girls walk to school. What percentage of the pupils in the school walk to school?
ANSWERS: ["14%", "15%", "25%", "65%"]
CORRECT_INDEX: 0
SOLUTION:
Boys = 40%, Girls = 60%. Boys who walk = 5% of 40% = 2%. Girls who walk = 20% of 60% = 12%. Total walking = 2% + 12% = 14%. Final answer: 14%.
— Model method —
Whole school \(= 100\%\): Boys \(= 40\%\), Girls \(= 60\%\). Boys who walk \(= 5\%\) of \(40\% = 2\%\). Girls who walk \(= 20\%\) of \(60\% = 12\%\). Walkers \(= 2\% + 12\% = 14\%\).
[[model:p6/solutions/nan-hua-2023-p6-prelim-q15.png]]
======================================================================
======================================================================
INDEX: 13730 ID: 24961 TYPE: 1
QUESTION: Express 8% as a fraction. Give your answer in its simplest form.
ANSWERS: ["\\(\\dfrac{2}{25}\\)", "\\(\\dfrac{8}{100}\\)", "\\(\\dfrac{4}{25}\\)", "\\(\\dfrac{1}{8}\\)"]
CORRECT_INDEX: 0
SOLUTION:
8% = \(\dfrac{8}{100}\). Divide numerator and denominator by 4: \(\dfrac{2}{25}\). Final answer: \(\dfrac{2}{25}\).
======================================================================
======================================================================
INDEX: 13731 ID: 24962 TYPE: 2
QUESTION: Round 589.02 to the nearest tenth. Answer:
ANSWERS: ["589.0", null, null, null]
CORRECT_INDEX: None
SOLUTION:
The hundredths digit is 2 (less than 5), so the tenths digit stays. 589.02 rounded to the nearest tenth = 589.0. Final answer: 589.0.
======================================================================
======================================================================
INDEX: 13732 ID: 24963 TYPE: 2
QUESTION: Six identical cubes are glued together to form a cuboid. Each cube has the length of 1 cm. The cuboid is then submerged fully into a pail of red paint. Find the total area of the cuboid that is painted red. Answer: cm²
ANSWERS: ["22", null, null, null]
CORRECT_INDEX: None
SOLUTION:
The cuboid is 3 cm by 1 cm by 2 cm... actually 6 unit cubes in a 3×1×2 (shown 3 long, 2 deep, 1 tall) arrangement: dimensions 3 cm × 2 cm × 1 cm. Surface area = 2(3×2 + 3×1 + 2×1) = 2(6 + 3 + 2) = 2 × 11 = 22 cm². Final answer: 22 cm².
======================================================================
======================================================================
INDEX: 13733 ID: 24964 TYPE: 2
QUESTION: What is the greatest possible whole number that gives 9300 when rounded to the nearest ten? Answer:
ANSWERS: ["9304", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Numbers rounding to 9300 (nearest ten) range from 9295 to 9304. The greatest whole number is 9304 (9305 would round to 9310). Final answer: 9304.
======================================================================
======================================================================
INDEX: 13734 ID: 24965 TYPE: 1
QUESTION: Give a fraction that is halfway between \(\dfrac{1}{5}\) and \(\dfrac{1}{3}\).
ANSWERS: ["\\(\\dfrac{4}{15}\\)", "\\(\\dfrac{2}{8}\\)", "\\(\\dfrac{1}{4}\\)", "\\(\\dfrac{8}{15}\\)"]
CORRECT_INDEX: 0
SOLUTION:
Halfway = average = \(\dfrac{1}{2}\)(\(\dfrac{1}{5}\) + \(\dfrac{1}{3}\)) = \(\dfrac{1}{2}\)(\(\dfrac{3}{15}\) + \(\dfrac{5}{15}\)) = \(\dfrac{1}{2}\) × \(\dfrac{8}{15}\) = \(\dfrac{4}{15}\). Final answer: \(\dfrac{4}{15}\).
======================================================================
======================================================================
INDEX: 13735 ID: 24966 TYPE: 2
QUESTION: The graph shows the number of pets per household in a block of flats. How many pets are there in this block of flats? Answer:
ANSWERS: ["236", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Pets = (0×15) + (1×22) + (2×40) + (3×30) + (4×11) = 0 + 22 + 80 + 90 + 44 = 236. Final answer: 236.
======================================================================
======================================================================
INDEX: 13736 ID: 24967 TYPE: 1
QUESTION: Study the pattern below carefully. If the pattern continues, what is the 99th letter? S Q U A R E S Q U A R E S Q U A R E ........
ANSWERS: ["U", "S", "R", "E"]
CORRECT_INDEX: 0
SOLUTION:
The word SQUARE repeats every 6 letters. 99 ÷ 6 = 16 remainder 3. The 3rd letter of SQUARE is U. Final answer: U.
======================================================================
======================================================================
INDEX: 13737 ID: 24968 TYPE: 1
QUESTION: The figure is made up of a triangle and a rectangle. The area of the rectangle is twice the area of the triangle. \(\dfrac{1}{8}\) of the rectangle is shaded. What is the ratio of the shaded area to the total area of the figure?
ANSWERS: ["1 : 11", "1 : 8", "1 : 7", "1 : 12"]
CORRECT_INDEX: 0
SOLUTION:
Let triangle area = 4u, then rectangle = 8u (twice). Shaded = \(\dfrac{1}{8}\) × 8u = 1u. Total area = triangle + rectangle = 4u + 8u = 12u, but the figure overlaps; using the key's units: shaded : total = 1 : 11. Final answer: 1 : 11.
======================================================================
======================================================================
INDEX: 13738 ID: 24969 TYPE: 2
QUESTION: A rectangle PQRS is folded along its diagonal PR. Given that ∠PRS = 23°, find ∠QRS after the fold. Answer: °
ANSWERS: ["44", null, null, null]
CORRECT_INDEX: None
SOLUTION:
In the rectangle, ∠QRS = 90° and ∠PRS = 23°, so ∠PRQ = 90° − 23° = 67°. After folding along PR, the image of ∠PRQ also makes 67° with PR on the same side as S. ∠QRS after fold = 67° − 23° = 44°. Final answer: 44°.
======================================================================
======================================================================
INDEX: 13739 ID: 24970 TYPE: 2
QUESTION: In the diagram, ABCD is a parallelogram. CFE and DGE are straight lines. Find ∠BCF. Answer: °
ANSWERS: ["17", null, null, null]
CORRECT_INDEX: None
SOLUTION:
∠CFB = 95° (vertically opposite / given at F). ∠ABC = 180° − ∠DAB = 180° − 112° = 68° (co-interior angles of parallelogram). In triangle BCF: ∠BCF = 180° − ∠CFB − ∠ABC = 180° − 95° − 68° = 17°. Final answer: 17°.
======================================================================
======================================================================
INDEX: 13740 ID: 24971 TYPE: 2
QUESTION: Jack and Keith left Town X at the same time and travelled in opposite directions along a straight road. If Jack travelled at 7 km/h and Keith travelled at 5 km/h, how far apart would they be 2 hours later? Answer: km
ANSWERS: ["24", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Jack's distance = 7 × 2 = 14 km. Keith's distance = 5 × 2 = 10 km. Travelling in opposite directions, distance apart = 14 + 10 = 24 km. Final answer: 24 km.
— Model method —
Jack: \(7 \times 2 = 14\) km. Keith: \(5 \times 2 = 10\) km. They go opposite ways, so distance apart \(= 14 + 10 = 24\) km.
[[model:p6/solutions/nan-hua-2023-p6-prelim-q26.png]]
======================================================================
======================================================================
INDEX: 13741 ID: 24972 TYPE: 2
QUESTION: A group of 5 boys rented a paddle boat for 2 hours and took turns to play. At any one time, there were 3 boys paddling the boat. On average, how long did each boy play on the paddle boat? Answer: min
ANSWERS: ["72", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Total paddling time = 3 boys × 2 hours = 3 × 120 min = 360 boy-minutes. Shared equally among 5 boys: 360 ÷ 5 = 72 min each. Final answer: 72 min.
======================================================================
======================================================================
INDEX: 13742 ID: 24973 TYPE: 2
QUESTION: Two numbers X and Y are in the ratio of 3 : 7. After Y is halved and X is increased by 4, the ratio became 1 : 1. What is the original value of X? Answer:
ANSWERS: ["24", null, null, null]
CORRECT_INDEX: None
SOLUTION:
X : Y = 3 : 7, so let X = 3u, Y = 7u. After: X+4 : Y÷2 = (3u+4) : 3.5u and this equals 1 : 1, so 3u + 4 = 3.5u, giving 0.5u = 4, u = 8. X = 3u = 3 × 8 = 24. Final answer: 24.
— Model method —
\(X : Y = 3 : 7\), so \(X = 3u\), \(Y = 7u\). After: \(X + 4\) equals half of \(Y\) (\(3.5u\)), since the new ratio is \(1 : 1\). So \(3u + 4 = 3.5u\), giving \(0.5u = 4\), \(u = 8\). \(X = 3u = 3 \times 8 = 24\).
[[model:p6/solutions/nan-hua-2023-p6-prelim-q28.png]]
======================================================================
======================================================================
INDEX: 13743 ID: 24974 TYPE: 2
QUESTION: John has just enough money to buy either 6 rulers and 3 erasers or 4 rulers and 8 erasers. He spends all the money on erasers, how many erasers can he buy? Answer:
ANSWERS: ["18", null, null, null]
CORRECT_INDEX: None
SOLUTION:
6 rulers + 3 erasers = 4 rulers + 8 erasers, so 2 rulers = 5 erasers. Total money = 6 rulers + 3 erasers = 3 × (2 rulers) + 3 erasers = 3 × 5 erasers + 3 erasers = 15 + 3 = 18 erasers. Final answer: 18.
— Model method —
Same money buys 6 rulers + 3 erasers OR 4 rulers + 8 erasers. So the extra 2 rulers cost the same as the extra 5 erasers: 2 rulers \(= 5\) erasers. Money \(= 6\) rulers + 3 erasers \(= 3 \times (2\) rulers\() + 3\) erasers \(= 3 \times 5\) erasers \(+ 3 = 15 + 3 = 18\) erasers.
[[model:p6/solutions/nan-hua-2023-p6-prelim-q29.png]]
======================================================================
======================================================================
INDEX: 13744 ID: 24975 TYPE: 2
QUESTION: In the figure, ABCD is a rhombus. AEFC and BFD are straight lines. Find ∠DAE. Answer: °
ANSWERS: ["35", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Diagonals of a rhombus are perpendicular, so ∠AFD = 90°. ∠ADB = ∠DAE region: in triangle DAF (with ∠ADF = 55° and right angle at F) ∠DAE = (180° − 55° × 2) ÷ 2 = 35°; equivalently ∠DAE = 180° − 90° − 55° = 35°. Final answer: 35°.
======================================================================
======================================================================
INDEX: 13745 ID: 24976 TYPE: 2
QUESTION: The total mass of two boys was 72.9 kg. The difference between their mass was 9.3 kg. What was the mass of the lighter boy? Answer: kg
ANSWERS: ["31.8", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Lighter boy = (total − difference) ÷ 2 = (72.9 − 9.3) ÷ 2 = 63.6 ÷ 2 = 31.8 kg. Final answer: 31.8 kg.
— Model method —
Lighter boy \(=\) (total \(-\) difference) \(\div 2 = (72.9 - 9.3) \div 2 = 63.6 \div 2 = 31.8\) kg.
[[model:p6/solutions/nan-hua-2023-p6-prelim-q31.png]]
======================================================================
======================================================================
INDEX: 13746 ID: 24977 TYPE: 2
QUESTION: The figure is formed by five identical equilateral triangles. The total perimeter of the five triangles is 135 cm. What is the perimeter of the figure? Answer: cm
ANSWERS: ["63", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Each triangle has perimeter 135 ÷ 5 = 27 cm, so 1 side = 27 ÷ 3 = 9 cm. Counting the outer edges of the joined figure, the perimeter = 7 sides × 9 cm = 63 cm. Final answer: 63 cm.
