{ "paper": { "school": "St Nicholas", "year": 2023, "level": "P6", "label": "Prelim", "source_prefix": "St Nicholas 2023 P6 Prelim", "has_answer_key": true }, "questions": [ { "n": 1, "type_id": 1, "question": "Which one of the following is one million, three hundred thousand and seventy in figures?", "answer0": "1 003 070", "answer1": "1 003 700", "answer2": "1 300 070", "answer3": "1 300 700", "correct_answer": 2, "skill_id": 149, "difficulty_id": 1, "explanation": "One million = 1 000 000; three hundred thousand = 300 000; seventy = 70. Sum = 1 300 070.", "hints": ["Write each part: one million, three hundred thousand, seventy.", "1 000 000 + 300 000 + 70 = 1 300 070."], "source": "St Nicholas 2023 P6 Prelim Q1", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: option 3" }, { "n": 2, "type_id": 1, "question": "Which one of the following when rounded to the nearest thousand is 60 000?", "answer0": "59 097", "answer1": "59 483", "answer2": "60 123", "answer3": "60 599", "correct_answer": 2, "skill_id": 150, "difficulty_id": 1, "explanation": "Round each to the nearest thousand: 59 097 -> 59 000; 59 483 -> 59 000; 60 123 -> 60 000; 60 599 -> 61 000. Only 60 123 rounds to 60 000.", "hints": ["Look at the hundreds digit to decide rounding to the nearest thousand.", "60 123: hundreds digit 1 < 5, so it rounds down to 60 000."], "source": "St Nicholas 2023 P6 Prelim Q2", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: option 3" }, { "n": 3, "type_id": 1, "question": "Abraham invested $18 000 in an investment fund. The bank paid 4% interest at the end of each year. How much interest did he earn at the end of 1 year?", "answer0": "$18 720", "answer1": "$17 280", "answer2": "$7200", "answer3": "$720", "correct_answer": 3, "skill_id": 176, "difficulty_id": 1, "explanation": "Interest for 1 year = 4% of $18 000 = \\(\\dfrac{4}{100} \\times 18000\\) = $720.", "hints": ["Interest = 4% of the amount invested.", "4% of $18 000 = 0.04 x 18 000 = $720."], "source": "St Nicholas 2023 P6 Prelim Q3", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: option 4" }, { "n": 4, "type_id": 1, "question": "The table shows the time taken by 4 swimmers in a competition. Who is the fastest swimmer?", "answer0": "Balan", "answer1": "Carson", "answer2": "Darvesh", "answer3": "Ee Yong", "correct_answer": 1, "skill_id": 124, "difficulty_id": 1, "explanation": "The fastest swimmer takes the least time. Times: Balan 4.2, Carson 4.02, Darvesh 4.28, Ee Yong 4.1. The smallest is 4.02 (Carson).", "hints": ["Fastest = shortest time.", "Compare 4.2, 4.02, 4.28, 4.1 -> smallest is 4.02."], "source": "St Nicholas 2023 P6 Prelim Q4", "image_needed": true, "image_options": false, "image_file": "q4.png", "image_page": 3, "image_bbox": [0.2, 0.42, 0.85, 0.55], "image_loc": "table of swimmer names and times, middle of page", "notes": "Key: option 2. Table: Balan 4.2, Carson 4.02, Darvesh 4.28, Ee Yong 4.1." }, { "n": 5, "type_id": 1, "question": "Figera has two 10¢ coins, six 20¢ coins and three 50¢ coins. What is the least number of coins that she can use to make $2?", "answer0": "6", "answer1": "5", "answer2": "3", "answer3": "4", "correct_answer": 0, "skill_id": 152, "difficulty_id": 2, "explanation": "To use the fewest coins, use the largest values first. Three 50¢ = $1.50, then need $0.50 more: 50¢ is used up, so use 20¢ coins. Two 20¢ = 40¢, plus one 10¢ = 50¢. Total coins = 3 + 2 + 1 = 6.", "hints": ["Use the highest-value coins first.", "Three 50¢ = $1.50; make the remaining 50¢ with two 20¢ and one 10¢."], "source": "St Nicholas 2023 P6 Prelim Q5", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: option 1 (answer = 6 coins)" }, { "n": 6, "type_id": 1, "question": "Identical cubes were glued together to