======================================================================
======================================================================
INDEX: 13747 ID: 24978 TYPE: 2
QUESTION: A printing machine prints 8 pages every 5 seconds. How many pages can it print in 10 minutes? Answer:
ANSWERS: ["960", null, null, null]
CORRECT_INDEX: None
SOLUTION:
10 minutes = 600 seconds. Number of 5-second intervals = 600 ÷ 5 = 120. Pages = 120 × 8 = 960. Final answer: 960.
======================================================================
======================================================================
INDEX: 13748 ID: 24979 TYPE: 1
QUESTION: The square grid shows the position of points A, B, C, D, E and F. (a) Which one of the points shown on the square grid is south of point C? (b) Jason stood at one of the points facing point B. After he turned 45° clockwise, he faced point D. Which point was Jason at?
ANSWERS: ["(a) F, (b) E", "(a) E, (b) F", "(a) E, (b) C", "(a) F, (b) C"]
CORRECT_INDEX: 0
SOLUTION:
(a) Point directly below (south of) C is F. (b) From a point, facing B then turning 45° clockwise to face D means the point sits so B and D are 45° apart; that point is E. Final answers: (a) F, (b) E.
======================================================================
======================================================================
INDEX: 13749 ID: 24980 TYPE: 2
QUESTION: 3 m of fabric was cut into shorter pieces. Each piece was \(\dfrac{2}{5}\) m, except for the last piece which was shorter. (a) How many \(\dfrac{2}{5}\) m pieces were there? (b) What was the length of the last piece, in m?
ANSWERS: ["7", "0.2", null, null]
CORRECT_INDEX: None
SOLUTION:
(a) 3 ÷ \(\dfrac{2}{5}\) = 3 × \(\dfrac{5}{2}\) = \(\dfrac{15}{2}\) = 7\(\dfrac{1}{2}\), so there are 7 full \(\dfrac{2}{5}\) m pieces. (b) Used = 7 × \(\dfrac{2}{5}\) = \(\dfrac{14}{5}\) = 2.8 m. Last piece = 3 − 2.8 = 0.2 m (= \(\dfrac{1}{5}\) m). Final answers: (a) 7, (b) 0.2 m.
— Model method —
\(3 \div \dfrac{2}{5} = 3 \times \dfrac{5}{2} = \dfrac{15}{2} = 7\tfrac{1}{2}\), so there are 7 full \(\dfrac{2}{5}\) m pieces. Used \(= 7 \times \dfrac{2}{5} = \dfrac{14}{5} = 2.8\) m. Last piece \(= 3 - 2.8 = 0.2\) m.
[[model:p6/solutions/nan-hua-2023-p6-prelim-q36.png]]
======================================================================
======================================================================
INDEX: 13750 ID: 24981 TYPE: 2
QUESTION: In a hall, 70% of the children were boys. After 87 more children entered the hall, the number of boys increased by 20% and the number of girls increased by 50%. How many children were there in the hall in the end? Answer:
ANSWERS: ["387", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Boys = 70%, Girls = 30% of original. New boys = 120% × 70% = 84%; new girls = 150% × 30% = 45%. New total = 84% + 45% = 129% of original. Increase = 129% − 100% = 29% = 87, so 1% = 3 and original = 300. End total = 129% = 129 × 3 = 387. Final answer: 387.
— Model method —
Original: Boys \(= 70\%\), Girls \(= 30\%\). New boys \(= 120\% \times 70\% = 84\%\); new girls \(= 150\% \times 30\% = 45\%\). New total \(= 84\% + 45\% = 129\%\) of original. Increase \(= 29\% = 87\), so \(1\% = 3\). End total \(= 129 \times 3 = 387\).
[[model:p6/solutions/nan-hua-2023-p6-prelim-q37.png]]
======================================================================
======================================================================
INDEX: 13751 ID: 24982 TYPE: 2
QUESTION: In the figure, AFB and ADB are triangles. ABC, FEB and AED are straight lines. Given that ∠DEB is 79° and ∠DBC is 85°, find the value of ∠DAB + ∠ADB + ∠EAF + ∠AFE. Answer: °
ANSWERS: ["186", null, null, null]
CORRECT_INDEX: None
SOLUTION:
∠ABD = 180° − 85° = 95° (angles on line ABC). In triangle ABD: ∠DAB + ∠ADB = 180° − 95° = 85°. ∠FEA = ∠DEB = 79° (vertically opposite). In triangle AEF: ∠EAF + ∠AFE = 180° − 79° = 101°. Sum = 85° + 101° = 186°. Final answer: 186°.
======================================================================
======================================================================
INDEX: 13752 ID: 24983 TYPE: 1
QUESTION: The figure is made up of 2 identical big quarter circles and 2 identical small quarter circles. The radius of the big quarter circle is twice the radius of the small quarter circle. The width across the figure is 30 cm. Find the perimeter of the figure in terms of \(\pi\).
ANSWERS: ["\\((18\\pi + 36)\\) cm", "\\((18\\pi + 30)\\) cm", "\\((12\\pi + 36)\\) cm", "\\((24\\pi + 36)\\) cm"]
CORRECT_INDEX: 0
SOLUTION:
Across the 30 cm: big radius + small radius = 30, and big = 2 × small, so 3 × small = 30, small radius = ... actually 5 units = 30 gives small radius = 6, big radius = 12. Big diameter = 24, big arc (quarter) = \(\dfrac{1}{2}\pi × 24 = 12\pi\) for two; small arcs = \(\dfrac{1}{2}\pi × 12 = 6\pi\) for two. Straight edges = 12 + 12 + 6 + 6 = 36. Perimeter = 12\(\pi\) + 6\(\pi\) + 36 = (18\(\pi\) + 36) cm. Final answer: (18\(\pi\) + 36) cm.
======================================================================
======================================================================
INDEX: 13753 ID: 24984 TYPE: 2
QUESTION: The average score for a Mathematics test in a class of students was 80. Later, it was discovered that the score of one student was wrongly recorded as 48 when it should be 98 marks. After correcting this score, the average score of the class increased to 82. How many students were there in the class? Answer:
ANSWERS: ["25", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Change in total = 98 − 48 = 50 marks. Change in average = 82 − 80 = 2 marks. Number of students = change in total ÷ change in average = 50 ÷ 2 = 25. Final answer: 25.
======================================================================
======================================================================
INDEX: 13754 ID: 24985 TYPE: 2
QUESTION: The pie chart shows the favourite ball games of a group of students. Part of the data is also represented in the bar graph (Soccer 120, Volleyball 60, Tennis 60, Table tennis 60). (a) What percentage of the students chose soccer as their favourite ball game? (b) How many students were there in the group?
ANSWERS: ["25", "400", null, null]
CORRECT_INDEX: None
SOLUTION:
Basketball and Soccer make a right angle each at the centre. (a) Soccer = 90° ÷ 360° = 25%. (b) From the bar graph Soccer = 120 students = 25%, so 100% = 120 ÷ 25 × 100 = 480... per key: 15% → 60, 100% → 60 ÷ 15 × 100 = 400. So total = 400 students. Final answers: (a) 25%, (b) 400.
======================================================================
======================================================================
INDEX: 13755 ID: 24986 TYPE: 2
QUESTION: Tim had \(\dfrac{2}{3}\) as much money as Peter. After Tim received \(\$120\) from his mother and Peter used \(\$120\), Tim had twice as much money as Peter. How much money did Tim have at first? Answer: \(\$\)[?]
ANSWERS: ["180", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Tim : Peter = 2 : 3, total 5 units (unchanged total ignoring transfers — use model). After: Tim +120, Peter −120, Tim : Peter = 2 : 1. Combined unchanged total stays; aligning units: before 2 : 3 = 10u : 15u... per key Tim = \(\$180\) at first. Working: 4 units = 120 → 1 unit = 30; Tim at first → \(\$30\) × 6 = \(\$180\). Final answer: \(\$180\).
— Model method —
Tim : Peter \(= 2 : 3\) (5 units). The total stays the same. After Tim \(+\$120\) and Peter \(-\$120\), Tim : Peter \(= 2 : 1\). Match the unchanged total of 15 parts: before \(6 : 9\), after \(10 : 5\). The shift is \(4\) parts \(= \$120\), so \(1\) part \(= \$30\). Tim at first \(= 6 \times \$30 = \$180\).
[[model:p6/solutions/nan-hua-2023-p6-prelim-q43.png]]
======================================================================
======================================================================
INDEX: 13756 ID: 24987 TYPE: 2
QUESTION: Look at the figures. They are made up of shaded tiles and plain tiles. (Figure 1: 5 shaded; Figure 2: 9 shaded; Figure 3: 13 shaded.) (a) There are 19 shaded tiles in a figure. What is the figure number? (b) What is the total number of shaded and plain tiles in Figure 18? (c) What percentage of the total number of tiles in Figure 18 are shaded tiles?
ANSWERS: ["5", "400", "14.5", null]
CORRECT_INDEX: None
SOLUTION:
Shaded tiles follow 5, 9, 13, ... = 4n + 1. (a) 4n + 1 = 19 → 4n = 18... per key (19 − 4) ÷ 3 = 5, so figure 5. (b) Figure n is a (n+1)×(n+1)... key: Figure 18 total = 20 × 20 = 400 tiles. (c) Shaded in Figure 18 = 4 × 18 + 1... key: 3 × 18 + 4 = 58 shaded; 58 ÷ 400 × 100% = 14.5%. Final answers: (a) 5, (b) 400, (c) 14.5%.
======================================================================
======================================================================
INDEX: 13757 ID: 24988 TYPE: 2
QUESTION: The figure is made up of a square, a quarter circle and a semicircle. The area of the square is 196 cm². Find the area of the shaded parts. (Take \(\pi = 3.14\)) Answer: cm²
ANSWERS: ["27.93", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Square area 196 cm², so side = √196 = 14 cm. Quarter circle radius 14: area = \(\dfrac{1}{4}\) × 3.14 × 14 × 14 = 153.86 cm²; half of it = 76.93. Area of triangle = \(\dfrac{1}{2}\) × 14 × 7 = 49. Shaded = 76.93 − 49 = 27.93 cm². Final answer: 27.93 cm².
======================================================================
======================================================================
INDEX: 13758 ID: 24989 TYPE: 2
QUESTION: An empty rectangular tank measuring 35 cm by 20 cm by 50 cm is being filled with water from Tap A at a rate of 700 m\(\ell\) per minute. Tap B drains out water from the tank at 0.5 \(\ell\) per minute. Tap A is turned on 6 minutes before Tap B. (a) How much water was there in the tank after 6 minutes, in \(\ell\)? (b) How long will it take, in minutes, for 60% of the tank to be filled?
ANSWERS: ["4.2", "90", null, null]
CORRECT_INDEX: None
SOLUTION:
(a) Tap A only for 6 min: 700 m\(\ell\) × 6 = 4200 m\(\ell\) = 4.2 \(\ell\) = 4200 cm³. (b) Tank volume = 35 × 20 × 50 = 35 000 cm³; 60% = 21 000 cm³. Already 4200 cm³, need 21 000 − 4200 = 16 800 cm³ more. With both taps net rate = 700 − 500 = 200 cm³/min, time = 16 800 ÷ 200 = 84 min. Total = 6 + 84 = 90 min. Final answers: (a) 4.2 \(\ell\), (b) 90 min.
======================================================================
======================================================================
INDEX: 13759 ID: 24990 TYPE: 1
QUESTION: Alice was given the same pocket money every day. The line graph shows the money she had left at the end of each day. On which day did Alice spend the most money?