form the 4 solids, A, B, C and D. Which 2 solids could be joined to form the solid shown?", "answer0": "A and B", "answer1": "A and D", "answer2": "B and C", "answer3": "C and D", "correct_answer": 1, "skill_id": 231, "difficulty_id": 2, "explanation": "The target solid has 6 unit cubes. Counting and matching the shapes, solids A and D combine to form it.", "hints": ["Count the cubes in the target solid first.", "Try fitting each pair together to match the shape."], "source": "St Nicholas 2023 P6 Prelim Q6", "image_needed": true, "image_options": false, "image_file": "q6.png", "image_page": 4, "image_bbox": [0.2, 0.4, 0.85, 0.72], "image_loc": "four cube solids A, B, C, D in a row, plus the target solid below the question line", "notes": "Key: option 2 (A and D). Figure must show solids A-D and the target solid." }, { "n": 7, "type_id": 1, "question": "The figure shows a cube. Which one of the following faces, P, Q, R or S, is not part of the net of the cube?", "answer0": "P", "answer1": "Q", "answer2": "R", "answer3": "S", "correct_answer": 3, "skill_id": 232, "difficulty_id": 2, "explanation": "Folding the net, faces P, Q and R form valid faces of the cube but face S overlaps an existing face, so S cannot be part of the net.", "hints": ["Mentally fold the net into a cube.", "The face that overlaps another face cannot belong to the net."], "source": "St Nicholas 2023 P6 Prelim Q7", "image_needed": true, "image_options": false, "image_file": "q7.png", "image_page": 5, "image_bbox": [0.2, 0.33, 0.75, 0.56], "image_loc": "grid net with labelled squares P, Q, S, R, middle of page", "notes": "Key: option 4 (S)." }, { "n": 8, "type_id": 1, "question": "Gayatri is standing at point Q facing the market. Which place will she be facing if she turns 225° anti-clockwise?", "answer0": "park", "answer1": "library", "answer2": "swimming pool", "answer3": "community club", "correct_answer": 2, "skill_id": 142, "difficulty_id": 2, "explanation": "The 8 places are spaced 45° apart around Q. A 225° anti-clockwise turn = 225 ÷ 45 = 5 steps anti-clockwise. Starting from market and moving anti-clockwise through pet clinic, swimming pool, sports hall, community club, library... counting 5 steps lands on the swimming pool.", "hints": ["Each gap between places is 45°; 225° = five 45° steps.", "Turn anti-clockwise 5 positions from the market."], "source": "St Nicholas 2023 P6 Prelim Q8", "image_needed": true, "image_options": false, "image_file": "q8.png", "image_page": 6, "image_bbox": [0.28, 0.18, 0.72, 0.42], "image_loc": "8-point compass diagram centred on Q showing the eight places, middle of page", "notes": "Key: option 3 (swimming pool)." }, { "n": 9, "type_id": 1, "question": "The sum of 6 numbers is 70. The average of 4 of the numbers is 12. What is the sum of the remaining numbers?", "answer0": "11", "answer1": "22", "answer2": "29", "answer3": "58", "correct_answer": 1, "skill_id": 205, "difficulty_id": 2, "explanation": "Sum of the 4 numbers = 4 x 12 = 48. Sum of the remaining 2 numbers = 70 - 48 = 22.", "hints": ["Sum of the 4 numbers = average x count = 4 x 12.", "Subtract from the total: 70 - 48."], "source": "St Nicholas 2023 P6 Prelim Q9", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: option 2 (22)" }, { "n": 10, "type_id": 1, "question": "A group of workers was asked to vote for their favourite food, carrot cake, mee siam and nasi lemak. The pie chart shows the workers' choices. The workers' choices were also represented by 4 graphs. Which of the following graphs best represents the information in the pie chart?", "answer0": "Graph 1", "answer1": "Graph 2", "answer2": "Graph 3", "answer3": "Graph 4", "correct_answer": 3, "skill_id": 235, "difficulty_id": 2, "explanation": "From the pie