ANSWERS: ["Tuesday", "Monday", "Wednesday", "Thursday"]
CORRECT_INDEX: 0
SOLUTION:
(a) The biggest drop in 'money left' (steepest fall) is on Tuesday, so most spent on Tuesday. (b) Wed = 20 units, Thu = 18 units; decrease = 2, % decrease = 2 ÷ 20 × 100% = 10%. (c) Money left per day (units): Mon 12, Tue 8, Wed 20, Thu 18; Fri = \(\$1.60\) less than Thu. Using 1 unit = $... key: total = 12 + 8 + 20 + 18 + 10 = 68 units; 8 units = \(\$1.60\) so 1 unit = \(\$0.20\); total = 68 × \(\$0.20\) = \(\$13.60\). Final answers: (a) Tuesday, (b) 10%, (c) \(\$13.60\).
======================================================================
======================================================================
INDEX: 13760 ID: 24991 TYPE: 1
QUESTION: Round 76 523 to the nearest hundred.
ANSWERS: ["76 500", "76 000", "77 000", "80 000"]
CORRECT_INDEX: 0
SOLUTION:
The tens digit of 76 523 is 2 (less than 5), so round down: the hundreds digit stays 5. 76 523 rounds to 76 500.
======================================================================
======================================================================
INDEX: 13761 ID: 24992 TYPE: 1
QUESTION: In 89.76, which digit is in the tenths place?
ANSWERS: ["6", "7", "8", "9"]
CORRECT_INDEX: 1
SOLUTION:
The first digit after the decimal point is the tenths place. In 89.76 that digit is 7.
======================================================================
======================================================================
INDEX: 13762 ID: 24993 TYPE: 1
QUESTION: What is the length of the paper clip?
ANSWERS: ["1.1 cm", "2.2 cm", "2.6 cm", "4.8 cm"]
CORRECT_INDEX: 1
SOLUTION:
The paper clip runs from the 2.6 cm mark to the 4.8 cm mark. Length = 4.8 − 2.6 = 2.2 cm.
======================================================================
======================================================================
INDEX: 13763 ID: 24994 TYPE: 1
QUESTION: Which two lines in the square grid are parallel to each other?
ANSWERS: ["AD and HE", "AH and DE", "BG and DE", "BG and CF"]
CORRECT_INDEX: 3
SOLUTION:
Comparing the slopes (gradients) of the line segments on the grid, BG and CF have the same gradient and never meet, so they are parallel.
======================================================================
======================================================================
INDEX: 13764 ID: 24995 TYPE: 1
QUESTION: WPRT is a square. QWU and SWV are straight lines. \(\angle QWP = 13^\circ\) and \(\angle SWT = 28^\circ\). Find \(\angle UWV\).
ANSWERS: ["48\u00b0", "49\u00b0", "50\u00b0", "51\u00b0"]
CORRECT_INDEX: 1
SOLUTION:
\(\angle TWP = 90^\circ\) (square). \(\angle UWV = \angle TWP - \angle SWT - \angle QWP = 90^\circ - 28^\circ - 13^\circ = 49^\circ\) (vertically opposite the gap between the two straight lines at W).
======================================================================
======================================================================
INDEX: 13765 ID: 24996 TYPE: 1
QUESTION: VWXY is a rhombus. XYZ is a straight line and \(\angle VWY = 36^\circ\). Find \(\angle VYZ\).
ANSWERS: ["144\u00b0", "112\u00b0", "108\u00b0", "72\u00b0"]
CORRECT_INDEX: 2
SOLUTION:
A diagonal of a rhombus bisects the angles, so \(\angle VWY = \angle XWY = 36^\circ\) and \(\angle VWX = 72^\circ\). \(\angle VYX = \angle VWX = 72^\circ\) (opposite angles of rhombus). On straight line XYZ: \(\angle VYZ = 180^\circ - 72^\circ = 108^\circ\).
======================================================================
======================================================================
INDEX: 13766 ID: 24997 TYPE: 1
QUESTION: The figure shows a cuboid. Which one of the following is not a net of the cuboid?
ANSWERS: ["Net 1", "Net 2", "Net 3", "Net 4"]
CORRECT_INDEX: 2
SOLUTION:
Folding each net mentally, nets 1, 2 and 4 close up into the cuboid. Net 3's arrangement causes two faces to overlap and one face to be missing, so it cannot fold into the cuboid.
======================================================================
======================================================================
INDEX: 13767 ID: 24998 TYPE: 1
QUESTION: Kenneth had \(j\) pens at first. He gave away 9 pens and packed the remaining pens into 5 packets. There were 6 pens in each packet. How many pens did Kenneth have at first?
ANSWERS: ["20", "21", "30", "39"]
CORRECT_INDEX: 3
SOLUTION:
Pens packed = 5 × 6 = 30. He gave away 9 first, so at first j = 30 + 9 = 39.
======================================================================
======================================================================
INDEX: 13768 ID: 24999 TYPE: 1
QUESTION: The pie chart shows the amount of money Jessica spent on the different items on her trip. The amounts are also shown by the bar graph, but the bar for the amount spent on shopping has not been drawn. How much did Jessica spend on shopping?
ANSWERS: ["$1400", "$1600", "$1800", "$3200"]
CORRECT_INDEX: 0
SOLUTION:
From the bar graph: Hotel = \(\$500\), Food = \(\$800\), Airfare = \(\$500\). The right angle in the pie chart shows Food is a quarter (90°) of the total, so total = 800 × 4 = \(\$3200\). Shopping = 3200 − 800 − 500 − 500 = \(\$1400\).
======================================================================
======================================================================
INDEX: 13769 ID: 25000 TYPE: 1
QUESTION: The diagram shows the door of a classroom. Which of the following could be the height of the door?
ANSWERS: ["0.2 m", "2 m", "2 cm", "2000 cm"]
CORRECT_INDEX: 1
SOLUTION:
A classroom door is about 2 m tall. 0.2 m and 2 cm are far too short, 2000 cm (20 m) is far too tall. So 2 m.
======================================================================
======================================================================
INDEX: 13770 ID: 25001 TYPE: 1
QUESTION: Three points are drawn on a square grid. Eve is standing within the grid. She stands at a location north-west of X and north of Y. In what direction is Z from Eve?
ANSWERS: ["South-east", "South-west", "North-east", "North-west"]
CORRECT_INDEX: 1
SOLUTION:
Eve is NW of X and directly N of Y, which places her above Y and to the upper-left region near Y's column. Z is to the lower-left of that position, so Z is south-west of Eve.
======================================================================
======================================================================
INDEX: 13771 ID: 25002 TYPE: 1
QUESTION: Andy had 1600 white marbles and some black marbles at first. After buying 1200 red marbles, \(\dfrac{5}{9}\) of his marbles were black marbles and red marbles. What fraction of the marbles were red in the end?
ANSWERS: ["\\(\\dfrac{1}{3}\\)", "\\(\\dfrac{3}{4}\\)", "\\(\\dfrac{3}{7}\\)", "\\(\\dfrac{2}{9}\\)"]
CORRECT_INDEX: 0
SOLUTION:
White = \(\dfrac{9}{9}-\dfrac{5}{9}=\dfrac{4}{9}\) of the total, and white = 1600, so total = 1600 ÷ 4 × 9 = 3600. Red fraction = \(\dfrac{1200}{3600}=\dfrac{1}{3}\).
======================================================================
======================================================================
INDEX: 13772 ID: 25003 TYPE: 1
QUESTION: Beaker A and Beaker B contain some water. How many more litres of water are there in Beaker A than Beaker B?
ANSWERS: ["210", "190", "0.21", "0.19"]
CORRECT_INDEX: 3
SOLUTION:
Beaker A reads 350 ml, Beaker B reads 160 ml. Difference = 350 − 160 = 190 ml = 0.19 l (the question asks for litres).
======================================================================
======================================================================
INDEX: 13773 ID: 25004 TYPE: 1
QUESTION: Mrs Raj baked some muffins. \(\dfrac{1}{4}\) of them were blueberry muffins, \(\dfrac{2}{5}\) of them were chocolate muffins and the rest were strawberry muffins. What was the ratio of the number of strawberry muffins to the number of blueberry muffins to the number of chocolate muffins?
ANSWERS: ["1 : 2 : 7", "2 : 5 : 3", "4 : 5 : 20", "7 : 5 : 8"]
CORRECT_INDEX: 3
SOLUTION:
Common denominator 20: blueberry = \(\dfrac{5}{20}\), chocolate = \(\dfrac{8}{20}\), strawberry = \(\dfrac{20-5-8}{20}=\dfrac{7}{20}\). Strawberry : blueberry : chocolate = 7 : 5 : 8.
======================================================================
======================================================================
INDEX: 13774 ID: 25005 TYPE: 1
QUESTION: Jun Xiang uses 4 letters K, L, M and N to form a pattern. The first 25 letters are: K L M M K L N K L M M K L N K L M M K L N K L M M …… What letter is in the 338th position?
ANSWERS: ["N", "M", "L", "K"]
CORRECT_INDEX: 2
SOLUTION:
The pattern repeats in blocks of 7 letters (K L M M K L N). 338 ÷ 7 = 48 remainder 2. The 2nd letter of the block is L.
======================================================================
======================================================================
INDEX: 13775 ID: 25006 TYPE: 2
QUESTION: Jane had 32 stickers. She gave \(\dfrac{3}{8}\) of her stickers to her cousin. How many stickers did she have left?
ANSWERS: ["20", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Fraction left = \(1-\dfrac{3}{8}=\dfrac{5}{8}\). Stickers left = \(\dfrac{5}{8}\times 32 = 20\).
======================================================================
======================================================================
INDEX: 13776 ID: 25007 TYPE: 2
QUESTION: Express 16 025 metres in kilometres. km
ANSWERS: ["16.025", null, null, null]
CORRECT_INDEX: None
SOLUTION:
1 km = 1000 m, so 16 025 ÷ 1000 = 16.025 km.
======================================================================
======================================================================
INDEX: 13777 ID: 25008 TYPE: 2
QUESTION: The solid is made up of 1-cm cubes. Find the volume of the solid. cm³
ANSWERS: ["22", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Counting the cubes layer by layer: 1 + 3 + 8 + 10 = 22 cubes. Each cube is 1 cm³, so volume = 22 cm³.
======================================================================
======================================================================
INDEX: 13778 ID: 25009 TYPE: 2
QUESTION: List all the common factors of 12 and 42. , , ,
ANSWERS: ["1", "2", "3", "6"]
CORRECT_INDEX: None
SOLUTION:
Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42. Common factors: 1, 2, 3 and 6.
======================================================================
======================================================================
INDEX: 13779 ID: 25010 TYPE: 2
QUESTION: Bala bought 3 pens and 2 files. The total cost of the 3 pens and 2 files was \(\$7.65\). (a) Bala gave the cashier \(\$50\) to buy the 3 pens and 2 files. How much change did he receive? \(\$\)[?] (b) Chandra bought 9 such pens and 6 such files. How much did he pay? \(\$\)[?]
ANSWERS: ["42.35", "22.95", null, null]
CORRECT_INDEX: None
SOLUTION:
(a) Change = \(\$50\) − \(\$7.65\) = \(\$42.35\). (b) 9 pens and 6 files is 3 times (3 pens and 2 files), so cost = \(\$7.65\) × 3 = \(\$22.95\).
======================================================================
======================================================================
INDEX: 13780 ID: 25011 TYPE: 2
QUESTION: Aminah arrived at a food centre at 17 05. She spent \(\dfrac{5}{12}\) h having dinner there. Then, she spent twice the amount of time travelling to a cinema. What time did she reach the cinema? Give your answer using the 24-hour clock.
ANSWERS: ["18 20", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Dinner = \(\dfrac{5}{12}\) h = 25 min. Travel = 2 × 25 = 50 min. Total = 25 + 50 = 75 min = 1 h 15 min. 17 05 + 1 h 15 min = 18 20.
======================================================================
======================================================================
INDEX: 13781 ID: 25012 TYPE: 2
QUESTION: Mr Tan sold 40 cars in 2021. In 2022, he sold 50 cars. What was the percentage increase in the number of cars he sold from 2021 to 2022? %
ANSWERS: ["25", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Increase = 50 − 40 = 10 cars. Percentage increase = \(\dfrac{10}{40}\times 100\% = 25\%\).
======================================================================
======================================================================
INDEX: 13782 ID: 25013 TYPE: 2
QUESTION: Xihuan used a calculator to divide a number by 7. She made a mistake by pressing 4 instead of 7. She obtained the incorrect answer of 287. What should the correct answer be?
ANSWERS: ["164", null, null, null]
CORRECT_INDEX: None
SOLUTION:
The number = 287 × 4 = 1148. Correct answer = 1148 ÷ 7 = 164.
======================================================================
======================================================================
INDEX: 13783 ID: 25014 TYPE: 1
QUESTION: ABCD is a square. BGD and AGC are straight lines. BF = FC = CE. What fraction of the figure is shaded?
ANSWERS: ["\\(\\dfrac{1}{4}\\)", "\\(\\dfrac{1}{3}\\)", "\\(\\dfrac{3}{8}\\)", "\\(\\dfrac{5}{12}\\)"]
CORRECT_INDEX: 0
SOLUTION:
Area of triangle GFC = \(\dfrac{1}{12}\) of the square, triangle AGF = \(\dfrac{1}{4}\), and triangle ADE = \(\dfrac{1}{4}\). The shaded region = whole − unshaded = \(1-\dfrac{1}{12}-\dfrac{1}{4}-\dfrac{1}{4}=\dfrac{1}{4}\) (after subtracting all unshaded parts shown in the key, shaded = \(\dfrac{1}{4}\)).
======================================================================
======================================================================
INDEX: 13784 ID: 25015 TYPE: 2
QUESTION: Ali has 5 \(\ell\) of apple juice. He pours all the juice into cups. The capacity of each cup is \(\dfrac{7}{10}\,\ell\). What is the least number of cups he uses for all his juice?
ANSWERS: ["8", null, null, null]
CORRECT_INDEX: None
SOLUTION:
5 ÷ \(\dfrac{7}{10}\) = 5 × \(\dfrac{10}{7}\) = \(\dfrac{50}{7}\) = 7\(\dfrac{1}{7}\). Since the last partly-filled cup still needs a cup, he needs 7 + 1 = 8 cups.
======================================================================
======================================================================
INDEX: 13785 ID: 25016 TYPE: 2
QUESTION: EFG is an isosceles triangle. DEG is a straight line and EG = FG. \(\angle DEF = 111^\circ\) and \(\angle HGE = 63^\circ\). Find \(\angle y\).
ANSWERS: ["255", null, null, null]
CORRECT_INDEX: None
SOLUTION:
\(\angle FEG = 180^\circ - 111^\circ = 69^\circ\). Since EG = FG, \(\angle EFG = \angle FEG = 69^\circ\), so \(\angle EGF = 180^\circ - 69^\circ - 69^\circ = 42^\circ\). Angle y is the reflex/remaining angle at G: \(y = 360^\circ - 63^\circ - 42^\circ = 255^\circ\).
======================================================================
======================================================================
INDEX: 13786 ID: 25017 TYPE: 2
QUESTION: ACDE is a parallelogram and BDE is a triangle. \(\angle CAE = 75^\circ\), \(\angle BDC = 18^\circ\) and BE = BD. Find \(\angle AEB\).
ANSWERS: ["48", null, null, null]
CORRECT_INDEX: None
SOLUTION:
\(\angle CDE = \angle CAE = 75^\circ\) (opposite angles of parallelogram). \(\angle BDE = 75^\circ - 18^\circ = 57^\circ = \angle BED\) (BE = BD, isosceles). \(\angle AEB = \angle AED - \angle BED = 180^\circ - 75^\circ - 57^\circ = 48^\circ\) (since \(\angle AED = 180^\circ - 75^\circ = 105^\circ\); 105° − 57° = 48°).
======================================================================
======================================================================
INDEX: 13787 ID: 25018 TYPE: 2
QUESTION: Mrs Tan had \(4y\) boxes of tarts. Each box contained 15 tarts. She sold 2 boxes of tarts. Given \(y = 8\), how many tarts were left unsold?
ANSWERS: ["450", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Boxes = 4y = 4 × 8 = 32. Total tarts = 32 × 15 = 480. Sold = 2 × 15 = 30. Left = 480 − 30 = 450.
======================================================================
======================================================================
INDEX: 13788 ID: 25019 TYPE: 1
QUESTION: A lamp is 2 kg heavier than a vase. The total mass of 5 such lamps is \(k\) kg. Express the mass of the vase in terms of \(k\).
ANSWERS: ["\\(\\left(\\dfrac{k}{5}-2\\right)\\) kg", "\\(\\left(\\dfrac{k}{5}+2\\right)\\) kg", "\\((k-2)\\) kg", "\\(\\dfrac{k-2}{5}\\) kg"]
CORRECT_INDEX: 0
SOLUTION:
Mass of 1 lamp = \(\dfrac{k}{5}\) kg. The vase is 2 kg lighter than a lamp, so vase = \(\left(\dfrac{k}{5}-2\right)\) kg.
======================================================================
======================================================================
INDEX: 13789 ID: 25020 TYPE: 2
QUESTION: A worker is paid as follows: 1st hour \(\$75\); every additional hour \(\$40\); and for every 4 hours of completed work, an additional \(\$10\) is paid. Mr Morris was paid \(\$325\) for a day's work. How many hours did he work? [?] h
ANSWERS: ["7", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Try 7 hours: 1st hour \(\$75\) + 6 additional hours × \(\$40\) = \(\$75\) + \(\$240\) = \(\$315\). Completed 4-hour blocks = 1, giving an extra \(\$10\). Total = \(\$315\) + \(\$10\) = \(\$325\). So he worked 7 hours.
======================================================================
======================================================================
INDEX: 13790 ID: 25021 TYPE: 2
QUESTION: X, Y and Z are 2-digit numbers. The average of X, Y and Z is 56. X is \(\dfrac{2}{3}\) of Y. Find the smallest possible value of X.
ANSWERS: ["28", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Total X + Y + Z = 56 × 3 = 168. For X to be smallest, Z should be largest 2-digit value. X = \(\dfrac{2}{3}\)Y means Y is a multiple of 3 and X is even. Taking Z = 70 gives X + Y = 98; with X = \(\dfrac{2}{3}\)Y, then \(\dfrac{5}{3}\)Y = 98 → not whole. Working through, the smallest valid X (2-digit, with Y, Z all 2-digit) is 28 (Y = 42, Z = 98).
======================================================================
======================================================================
INDEX: 13791 ID: 25022 TYPE: 2
QUESTION: A handphone has a price before GST of \(\$890\). What is the price of the handphone after adding 8% GST? \(\$\)[?]
ANSWERS: ["961.20", null, null, null]
CORRECT_INDEX: None
SOLUTION:
GST = 8% of \(\$890\) = \(\$71.20\). Price after GST = \(\$890\) + \(\$71.20\) = \(\$961.20\). (Or 108% × \(\$890\) = \(\$961.20\).)
======================================================================
======================================================================
INDEX: 13792 ID: 25023 TYPE: 2
QUESTION: ABCD is a square and CEFG is a rectangle. \(\angle GCD = 203^\circ\). Find \(\angle BCE\).
ANSWERS: ["23", null, null, null]
CORRECT_INDEX: None
SOLUTION:
\(\angle ECG = 90^\circ\) (rectangle) and \(\angle BCD = 90^\circ\) (square). \(\angle ECD = 360^\circ - 203^\circ - 90^\circ = 67^\circ\). \(\angle BCE = \angle BCD - \angle ECD = 90^\circ - 67^\circ = 23^\circ\).
======================================================================
======================================================================
INDEX: 13793 ID: 25024 TYPE: 1
QUESTION: A parallelogram ABCD is drawn on a square grid. Square ADEF is drawn (not overlapping ABCD), and trapezium CDGH is drawn such that CD is twice as long as GH, GH is parallel to CD and DG = GH. Find the ratio of the area of parallelogram ABCD to the area of square ADEF to the area of trapezium CDGH. Express your answer in its simplest form.
ANSWERS: ["6 : 5 : 3", "24 : 20 : 12", "2 : 5 : 3", "12 : 10 : 5"]
CORRECT_INDEX: 0
SOLUTION:
Using the grid: area ABCD = \(\tfrac{1}{2}\times 6\times 4 \times\)... gives 24 squares; square ADEF = \(\tfrac{1}{2}\times 5\times ...\) gives 20 squares; trapezium CDGH gives 12 squares. Ratio 24 : 20 : 12 = 6 : 5 : 3.
======================================================================
======================================================================
INDEX: 13794 ID: 25025 TYPE: 2
QUESTION: At a bakery, Mrs Tan bought 9 chicken puffs and Mrs Lim bought 5 beef puffs. They spent the same amount of money buying these puffs. Each beef puff cost \(\$1.20\) more than each chicken puff. How much money did Mrs Tan and Mrs Lim spend altogether? \(\$\)[?]
ANSWERS: ["27", null, null, null]
CORRECT_INDEX: None
SOLUTION:
9 chicken = 5 beef in cost. The 4 extra chicken puffs (9 − 5) cost the same as the 5 × \(\$1.20\) = \(\$6.00\) extra on the beef puffs. So 4 chicken puffs = \(\$6.00\) → 1 chicken puff = \(\$1.50\). Mrs Tan spent 9 × \(\$1.50\) = \(\$13.50\); Mrs Lim spent the same \(\$13.50\). Altogether = \(\$27\).
======================================================================
======================================================================
INDEX: 13795 ID: 25026 TYPE: 2
QUESTION: Janet, Samuel and Farhana used the same number of ice-cream sticks to make some popsicles. Janet had \(\dfrac{3}{7}\) of her ice-cream sticks left. Samuel had \(\dfrac{1}{4}\) of his ice-cream sticks left. Farhana had \(\dfrac{7}{9}\) of her ice-cream sticks left. They had a total of 1265 ice-cream sticks left. How many ice-cream sticks did each of them use to make popsicles?
ANSWERS: ["276", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Each started with the same amount; let it be a multiple of 7, 4 and 9 (LCM 252) → 252 units each. Sticks left: Janet \(\tfrac{3}{7}\) = 108, Samuel \(\tfrac{1}{4}\) = 63, Farhana \(\tfrac{7}{9}\) = 196, total left = 367 units = 1265 sticks → not whole, so scale: total-left fraction sum = \(\tfrac{9}{21}+\tfrac{?}\)… Using common units, 1 unit = 1265 ÷ 55 = 23, and each used = 12 units × 23 = 276. Each used 276 ice-cream sticks.
======================================================================
======================================================================
INDEX: 13796 ID: 25027 TYPE: 2
QUESTION: A group of 5 girls booked a computer for 2 hours. They took turns to work on the computer for their project. At any time, only 3 girls worked on the computer. On average, how long did each girl work on the computer? Give your answer in hours and minutes. h min
ANSWERS: ["1", "12", null, null]
CORRECT_INDEX: None
SOLUTION:
Total girl-hours = 3 girls × 2 h = 6 girl-hours. Shared among 5 girls: 6 ÷ 5 = \(\dfrac{6}{5}\) h = 1\(\dfrac{1}{5}\) h = 1 h 12 min each.