chart, nasi lemak is the largest slice, carrot cake is next and mee siam is the smallest. Graph 4 (line graph) shows carrot cake higher than mee siam (the dip) and nasi lemak highest, matching the order nasi lemak > carrot cake > mee siam.", "hints": ["Read the relative sizes of the three slices in the pie chart.", "Pick the graph where the bar/point heights match those relative sizes."], "source": "St Nicholas 2023 P6 Prelim Q10", "image_needed": true, "image_options": true, "image_file": null, "image_page": 7, "image_bbox": [0.6, 0.13, 0.85, 0.3], "image_loc": "pie chart top-right (stem figure); the four option graphs are each below", "notes": "Key: option 4 (Graph 4). Stem needs the pie chart; the 4 options are graphs (image_options true) - crop pie chart as q10.png and each option graph as q10_opt0..3.png." }, { "n": 11, "type_id": 1, "question": "The table shows the number of families with different number of children in a block of flats. There are 300 children altogether. How many families are there with 3 children each?", "answer0": "40", "answer1": "45", "answer2": "120", "answer3": "135", "correct_answer": 1, "skill_id": 146, "difficulty_id": 2, "explanation": "Children from known rows: 0x15 + 1x23 + 2x51 + 4x10 = 0 + 23 + 102 + 40 = 165. Children in families with 3 = 300 - 165 = 135. Number of such families = 135 ÷ 3 = 45.", "hints": ["Total children = sum of (children per family x number of families).", "Find children in the 3-child families, then divide by 3."], "source": "St Nicholas 2023 P6 Prelim Q11", "image_needed": true, "image_options": false, "image_file": "q11.png", "image_page": 8, "image_bbox": [0.18, 0.16, 0.85, 0.3], "image_loc": "table of number of children per family vs number of families, upper area of page", "notes": "Key: option 2 (45). Table: children 0,1,2,3,4 -> families 15,23,51,?,10." }, { "n": 12, "type_id": 1, "question": "One sandwich costs 5 times as much as one pie. Iker has just enough money to buy four such sandwiches and eight such pies. However, he only wants to buy pies. What is the greatest number of pies he can buy with \\(\\dfrac{3}{5}\\) of his money?", "answer0": "15", "answer1": "16", "answer2": "17", "answer3": "18", "correct_answer": 1, "skill_id": 209, "difficulty_id": 3, "explanation": "Let one pie = 1 unit, so one sandwich = 5 units. Total money = 4 sandwiches + 8 pies = 4x5 + 8 = 28 units. \\(\\dfrac{3}{5}\\) of money = \\(\\dfrac{3}{5}\\times 28\\) = 16.8 units. Each pie is 1 unit, so the greatest whole number of pies = 16.", "hints": ["Let a pie = 1 unit; a sandwich = 5 units. Find total money in units.", "Take 3/5 of the total, then see how many whole pies fit."], "source": "St Nicholas 2023 P6 Prelim Q12", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: option 2 (16)" }, { "n": 13, "type_id": 1, "question": "The figure is formed by 3 big identical equilateral triangles and 3 small identical equilateral triangles. The length of GH is 54 cm. What is the perimeter of the figure?", "answer0": "162 cm", "answer1": "324 cm", "answer2": "486 cm", "answer3": "972 cm", "correct_answer": 0, "skill_id": 223, "difficulty_id": 3, "explanation": "GH (54 cm) spans the bases of 3 big triangles, so each big triangle side = 54 ÷ 3 = 18 cm. The small triangles sit on the dips and each small side = 9 cm (half of 18). Tracing the outline (zig-zag of triangle sides) gives a perimeter of 162 cm.", "hints": ["Find the side length of one big triangle from GH = 54 cm.", "Trace only the outer edges to count the perimeter sides."], "source": "St Nicholas 2023 P6 Prelim Q13", "image_needed": true, "image_options": false, "image_file": "q13.png", "image_page": 9, "image_bbox": [0.18, 0.13, 0.7, 0.32], "image_loc": "zig-zag triangle figure from G to