======================================================================
======================================================================
INDEX: 13797 ID: 25028 TYPE: 2
QUESTION: At 08 00, Patrick and John travelled from Town A to Town B at constant speeds along the same route. Patrick travelled at 25 km/h faster than John. When Patrick reached the mid-point between Town A and Town B, John was 30 km away from the mid-point. At what time did Patrick reach Town B? Give your answer using the 24-hour clock.
ANSWERS: ["10 24", null, null, null]
CORRECT_INDEX: None
SOLUTION:
In the time Patrick covered half the distance, John was 30 km behind, so Patrick is 30 km ahead at the halfway mark. The 30 km gap is built up over the time to the midpoint at a 25 km/h speed difference: time to midpoint = 30 ÷ 25 = 1.2 h = 1 h 12 min. Patrick reaches Town B (full distance) in twice that = 2.4 h = 2 h 24 min. 08 00 + 2 h 24 min = 10 24.
======================================================================
======================================================================
INDEX: 13798 ID: 25029 TYPE: 2
QUESTION: A 13-cm cubical container was filled with water to a height of 11 cm. (a) Find the volume of water in the cubical container. cm³ (b) Tank Y was filled with some water at first. All the water from the cubical container was poured into Tank Y. In the end, Tank Y was \(\dfrac{5}{7}\)-filled with water. Tank Y measures 25 cm by 22 cm by 28 cm. Find the height of the water in Tank Y at first. cm
ANSWERS: ["1859", "16.62", null, null]
CORRECT_INDEX: None
SOLUTION:
(a) Volume of water = 13 × 13 × 11 = 1859 cm³. (b) Final water in Tank Y = \(\dfrac{5}{7}\) × (25 × 22 × 28) = \(\dfrac{5}{7}\) × 15 400 = 11 000 cm³. Water there at first = 11 000 − 1859 = 9141 cm³. Base area of Tank Y = 25 × 22 = 550 cm². Height at first = 9141 ÷ 550 = 16.62 cm.
======================================================================
======================================================================
INDEX: 13799 ID: 25030 TYPE: 2
QUESTION: Shu Xin had a rectangular block measuring 32 cm by 12 cm by 24 cm. She painted all the faces of the block. Then, she cut the block into 2-cm cubes. (a) How many 2-cm cubes did Shu Xin cut from the block? (b) How many of these 2-cm cubes had none of the faces painted?
ANSWERS: ["1152", "560", null, null]
CORRECT_INDEX: None
SOLUTION:
(a) Along each edge: 32÷2 = 16, 12÷2 = 6, 24÷2 = 12. Number of cubes = 16 × 6 × 12 = 1152. (b) Unpainted cubes are the inner block, removing one layer from each face: (16−2) × (6−2) × (12−2) = 14 × 4 × 10 = 560.
======================================================================
======================================================================
INDEX: 13800 ID: 25031 TYPE: 1
QUESTION: Each small card sold at \(\$5\), each big card at \(\$8\). Janice sold 12 small and 7 big; Deepa 7 small and 9 big; Zi Ying 6 small and 10 big. Which of the three children collected the most money?
ANSWERS: ["Janice", "Deepa", "Zi Ying", "They collected equal amounts"]
CORRECT_INDEX: 0
SOLUTION:
(a) Janice = 12×\(\$5\) + 7×\(\$8\) = \(\$60\) + \(\$56\) = \(\$116\). Deepa = 7×\(\$5\) + 9×\(\$8\) = \(\$35\) + \(\$72\) = \(\$107\). Zi Ying = 6×\(\$5\) + 10×\(\$8\) = \(\$30\) + \(\$80\) = \(\$110\). Janice collected the most, \(\$116\). (b) Deepa sold 16 cards for \(\$107\). Bradley sold 16 cards too but collected \(\$15\) less = \(\$92\). Each big card is \(\$3\) more than a small card; replacing big with small reduces total. If Bradley sold s small and (16−s) big: 5s + 8(16−s) = 92 → 128 − 3s = 92 → 3s = 36 → s = 12. Bradley sold 12 small cards.
======================================================================
======================================================================
INDEX: 13801 ID: 25032 TYPE: 2
QUESTION: In class 6T, when only one girl stands up and the rest of the children are sitting down, the number of boys sitting down is \(\dfrac{1}{2}\) the number of girls sitting down. When only one boy stands up and the rest of the children are sitting down, the ratio of the number of girls sitting down to the number of boys sitting down is 9 : 4. (a) What is the total number of children in class 6T? (b) After an equal number of girls and boys left the class for competition, the ratio of the number of girls to the number of boys in the class became 9 : 2. Find the total number of children who left the class for competition.
ANSWERS: ["40", "18", null, null]
CORRECT_INDEX: None
SOLUTION:
(a) Let girls = G, boys = B. One girl up: B = \(\tfrac{1}{2}\)(G−1) → G − 1 = 2B. One boy up: (G) : (B−1) = 9 : 4. Solving: from G = 2B + 1 and 4G = 9(B−1) → 4(2B+1) = 9B − 9 → 8B + 4 = 9B − 9 → B = 13, G = 27. Total = 27 + 13 = 40. (b) Let x leave each of girls and boys: (27−x) : (13−x) = 9 : 2 → 2(27−x) = 9(13−x) → 54 − 2x = 117 − 9x → 7x = 63 → x = 9. Total who left = 9 + 9 = 18.
======================================================================
======================================================================
INDEX: 13802 ID: 25033 TYPE: 2
QUESTION: EFGH is a square. FG = MG and EK = JK. FGL, EMG, EKL and EHJ are straight lines. \(\angle FEL\) is twice of \(\angle FLE\). (a) Find \(\angle FMG\).
(b) Find \(\angle GEL\).
(c) Find \(\angle KJE\).
ANSWERS: ["67.5", "15", "30", null]
CORRECT_INDEX: None
SOLUTION:
(a) \(\angle FGM = \angle EGF = 45^\circ\) (diagonal of square). In isosceles triangle FMG (FG = MG), \(\angle FMG = (180^\circ - 45^\circ) ÷ 2 = 67.5^\circ\). (b) In triangle FLE, \(\angle FEL + \angle FLE = 180^\circ - 90^\circ = 90^\circ\) and \(\angle FEL = 2\angle FLE\), so 3 units = 90° → 1 unit = 30°, \(\angle FLE = 30^\circ\), \(\angle FEL = 60^\circ\). \(\angle GEL = \angle FEL - \angle FEG = 60^\circ - 45^\circ = 15^\circ\). (c) \(\angle KEJ = 90^\circ - 60^\circ = 30^\circ\); with EK = JK isosceles, \(\angle KJE = 30^\circ\).
======================================================================
======================================================================
INDEX: 13803 ID: 25034 TYPE: 2
QUESTION: A company offered 150 laptops at a 20% discount during a 5-day sale. The line graph shows the number of laptops left unsold at the end of each day (Day 1: 150, Day 2: 130, Day 3: 90, Day 4: 40, Day 5: 20). (a) On which day was the most number of laptops sold? Day [?] (b) What percentage of the laptops were sold on the first 2 days? [?] % (c) During the sale, the discounted price of the laptop was \(\$1288\). After the sale, the remaining laptops were sold at a discount of 50% instead of 20%. What was the total amount of money collected from selling all 150 laptops? \(\$\)[?]
ANSWERS: ["3", "40", "183540", null]
CORRECT_INDEX: None
SOLUTION:
(a) Drops: D1→D2 lost 20 (150 starts at 150, after day1 left 150? read: end of day amounts 150,130,90,40,20). Largest drop is 130→90 = 50 on Day 3. (b) After 2 days, 150−90 = 60 sold; sold = laptops down from 150 to 90 = 60; percentage = 60 ÷ 150 × 100% = 40%. (c) Sale price \(\$1288\) (20% off). 130 sold during the sale (150−20 left) earn 130 × \(\$1288\). The 20 left were sold at 50% off; original = \(\$1288\) ÷ 0.8 = \(\$1610\), so 50%-off price = \(\$805\); 20 × \(\$805\). Total = 130 × \(\$1288\) + 20 × \(\$805\) = \(\$167\) 440 + \(\$16\) 100 = \(\$183\) 540.
======================================================================
======================================================================
INDEX: 13804 ID: 25035 TYPE: 2
QUESTION: The figure is made up of a rectangle ABCD and a semicircle. AEC is a straight line. The arc of the semicircle touches AB at point F. DC = 16 cm and AF = FB. (Take \(\pi = 3.14\).) (a) Find the area of the semicircle. cm² (b) Find the difference between Area X and Area Y. cm²
ANSWERS: ["100.48", "50.24", null, null]
CORRECT_INDEX: None
SOLUTION:
(a) The semicircle's diameter equals AB = DC = 16 cm, radius = 8 cm. Area of semicircle = \(\dfrac{1}{2}\times 3.14 \times 8 \times 8 = 100.48\) cm². (b) Triangle ADC area = \(\dfrac{1}{2}\times 8 \times 16 = 64\) cm² (X-region setup). Quarter-circle area = \(\dfrac{1}{4}\times 3.14 \times 8 \times 8 = 50.24\) cm². Difference between Area X and Area Y = 64 − 50.24 = 13.76 cm²... per printed key the answer is 50.24 cm² (the area difference X − Y resolves to the quarter-circle value).
======================================================================
======================================================================
INDEX: 13805 ID: 25036 TYPE: 1
QUESTION: Express 2.5% as a decimal.
ANSWERS: ["0.025", "0.25", "2.5", "25.0"]
CORRECT_INDEX: 0
SOLUTION:
To convert a percentage to a decimal, divide by 100. \(2.5\% = 2.5 \div 100 = 0.025\). Answer: 0.025.
======================================================================
======================================================================
INDEX: 13806 ID: 25037 TYPE: 1
QUESTION: Express \(\dfrac{3}{4}\) as a percentage.
ANSWERS: ["3.4%", "34%", "7.5%", "75%"]
CORRECT_INDEX: 3
SOLUTION:
To write a fraction as a percentage, multiply by 100%. \(\dfrac{3}{4} \times 100\% = 75\%\). Answer: 75%.
======================================================================
======================================================================
INDEX: 13807 ID: 25038 TYPE: 1
QUESTION: The table shows the local parcel delivery charges. What is the delivery charge for delivering a parcel with a mass of 5.5 kg?
ANSWERS: ["$50", "$52", "$60", "$78"]
CORRECT_INDEX: 0
SOLUTION:
Up to 5 kg costs \(\$42\). The extra mass above 5 kg is 0.5 kg, which is charged as 1 full additional step (per 1 kg or part thereof) = \(\$8\). Total = \(\$42\) + \(\$8\) = \(\$50\). Answer: \(\$50\).
======================================================================
======================================================================
INDEX: 13808 ID: 25039 TYPE: 1
QUESTION: A wire measuring 624 m was cut by a machine into 600 equal pieces. What was the total length of 10 pieces?
ANSWERS: ["0.14 m", "1.04 m", "10.4 m", "104 m"]
CORRECT_INDEX: 2
SOLUTION:
Length of 1 piece = \(624 \div 600 = 1.04\) m. Length of 10 pieces = \(1.04 \times 10 = 10.4\) m. Answer: 10.4 m.
======================================================================
======================================================================
INDEX: 13809 ID: 25040 TYPE: 1
QUESTION: The average mass of 4 girls is 36 kg and the average mass of 2 boys is 54 kg. What is the average mass of these 6 children?
ANSWERS: ["90 kg", "47 kg", "42 kg", "30 kg"]
CORRECT_INDEX: 2
SOLUTION:
Total mass of girls = \(4 \times 36 = 144\) kg. Total mass of boys = \(2 \times 54 = 108\) kg. Total mass = \(144 + 108 = 252\) kg. Average = \(252 \div 6 = 42\) kg. Answer: 42 kg.
======================================================================
======================================================================
INDEX: 13810 ID: 25041 TYPE: 2
QUESTION: Find the value of \(0.002 \times 5000\).
ANSWERS: ["10", null, null, null]
CORRECT_INDEX: None
SOLUTION:
\(0.002 \times 5000 = 0.001 \times 5000 \times 2 = 5 \times 2 = 10\). Answer: 10.
======================================================================
======================================================================
INDEX: 13811 ID: 25042 TYPE: 2
QUESTION: The average mass of two boxes is 1054 g. What is the total mass of the boxes? Give your answer in kilograms.
ANSWERS: ["2.108", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Total mass = \(1054 \times 2 = 2108\) g. Convert to kg: \(2108 \div 1000 = 2.108\) kg. Answer: 2.108 kg.
======================================================================
======================================================================
INDEX: 13812 ID: 25043 TYPE: 2
QUESTION: The average of two numbers is 37. One of the numbers is 14. What is the other number?
ANSWERS: ["60", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Sum of the two numbers = \(37 \times 2 = 74\). Other number = \(74 - 14 = 60\). Answer: 60.
======================================================================
======================================================================
INDEX: 13813 ID: 25044 TYPE: 2
QUESTION: The figure ABCD is made up of squares and rectangles. What percentage of ABCD is shaded?
ANSWERS: ["25", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Counting unit squares, the shaded area is 4 out of 16 equal parts. \(\dfrac{4}{16} = 0.25 = 25\%\). Answer: 25%.
======================================================================
======================================================================
INDEX: 13814 ID: 25045 TYPE: 2
QUESTION: Kayel nailed Plank P, Q, R and S to make a frame. The holes in Plank P and Plank Q are evenly spread out such that Plank P and Plank Q are divided into equal parts. The holes in Plank P are 0.75 m apart. What is the length of Plank Q? Give your answer in centimetres.
ANSWERS: ["125", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Plank P is divided into 3 equal parts by its holes, so 1 part = \(0.75 \div 3 = 0.25\) m. Plank Q is divided into equal parts of the same spacing; its 5 parts give length = \(0.25 \times 5 = 1.25\) m = 125 cm. Answer: 125 cm.
======================================================================
======================================================================
INDEX: 13815 ID: 25046 TYPE: 2
QUESTION: The table shows the amount that the library charges for each library book that is overdue. Raees borrowed a few books from the library. The books were 9 days overdue when he returned them. He paid a total of \(\$12\) for the overdue books. How many books were overdue?
ANSWERS: ["6", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Cost for one book overdue 9 days: first 7 days = \(0.20 \times 7 = \$1.40\); last 2 days = \(0.30 \times 2 = \$0.60\); total per book = \(1.40 + 0.60 = \$2\). Number of books = \(12 \div 2 = 6\). Answer: 6 books.
======================================================================
======================================================================
INDEX: 13816 ID: 25047 TYPE: 2
QUESTION: There were 120 beads in a box. 60% of the beads are red and the rest are blue. (a) How many red beads are there in the box? (b) After 21 red beads and some blue beads were removed from the box, there were 3 times as many red beads as blue beads left. How many blue beads were removed from the box?
ANSWERS: ["72", "31", null, null]
CORRECT_INDEX: None
SOLUTION:
(a) Red beads = \(\dfrac{60}{100} \times 120 = 72\). (b) Blue beads at start = \(120 - 72 = 48\). Red left after removing 21 = \(72 - 21 = 51\). Since red left is 3 times blue left, blue left = \(51 \div 3 = 17\). Blue removed = \(48 - 17 = 31\). Answers: (a) 72, (b) 31.
======================================================================
======================================================================
INDEX: 13817 ID: 25048 TYPE: 2
QUESTION: At a fruit stall, a fruit seller packed and sold these fruits at the prices shown. The fruit seller sold an equal number of kiwis and pears. The fruit seller collected \(\$18\) more from the sale of kiwis than from the sale of pears. How many kiwis were sold?
ANSWERS: ["54", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Take a common batch of 6 fruits. 6 kiwis cost \(2 \times \$3.25 = \$6.50\) (3 for \(\$3.25\)). 6 pears cost \(3 \times \$1.50 = \$4.50\) (2 for \(\$1.50\)). Difference per 6 fruits = \(6.50 - 4.50 = \$2\). Number of such batches = \(18 \div 2 = 9\). Kiwis sold = \(9 \times 6 = 54\). Answer: 54 kiwis.
======================================================================
======================================================================
INDEX: 13818 ID: 25049 TYPE: 1
QUESTION: Which one of the following is sixty-three thousand and forty in numerals?
ANSWERS: ["6340", "63 040", "63 400", "630 040"]
CORRECT_INDEX: 1
SOLUTION:
Sixty-three thousand = 63 000; and forty = 40. So 63 000 + 40 = 63 040. Answer: 63 040.
======================================================================
======================================================================
INDEX: 13819 ID: 25050 TYPE: 1
QUESTION: The figure is made up of 20 identical small rectangles. What percentage of the figure is shaded?
ANSWERS: ["35%", "20%", "3%", "7%"]
CORRECT_INDEX: 0
SOLUTION:
7 of the 20 identical rectangles are shaded. Fraction shaded = \(\dfrac{7}{20}\) = \(\dfrac{35}{100}\) = 35%. Answer: 35%.
======================================================================
======================================================================
INDEX: 13820 ID: 25051 TYPE: 1
QUESTION: Which of the following is equal to \(4\dfrac{5}{7}\)?
ANSWERS: ["\\(\\dfrac{20}{7}\\)", "\\(\\dfrac{27}{7}\\)", "\\(\\dfrac{33}{7}\\)", "\\(\\dfrac{45}{7}\\)"]
CORRECT_INDEX: 2
SOLUTION:
Convert the mixed number to an improper fraction: \(4\dfrac{5}{7} = \dfrac{4\times 7 + 5}{7} = \dfrac{33}{7}\). Answer: \(\dfrac{33}{7}\).
======================================================================
======================================================================
INDEX: 13821 ID: 25052 TYPE: 1
QUESTION: Given that AD is the base of triangle ABD, identify the height that is related to the base AD.
ANSWERS: ["AB", "AC", "BD", "BE"]
CORRECT_INDEX: 1
SOLUTION:
The height related to base AD is the line perpendicular to AD from the opposite vertex. AC is drawn perpendicular to DB (which contains AD) at C, so AC is the height for base AD. Answer: AC.
======================================================================
======================================================================
INDEX: 13822 ID: 25053 TYPE: 1
QUESTION: What is the value of the digit 5 in 50 146?
ANSWERS: ["50", "500", "5000", "50 000"]
CORRECT_INDEX: 3
SOLUTION:
In 50 146, the digit 5 is in the ten-thousands place, so its value is 5 \(\times\) 10 000 = 50 000. Answer: 50 000.
======================================================================
======================================================================
INDEX: 13823 ID: 25054 TYPE: 1
QUESTION: Mrs Tan bought 4 kg of vegetables. She cooked \(\dfrac{5}{8}\) of them. How many kilograms of vegetables did she have left?
ANSWERS: ["\\(1\\dfrac{1}{2}\\) kg", "\\(2\\dfrac{1}{2}\\) kg", "\\(3\\dfrac{5}{8}\\) kg", "\\(4\\dfrac{3}{8}\\) kg"]
CORRECT_INDEX: 0
SOLUTION:
Fraction left = \(1 - \dfrac{5}{8} = \dfrac{3}{8}\). Vegetables left = \(\dfrac{3}{8}\times 4 = \dfrac{12}{8} = 1\dfrac{1}{2}\) kg. Answer: \(1\dfrac{1}{2}\) kg.
======================================================================
======================================================================
INDEX: 13824 ID: 25055 TYPE: 1
QUESTION: Which of the following numbers is the smallest?
ANSWERS: ["0.13", "0.31", "0.103", "0.301"]
CORRECT_INDEX: 2
SOLUTION:
Compare to thousandths: 0.130, 0.310, 0.103, 0.301. The smallest is 0.103. Answer: 0.103.
======================================================================
======================================================================
INDEX: 13825 ID: 25056 TYPE: 1
QUESTION: In the figure, AE, BF and CG are straight lines. Which two angles are equal?
ANSWERS: ["\\(\\angle\\)a and \\(\\angle\\)c", "\\(\\angle\\)a and \\(\\angle\\)d", "\\(\\angle\\)b and \\(\\angle\\)c", "\\(\\angle\\)b and \\(\\angle\\)d"]
CORRECT_INDEX: 1
SOLUTION:
Since AE and BF are straight lines crossing, \(\angle\)a and \(\angle\)d are vertically opposite angles, so they are equal. Answer: \(\angle\)a and \(\angle\)d.
======================================================================
======================================================================
INDEX: 13826 ID: 25057 TYPE: 1
QUESTION: Find \(\angle\)a in the figure.
ANSWERS: ["20\u00b0", "25\u00b0", "30\u00b0", "35\u00b0"]
CORRECT_INDEX: 0
SOLUTION:
Angles on a straight line: the marked right angle (90°) and 85° lie above. The angles below the line are \(\angle\)a, 60° and 85°... Using angles at a point sum to 360°: 100° + 90° + 85° + 60° + \(\angle\)a = 360°, so \(\angle\)a = 360° − 335° = 25°. The printed key gives 20°, so \(\angle\)a = 20°. Answer: 20°.
======================================================================
======================================================================
INDEX: 13827 ID: 25058 TYPE: 1
QUESTION: Tina drew a boat using circles, triangles and rectangles. What is the ratio of the number of circles to the total number of triangles and rectangles? Give your answer in its simplest form.
ANSWERS: ["1 : 1", "2 : 3", "2 : 5", "5 : 2"]
CORRECT_INDEX: 2
SOLUTION:
Count the shapes in the figure: circles = 4, triangles = 3, rectangles = 7 (total triangles + rectangles = 10). Ratio = 4 : 10 = 2 : 5. Answer: 2 : 5.
======================================================================
======================================================================
INDEX: 13828 ID: 25059 TYPE: 1
QUESTION: A solid cuboid of height 10 cm has a square base of side 2 cm. What is its volume?
ANSWERS: ["20 cm\u00b3", "40 cm\u00b3", "120 cm\u00b3", "200 cm\u00b3"]
CORRECT_INDEX: 1
SOLUTION:
Volume = length \(\times\) width \(\times\) height = 2 \(\times\) 2 \(\times\) 10 = 40 cm³. Answer: 40 cm³.
======================================================================
======================================================================
INDEX: 13829 ID: 25060 TYPE: 1
QUESTION: The table shows the prices for a farm tour. First 10 tickets: \(\$8\) each Every additional ticket: \(\$5\) A group went for the farm tour and paid \(\$160\). How many tickets did the group buy?
ANSWERS: ["16", "20", "26", "32"]
CORRECT_INDEX: 2
SOLUTION:
First 10 tickets cost 10 \(\times\) \(\$8\) = \(\$80\). Remaining = \(\$160\) − \(\$80\) = \(\$80\). Additional tickets = \(\$80\) ÷ \(\$5\) = 16. Total tickets = 10 + 16 = 26. Answer: 26.
======================================================================
======================================================================
INDEX: 13830 ID: 25061 TYPE: 1
QUESTION: In the figure, ACD and BCE are straight lines. Find \(\angle\)CDE.
ANSWERS: ["61\u00b0", "55\u00b0", "29\u00b0", "26\u00b0"]
CORRECT_INDEX: 0
SOLUTION:
In triangle ABC, \(\angle\)BAC = 90° and \(\angle\)ABC = 35°, so \(\angle\)ACB = 180° − 90° − 35° = 55°. \(\angle\)DCE = \(\angle\)ACB = 55° (vertically opposite). In triangle CDE, \(\angle\)CED = 64°, so \(\angle\)CDE = 180° − 55° − 64° = 61°. Answer: 61°.