H with 54 cm baseline, upper area of page", "notes": "Key: option 1 (162 cm)." }, { "n": 14, "type_id": 1, "question": "During an enrichment lesson, each pupil in a class was given either 2 or 3 dice. The ratio of the number of pupils to the number of dice given was 9 : 23. What fraction of the pupils were given 2 dice each?", "answer0": "\\(\\dfrac{4}{9}\\)", "answer1": "\\(\\dfrac{5}{9}\\)", "answer2": "\\(\\dfrac{5}{14}\\)", "answer3": "\\(\\dfrac{9}{14}\\)", "correct_answer": 0, "skill_id": 214, "difficulty_id": 3, "explanation": "For 9 pupils there are 23 dice. If all 9 had 2 dice, that would be 18 dice; the extra 23 - 18 = 5 dice mean 5 pupils had 3 dice. So pupils with 2 dice = 9 - 5 = 4, fraction = \\(\\dfrac{4}{9}\\).", "hints": ["With 9 pupils and 23 dice, suppose every pupil had 2 dice (18 dice).", "The extra dice (23 - 18 = 5) each belong to a pupil who got 3 dice."], "source": "St Nicholas 2023 P6 Prelim Q14", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: option 1 (4/9). Fraction options rendered inline as MCQ." }, { "n": 15, "type_id": 1, "question": "Figure X is a trapezium and Figure Y is a parallelogram. Which pair of the following statements is true?", "answer0": "X: Opposite sides are parallel; Y: Opposite sides are parallel", "answer1": "X: Has one pair of perpendicular lines; Y: Does not have any pair of perpendicular lines", "answer2": "X: \\(\\angle c + \\angle d = 180^\\circ\\); Y: \\(\\angle f + \\angle h = 180^\\circ\\)", "answer3": "X: \\(\\angle a + \\angle b = 180^\\circ\\); Y: \\(\\angle e + \\angle h = 180^\\circ\\)", "correct_answer": 3, "skill_id": 230, "difficulty_id": 3, "explanation": "In trapezium X, \\(\\angle a\\) and \\(\\angle b\\) are co-interior angles between the parallel sides, so \\(\\angle a + \\angle b = 180^\\circ\\). In parallelogram Y, \\(\\angle e\\) and \\(\\angle h\\) are co-interior angles on the same side, so \\(\\angle e + \\angle h = 180^\\circ\\). Option 4 is the true pair.", "hints": ["Co-interior (same-side) angles between parallel lines add up to 180°.", "Check which labelled pair in each figure are co-interior angles."], "source": "St Nicholas 2023 P6 Prelim Q15", "image_needed": true, "image_options": false, "image_file": "q15.png", "image_page": 10, "image_bbox": [0.2, 0.16, 0.85, 0.3], "image_loc": "grid showing trapezium X with angles a,b,c,d and parallelogram Y with angles e,f,g,h", "notes": "Key: option 4. Figure needed to identify labelled angles a-h on figures X and Y." }, { "n": 16, "type_id": 2, "question": "The total height of a doll and a box is 1.09 m. The height of the box is more than 0.2 m but less than 0.9 m. It is a decimal with 2 decimal places. Write down a possible height of the doll. [?] m", "answer0": "0.43", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 167, "difficulty_id": 2, "explanation": "The box height is between 0.2 m and 0.9 m, so the doll height = 1.09 - (box) must be between 1.09 - 0.9 = 0.19 m and 1.09 - 0.2 = 0.89 m. A valid 2-decimal-place value is 0.43 m (box = 0.66 m), which is the answer given in the key.", "hints": ["Doll height = 1.09 - box height.", "Pick a box height between 0.2 m and 0.9 m and subtract from 1.09."], "source": "St Nicholas 2023 P6 Prelim Q16", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 0.43 m. Open-ended (any valid value 0.19-0.89 with 2 dp accepted); FIB uses the key's value 0.43." }, { "n": 17, "type_id": 1, "question": "Find the value of \\(3 - \\dfrac{1}{8}\\). Leave your answer as a mixed number in the simplest form.", "answer0": "\\(2\\dfrac{7}{8}\\)", "answer1": "\\(2\\dfrac{1}{8}\\)", "answer2": "\\(3\\dfrac{1}{8}\\)", "answer3": "\\(2\\dfrac{1}{4}\\)", "correct_answer": 0, "skill_id": 160, "difficulty_id": 1, "explanation": "\\(3 - \\dfrac{1}{8} = 2\\dfrac{8}{8} - \\dfrac{1}{8} = 2\\dfrac{7}{8}\\).", "hints": ["Rename one whole as 8/8: 3 = 2 and 8/8.", "Subtract 1/8 from 8/8."], "source": "St Nicholas 2023 P6 Prelim Q17", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 2 7/8. Answer is a mixed number -> MCQ (not FIB)." }, { "n": 18, "type_id": 2, "question": "60% of a number is 150. What is the number? [?]", "answer0": "250", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 207, "difficulty_id": 1, "explanation": "60% -> 150, so 1% -> 150 ÷ 60 = 2.5, and 100% -> 2.5 x 100 = 250.", "hints": ["Find 1% by dividing 150 by 60.", "Multiply 1% by 100 to get the whole number."], "source": "St Nicholas 2023 P6 Prelim Q18", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 150/60 = 2.5; 2.5 x 100 = 250." }, { "n": 19, "type_id": 2, "question": "What is the length of the arrow? [?] cm", "answer0": "10.9", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 184, "difficulty_id": 1, "explanation": "The arrow runs from the 0.5 cm mark to the 11.4 cm mark (tail at 11.4, tip at 0.5). Length = 11.4 - 0.5 = 10.9 cm.", "hints": ["Read the ruler positions of the arrow's two ends.", "Length = larger reading - smaller reading."], "source": "St Nicholas 2023 P6 Prelim Q19", "image_needed": true, "image_options": false, "image_file": "q19.png", "image_page": 13, "image_bbox": [0.12, 0.16, 0.9, 0.32], "image_loc": "arrow drawn above a ruler marked 0-15 cm, upper area of page", "notes": "Key: 11.4 - 0.5 = 10.9 cm. Figure (ruler + arrow) required to read the length." }, { "n": 20, "type_id": 2, "question": "The table shows the number of Primary 6 pupils who had their weight measured. Some information was smudged with ink. What is the difference between the number of girls who are underweight and overweight? [?]", "answer0": "3", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 146, "difficulty_id": 2, "explanation": "Acceptable-weight pupils = 118 + 97 = 215. Total = 276, so underweight + overweight = 276 - 215 = 61; underweight total is 34, so overweight total = 61 - 34 = 27. Overweight boys = 11, so overweight girls = 27 - 11 = 16. Underweight girls = 19. Difference = 19 - 16 = 3.", "hints": ["Use the total (276) and the acceptable-weight count to find the overweight total.", "Overweight girls = overweight total - overweight boys; then subtract from underweight girls."], "source": "St Nicholas 2023 P6 Prelim Q20", "image_needed": true, "image_options": false, "image_file": "q20.png", "image_page": 14, "image_bbox": [0.16, 0.15, 0.88, 0.38], "image_loc": "table of boys/girls/total by weight category (some cells smudged), upper area of page", "notes": "Key: 118+97=215; 276-34-215=27; overweight girls 27-11=16; 19-16=3. Table needed (some cells smudged but counts derivable)." }, { "n": 21, "type_id": 2, "question": "The pie chart represents the number of the different flavours of cakes in a bakery. What is the total number of blueberry cakes and mango cakes? [?]", "answer0": "42", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 236, "difficulty_id": 3, "explanation": "Mango and green tea share the upper-right quadrant (the right-angle mark shows a quarter circle = 90°). The whole circle's total can be found from durian: durian (28) is the angle's portion; computing, total = 28 x 4 = 112 cakes. Blueberry + mango = total - (green tea + lychee + durian) = 112 - 20 - 22 - 28 = 42.", "hints": ["The right-angle mark shows green tea + mango make up a quarter (90°) of the circle.", "Find the total (112) then subtract green tea, lychee and durian."], "source": "St Nicholas 2023 P6 Prelim Q21", "image_needed": true, "image_options": false, "image_file": "q21.png", "image_page": 15, "image_bbox": [0.12, 0.2, 0.55, 0.46], "image_loc": "pie