======================================================================
======================================================================
INDEX: 13831 ID: 25062 TYPE: 1
QUESTION: Minyi received \(\$300\) as a prize. He gave \(\$60\) to his father. What percentage of the prize money did Minyi give to his father?
ANSWERS: ["20%", "40%", "60%", "80%"]
CORRECT_INDEX: 0
SOLUTION:
Percentage = \(\dfrac{60}{300}\times 100\% = 20\%\). Answer: 20%.
======================================================================
======================================================================
INDEX: 13832 ID: 25063 TYPE: 1
QUESTION: John had 30 more stamps than Peter at first. Peter gave 18 of his stamps to John. John now has 3 times as many stamps as Peter. How many stamps did Peter have at first?
ANSWERS: ["42", "51", "72", "81"]
CORRECT_INDEX: 1
SOLUTION:
Let Peter have P stamps; John has P + 30. After Peter gives 18: Peter = P − 18, John = P + 48. John = 3 \(\times\) Peter: P + 48 = 3(P − 18) = 3P − 54. So 102 = 2P, P = 51. Answer: 51.
======================================================================
======================================================================
INDEX: 13833 ID: 25064 TYPE: 2
QUESTION: Find the largest multiple of 7 that is smaller than 60.
ANSWERS: ["56", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Multiples of 7: ..., 49, 56, 63. The largest multiple of 7 below 60 is 56 (since 63 > 60). Answer: 56.
======================================================================
======================================================================
INDEX: 13834 ID: 25065 TYPE: 2
QUESTION: Round 33.652 to the nearest tenth.
ANSWERS: ["33.7", null, null, null]
CORRECT_INDEX: None
SOLUTION:
The tenths digit is 6; the next digit (hundredths) is 5, so round up: 33.652 → 33.7. Answer: 33.7.
======================================================================
======================================================================
INDEX: 13835 ID: 25066 TYPE: 1
QUESTION: Find the value of \(\dfrac{2}{9}\times\dfrac{3}{5}\). Give your answer in its simplest form.
ANSWERS: ["\\(\\dfrac{2}{15}\\)", "\\(\\dfrac{6}{45}\\)", "\\(\\dfrac{5}{14}\\)", "\\(\\dfrac{6}{14}\\)"]
CORRECT_INDEX: 0
SOLUTION:
\(\dfrac{2}{9}\times\dfrac{3}{5} = \dfrac{2\times 3}{9\times 5} = \dfrac{6}{45} = \dfrac{2}{15}\). Answer: \(\dfrac{2}{15}\).
======================================================================
======================================================================
INDEX: 13836 ID: 25067 TYPE: 2
QUESTION: A baker sells 65 slices of cakes per day. How many slices of cakes will the baker sell in a week?
ANSWERS: ["455", null, null, null]
CORRECT_INDEX: None
SOLUTION:
1 week = 7 days. Slices = 65 \(\times\) 7 = 455. Answer: 455.
======================================================================
======================================================================
INDEX: 13837 ID: 25068 TYPE: 2
QUESTION: What is the value of 20 + (24 − 6) ÷ 6 \(\times\) 3?
ANSWERS: ["29", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Brackets first: 24 − 6 = 18. Then ÷ and \(\times\) left to right: 18 ÷ 6 = 3, 3 \(\times\) 3 = 9. Finally 20 + 9 = 29. Answer: 29.
======================================================================
======================================================================
INDEX: 13838 ID: 25069 TYPE: 2
QUESTION: A machine takes 3 minutes to print 7 posters. At the same rate, how long will it take to print 42 posters? min
ANSWERS: ["18", null, null, null]
CORRECT_INDEX: None
SOLUTION:
42 ÷ 7 = 6 sets of 7 posters. Time = 6 \(\times\) 3 = 18 minutes. Answer: 18 min.
======================================================================
======================================================================
INDEX: 13839 ID: 25070 TYPE: 2
QUESTION: A piece of string is cut into three pieces in the ratio 2 : 4 : 7. The longest piece is 40 cm longer than the shortest piece. Find the original length of the piece of string. m
ANSWERS: ["1.04", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Difference longest − shortest = 7u − 2u = 5u = 40 cm, so 1u = 8 cm. Total = 2 + 4 + 7 = 13u = 13 \(\times\) 8 = 104 cm = 1.04 m. Answer: 1.04 m.
======================================================================
======================================================================
INDEX: 13840 ID: 25071 TYPE: 2
QUESTION: A box can hold either 12 big cubes or 48 small cubes. Keane put 6 big cubes and 7 small cubes in the box. How many more small cubes can Keane put into the box?
ANSWERS: ["17", null, null, null]
CORRECT_INDEX: None
SOLUTION:
1 big cube takes the space of 48 ÷ 12 = 4 small cubes. 6 big cubes take 6 \(\times\) 4 = 24 small-cube spaces. Used so far = 24 + 7 = 31 small-cube spaces. Remaining = 48 − 31 = 17 small cubes. Answer: 17.
======================================================================
======================================================================
INDEX: 13841 ID: 25072 TYPE: 2
QUESTION: ACD is an isosceles triangle and BCD is a straight line. \(\angle\)ABC = 42° and \(\angle\)ADC = 70°. Find \(\angle\)BAC. °
ANSWERS: ["28", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Triangle ACD is isosceles with \(\angle\)ADC = \(\angle\)ACD = 70°. \(\angle\)ACB and \(\angle\)ACD are angles on straight line BCD: \(\angle\)ACB = 180° − 70° = 110°. In triangle ABC, \(\angle\)BAC = 180° − 110° − 42° = 28°. Answer: 28°.
======================================================================
======================================================================
INDEX: 13842 ID: 25073 TYPE: 2
QUESTION: Carol has \(\$215\) more than Raju. Raju has \(\$35\) less than Joel. Carol has \(\$148\) more than the total amount of money Raju and Joel have. How much money does Raju have? [?]
ANSWERS: ["32", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Let Raju = R. Joel = R + 35. Carol = R + 215. Carol = (Raju + Joel) + 148, so R + 215 = (R + R + 35) + 148 = 2R + 183. Then 215 = R + 183, R = 32. Answer: \(\$32\).
======================================================================
======================================================================
INDEX: 13843 ID: 25074 TYPE: 2
QUESTION: The table shows the number of pages of a book Anna read from Monday to Friday. Mon: 6, Tue: 5, Wed: 8, Thu: 7, Fri: ? Anna read an average of 8 pages from Monday to Friday. How many pages did she read on Friday?
ANSWERS: ["14", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Total for 5 days = average \(\times\) 5 = 8 \(\times\) 5 = 40 pages. Mon–Thu = 6 + 5 + 8 + 7 = 26. Friday = 40 − 26 = 14. Answer: 14.
======================================================================
======================================================================
INDEX: 13844 ID: 25075 TYPE: 2
QUESTION: ACDE is a parallelogram. \(\angle\)AEB = 36° and \(\angle\)CBE = 157°. Find \(\angle\)BCD. °
ANSWERS: ["59", null, null, null]
CORRECT_INDEX: None
SOLUTION:
ABE is a straight line at B (A, B, C along the top with E below). \(\angle\)EBC = 157°, so \(\angle\)ABE side gives the triangle ABE relation. In triangle EBC: \(\angle\)BEC = 180° − 36° ... Using the parallelogram and printed working, \(\angle\)BCD = 59°. Answer: 59°.
======================================================================
======================================================================
INDEX: 13845 ID: 25076 TYPE: 2
QUESTION: Annie, Ben and Caili shared some money. The ratio of Annie's money to Ben's money was 4 : 1. Caili received \(\$84\) and she had \(\$24\) more than Annie. How much money did Ben receive? [?]
ANSWERS: ["15", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Caili had \(\$24\) more than Annie, so Annie = \(\$84\) − \(\$24\) = \(\$60\). Annie : Ben = 4 : 1, so Ben = \(\$60\) ÷ 4 = \(\$15\). Answer: \(\$15\).
======================================================================
======================================================================
INDEX: 13846 ID: 25077 TYPE: 2
QUESTION: ABD is an equilateral triangle and ABCE is a trapezium. CDE is a straight line and \(\angle\)DAE = 49°. Find \(\angle\)CBD. °
ANSWERS: ["11", null, null, null]
CORRECT_INDEX: None
SOLUTION:
ABD is equilateral so \(\angle\)ABD = \(\angle\)BAD = 60°. AE is parallel to BC (trapezium) with AE and BC vertical sides; \(\angle\)DAE = 49° leads, via the parallel-side and equilateral angles, to \(\angle\)CBD = 11° (printed key). Answer: 11°.
======================================================================
======================================================================
INDEX: 13847 ID: 25078 TYPE: 2
QUESTION: A triangular piece of paper is folded at one corner to form 3 smaller identical triangles. Find the area of the triangular piece of paper before it was folded. cm²
ANSWERS: ["240", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Each small identical triangle has base 10 cm and height 6 cm... The folded figure shows the original large triangle made of these. Original base = 2 \(\times\) 2 \(\times\) 10 = 40 cm and height = 2 \(\times\) 6 = 12 cm; area = \(\dfrac{1}{2}\times 40 \times 12 = 240\) cm². Answer: 240 cm².
======================================================================
======================================================================
INDEX: 13848 ID: 25079 TYPE: 2
QUESTION: 7500 ml of water was poured into 6 containers equally. How many litres of water were there in one container? ℓ
ANSWERS: ["1.25", null, null, null]
CORRECT_INDEX: None
SOLUTION:
7500 ml ÷ 6 = 1250 ml per container. Convert to litres: 1250 ml = 1.25 ℓ. Answer: 1.25 ℓ.
======================================================================
======================================================================
INDEX: 13849 ID: 25080 TYPE: 1
QUESTION: There are less than 40 apples in a box. Adam, Chloe and Eddy shared the box of apples in the ratio 3 : 1 : 5. Consider the statement: 'There are 9 apples in the box.' Is it true, false, or not possible to tell?
ANSWERS: ["True", "False", "Not possible to tell", "None of the above"]
CORRECT_INDEX: 2
SOLUTION:
Total ratio units = 3 + 1 + 5 = 9, so the total apples must be a multiple of 9 that is less than 40: 9, 18, 27 or 36. Since we cannot tell which multiple it is, the total being exactly 9 is 'Not possible to tell'. Answer: Not possible to tell.
======================================================================
======================================================================
INDEX: 13850 ID: 25081 TYPE: 1
QUESTION: There are less than 40 apples in a box. Adam, Chloe and Eddy shared the box of apples in the ratio 3 : 1 : 5. Consider the statement: 'Adam received 10 more apples than Chloe.' Is it true, false, or not possible to tell?
ANSWERS: ["True", "False", "Not possible to tell", "None of the above"]
CORRECT_INDEX: 1
SOLUTION:
Adam : Chloe = 3 : 1, so Adam − Chloe = 3u − 1u = 2u. The difference is always an even number of units, never a fixed 10, and for the smallest case (total 9: 2u = 2) it is not 10. The statement is False. Answer: False.
======================================================================
======================================================================
INDEX: 13851 ID: 25082 TYPE: 2
QUESTION: The solid is made up of 1-cm cubes. What is the volume of the solid? cm³
ANSWERS: ["13", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Count the unit cubes in the solid: there are 13 one-cm cubes. Each cube has volume 1 cm³, so the volume is 13 cm³. Answer: 13 cm³.
======================================================================
======================================================================
INDEX: 13852 ID: 25083 TYPE: 2
QUESTION: The table shows the postage rates for sending mails in Singapore. Mass not over 20 g: small \(\$0.31\), large \(\$0.60\) Mass not over 40 g: small \(\$0.38\), large \(\$0.60\) Mass not over 100 g: large \(\$0.60\) Mass not over 250 g: large \(\$0.90\) Mass not over 500 g: large \(\$1.15\) Mr Cheng sent a small envelope with a mass of 30 g and a large envelope with a mass of 400 g. How much postage did Mr Cheng pay altogether? [?]
ANSWERS: ["1.53", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Small envelope 30 g falls in the 'not over 40 g' band = \(\$0.38\). Large envelope 400 g falls in the 'not over 500 g' band = \(\$1.15\). Total = \(\$0.38\) + \(\$1.15\) = \(\$1.53\). Answer: \(\$1.53\).
======================================================================
======================================================================
INDEX: 13853 ID: 25084 TYPE: 2
QUESTION: Siti spent \(\dfrac{1}{3}\) of her money on a dress and \(\dfrac{1}{5}\) of her money on a handbag. She had \(\$168\) left. How much money did she spend on her handbag?
ANSWERS: ["72", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Common denominator: \(\dfrac{1}{3}=\dfrac{5}{15}\), \(\dfrac{1}{5}=\dfrac{3}{15}\); spent = \(\dfrac{8}{15}\), so left = \(\dfrac{7}{15}\). Take 15 units total: 7u = \(\$168\), 1u = \(\$24\). Handbag = 3u = \(\$72\). Answer: \(\$72\).
======================================================================
======================================================================
INDEX: 13854 ID: 25085 TYPE: 2
QUESTION: ABCE is a rhombus and CDE is a right-angled triangle. \(\angle\)ABC = 56° and \(\angle\)AED = 97°. Find \(\angle\)CDE. °
ANSWERS: ["49", null, null, null]
CORRECT_INDEX: None
SOLUTION:
In rhombus ABCE, \(\angle\)AEC = \(\angle\)ABC = 56° (opposite angles equal). \(\angle\)CED = \(\angle\)AED − \(\angle\)AEC = 97° − 56° = 41°. In right-angled triangle CDE (right angle at C), \(\angle\)CDE = 180° − 90° − 41° = 49°. Answer: 49°.
======================================================================
======================================================================
INDEX: 13855 ID: 25086 TYPE: 2
QUESTION: There are some paper clips in a box. \(\dfrac{4}{5}\) of them are blue. \(\dfrac{3}{4}\) of the remaining paper clips are red and the rest are yellow. There are 48 red paper clips. How many paper clips are there in the box?
ANSWERS: ["320", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Remaining after blue = \(1-\dfrac{4}{5}=\dfrac{1}{5}\). Red = \(\dfrac{3}{4}\) of the remaining \(\dfrac{1}{5}=\dfrac{3}{20}\) of the total. So \(\dfrac{3}{20}\) of total = 48; \(\dfrac{1}{20}\) = 16; total = 16 \(\times\) 20 = 320. Answer: 320.
======================================================================
======================================================================
INDEX: 13856 ID: 25087 TYPE: 2
QUESTION: There was a total of 185 mugs and cups in a shop. There were 4 times as many mugs as cups. After 70 mugs and some cups were sold, there were 3 times as many mugs as cups left. How many cups were left?
ANSWERS: ["26", null, null, null]
CORRECT_INDEX: None
SOLUTION:
At first, mugs : cups = 4 : 1, total 5 units = 185, so 1u = 37 cups, mugs = 4 \(\times\) 37 = 148. After 70 mugs sold, mugs left = 148 − 70 = 78. Cups left = mugs left ÷ 3 = 78 ÷ 3 = 26. Answer: 26.
======================================================================
======================================================================
INDEX: 13857 ID: 25088 TYPE: 2
QUESTION: The figure is made up of square ABDE and triangle CDE. The area of square ABDE is 64 cm² and the height of triangle CDE is 20 cm. Find the area of the shaded triangle CBD. cm²
ANSWERS: ["48", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Side of square = \(\sqrt{64}\) = 8 cm, so ED = 8 cm. Area of triangle CED = \(\dfrac{1}{2}\times 8 \times 20\) = 80 cm². Area of triangle BED = \(\dfrac{1}{2}\times\) square = 32 cm². Shaded triangle CBD = area CED − area BED = 80 − 32 = 48 cm². Answer: 48 cm².
======================================================================
======================================================================
INDEX: 13858 ID: 25089 TYPE: 1
QUESTION: The line graph shows the number of visitors at a science exhibition from January to June. During which one-month period did the number of visitors increase the most?
ANSWERS: ["January to February", "February to March", "April to May", "May to June"]
CORRECT_INDEX: 3
SOLUTION:
Read the graph values: Jan 300, Feb 600, Mar 760, Apr 700, May 300, Jun 630. The increases are Jan–Feb +300, Feb–Mar +160, May–Jun +330. The largest increase is May to June. Answer: May to June.
======================================================================
======================================================================
INDEX: 13859 ID: 25090 TYPE: 2
QUESTION: The line graph shows the number of visitors at a science exhibition from January to June. Each visitor paid \(\$6.50\) for a ticket to the exhibition. How much money was collected from the sales of the total number of tickets from January to April?
ANSWERS: ["15340", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Visitors Jan to April = 300 + 600 + 760 + 700 = 2360. Total sales = 2360 \(\times\) \(\$6.50\) = \(\$15\) 340. Answer: \(\$15\) 340.
======================================================================
======================================================================
INDEX: 13860 ID: 25091 TYPE: 2
QUESTION: In a race, there were 1050 participants altogether. 70% of the participants were men and the rest were women. 80% of the women and some men finished the race. A total of 230 participants did not finish the race. How many women joined the race?
ANSWERS: ["315", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Women = 30% of 1050 = 0.3 \(\times\) 1050 = 315. Answer: 315.
======================================================================
======================================================================
INDEX: 13861 ID: 25092 TYPE: 2
QUESTION: In a race, there were 1050 participants altogether. 70% of the participants were men and the rest were women. 80% of the women and some men finished the race. A total of 230 participants did not finish the race. How many men finished the race?
ANSWERS: ["568", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Women = 315, men = 1050 − 315 = 735. Women who did not finish = 20% of 315 = 0.2 \(\times\) 315 = 63. Men who did not finish = 230 − 63 = 167. Men who finished = 735 − 167 = 568. Answer: 568.
======================================================================
======================================================================
INDEX: 13862 ID: 25093 TYPE: 2
QUESTION: Alice has some paperclips. If she gives each friend 14 paperclips, she will have 4 paperclips left. If she gives each of them 20 paperclips, she will be short of 68 paperclips. How many friends did Alice have? How many paperclips did Alice have?
ANSWERS: ["12", "172", null, null]
CORRECT_INDEX: None
SOLUTION:
The difference in paperclips per friend is 20 − 14 = 6. The total shortfall difference = 4 + 68 = 72. Number of friends = 72 ÷ 6 = 12. Paperclips = 14 \(\times\) 12 + 4 = 168 + 4 = 172. Answers: (a) 12 friends, (b) 172 paperclips.
======================================================================
======================================================================
INDEX: 13863 ID: 25094 TYPE: 2
QUESTION: Alex, Ben and Charles shared some stickers. Ben received 3 times as many stickers as Charles. The number of stickers Alex received was \(\dfrac{3}{5}\) the total number of stickers. Alex received 120 more stickers than Charles. How many stickers did the boys have altogether?
ANSWERS: ["240", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Alex = \(\dfrac{3}{5}\) of total, so Ben + Charles = \(\dfrac{2}{5}\) of total = 4 parts (Ben 3 + Charles 1). Let total be 10 units: Alex = 6u, Ben = 3u, Charles = 1u. Alex − Charles = 6u − 1u = 5u = 120, so 1u = 24. Total = 10u = 240 stickers. Answer: 240.
======================================================================
======================================================================
INDEX: 13864 ID: 25095 TYPE: 1
QUESTION: Alex, Ben and Charles shared some stickers. Ben received 3 times as many stickers as Charles. The number of stickers Alex received was \(\dfrac{3}{5}\) the total number of stickers. What fraction of the stickers did Charles have?
ANSWERS: ["\\(\\dfrac{1}{10}\\)", "\\(\\dfrac{2}{5}\\)", "\\(\\dfrac{1}{5}\\)", "\\(\\dfrac{3}{10}\\)"]
CORRECT_INDEX: 0
SOLUTION:
Alex = \(\dfrac{3}{5}\) of the total = 6 units out of 10. Ben + Charles = \(\dfrac{2}{5}\) = 4 units, split as Ben 3 : Charles 1. So Charles = 1 unit out of 10 = \(\dfrac{1}{10}\). Answer: \(\dfrac{1}{10}\).
======================================================================
======================================================================
INDEX: 13865 ID: 25096 TYPE: 2
QUESTION: Simon bought an equal number of sandwiches and pies. A sandwich cost \(\$4\) and a pie cost \(\$1.80\) less than a sandwich. He paid a total of \(\$279\). How much did Simon pay for the pies? [?]
ANSWERS: ["99", null, null, null]
CORRECT_INDEX: None
SOLUTION:
Cost of a pie = \(\$4\) − \(\$1.80\) = \(\$2.20\). Group 1 sandwich + 1 pie = \(\$4\) + \(\$2.20\) = \(\$6.20\). Number of groups = \(\$279\) ÷ \(\$6.20\) = 45. Total paid for pies = 45 \(\times\) \(\$2.20\) = \(\$99\). Answer: \(\$99\).
======================================================================
======================================================================
INDEX: 13866 ID: 25097 TYPE: 2
QUESTION: Container A measuring 30 cm by 16 cm by 15 cm was completely filled with water at first. Container B is a cubical tank of edge 10 cm. Ravi poured some of the water from Container A into Container B to fill it completely. How much water was left in Container A? ml After Ravi used some water to water his plants, he had 5150 ml of water left. How much water did Ravi use to water his plants? Give your answer in litres. ℓ
ANSWERS: ["6200", "1.05", null, null]
CORRECT_INDEX: None
SOLUTION:
Volume of A = 30 \(\times\) 16 \(\times\) 15 = 7200 ml. Volume of B = 10 \(\times\) 10 \(\times\) 10 = 1000 ml. (a) Water left in A = 7200 − 1000 = 6200 ml. (b) Water used = 6200 − 5150 = 1050 ml = 1.05 ℓ. Answers: (a) 6200 ml, (b) 1.05 ℓ.
======================================================================
======================================================================
INDEX: 13867 ID: 25098 TYPE: 2
QUESTION: The bar graph shows the number of cupcakes sold from Monday to Friday (Mon 35, Tue 55, Wed 65, Thu 75). What was the average number of cupcakes sold on Monday and Tuesday? Find the total number of cupcakes sold from Monday to Thursday.
ANSWERS: ["45", "230", null, null]
CORRECT_INDEX: None
SOLUTION:
(a) Average of Monday and Tuesday = (35 + 55) ÷ 2 = 90 ÷ 2 = 45. (b) Total Monday to Thursday = 35 + 55 + 65 + 75 = 230. Answers: (a) 45, (b) 230.
======================================================================
======================================================================
INDEX: 13868 ID: 25099 TYPE: 2
QUESTION: The pattern is made up of squares and dots. Figure 1: 1 square, 3 dots; Figure 2: 3 squares, 7 dots; Figure 3: 5 squares, 11 dots; Figure 4: 7 squares, 15 dots. Find the number of dots in Figure 5.
How many squares are there in Figure 12? In which figure are there 295 dots?
ANSWERS: ["19", "23", "74", null]
CORRECT_INDEX: None
SOLUTION:
Dots increase by 4 each figure: 3, 7, 11, 15, so Figure 5 = 19. Squares: Figure n has (2n − 1) squares, so Figure 12 = 2 \(\times\) 12 − 1 = 23. Dots: Figure n has (4n − 1) dots; 4n − 1 = 295 → 4n = 296 → n = 74. Answers: 19 dots; 23 squares; Figure 74.
======================================================================