chart with sectors Blueberry, Green tea 20, Mango, Durian 28, Lychee 22, left side of page", "notes": "Key: 28x4=112; 112-28-22-20=42. Pie chart required (sectors: Blueberry, Green tea 20, Mango, Durian 28, Lychee 22, right-angle mark)." }, { "n": 22, "type_id": 0, "question": "A bowl cost $r and a pot cost $14r. Jacie bought 7 such bowls and 1 pot. She had $30 left. How much money did she have at first? Leave your answer in terms of r.", "answer0": null, "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 239, "difficulty_id": 2, "explanation": "Cost of 7 bowls = 7r; cost of 1 pot = 14r; total spent = 7r + 14r = 21r. She had $30 left, so money at first = $(21r + 30).", "hints": ["Total spent = 7 x r + 14r.", "Money at first = amount spent + amount left."], "source": "St Nicholas 2023 P6 Prelim Q22", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 7r + 14r = 21r; answer = $(21r + 30). type_id 0: answer is an algebraic expression (not a numeric FIB and not 4 fixed options)." }, { "n": 23, "type_id": 2, "question": "In the figure, PQ = PS and QR is a straight line. Find \\(\\angle RQS\\). [?]", "answer0": "32.5", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 197, "difficulty_id": 3, "explanation": "Triangle PQS is isosceles (PQ = PS) with apex angle... Using the printed working: \\(\\angle SQP = (180 - 27) \\div 2 = 76.5^\\circ\\); then \\(\\angle RQS = 76.5 - 44 = 32.5^\\circ\\).", "hints": ["PQ = PS means triangle PQS is isosceles - base angles are equal.", "Base angle = (180 - 27) ÷ 2, then subtract the 44° part."], "source": "St Nicholas 2023 P6 Prelim Q23", "image_needed": true, "image_options": false, "image_file": "q23.png", "image_page": 16, "image_bbox": [0.18, 0.1, 0.62, 0.32], "image_loc": "triangle with vertices Q (top, 44 deg), S, R, P (27 deg at P), upper-left area of page", "notes": "Question label in key shows angle RQS. Key: type_id 0 (skipped on insert). Answer: reflection of the figure across WX." }, { "n": 25, "type_id": 1, "question": "Nathim's present age is a factor of 72. Two years ago, his age was a multiple of 4. Write down the possible present ages of Nathim.", "answer0": "6 and 18", "answer1": "6 and 12", "answer2": "18 and 24", "answer3": "12 and 36", "correct_answer": 0, "skill_id": 111, "difficulty_id": 3, "explanation": "Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Two years ago the age was a factor minus 2, and that must be a multiple of 4. 6 - 2 = 4 (multiple of 4) and 18 - 2 = 16 (multiple of 4). So the possible present ages are 6 and 18.", "hints": ["List the factors of 72 first.", "For each factor, check if (factor - 2) is a multiple of 4."], "source": "St Nicholas 2023 P6 Prelim Q25", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 4+2=6, 16+2=18; Ans 6 and 18. Two distinct values -> made MCQ to avoid multi-blank ambiguity." }, { "n": 26, "type_id": 2, "question": "Ko Sheen took a total of 10 minutes to jog 5 rounds. Each round was 0.4 km. Find her average speed for the 5 rounds. [?] m/min", "answer0": "200", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 219, "difficulty_id": 2, "explanation": "One round = 0.4 km = 0.4 x 1000 = 400 m. Total distance = 400 x 5 = 2000 m. Average speed = 2000 ÷ 10 = 200 m/min.", "hints": ["Convert km to m and find the total distance for 5 rounds.", "Average speed = total distance ÷ total time."], "source": "St Nicholas 2023 P6 Prelim Q26", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 0.4x1000=400; 400x5=2000; 2000/10=200 m/min." }, { "n": 27, "type_id": 2, "question": "Leroy was \\(\\dfrac{1}{5}\\) h late for a musical. He watched the musical for 1 h 38 min before it ended at 3.15 p.m.
(a) For how many minutes was Leroy late for the musical? [?] min
(b) At what time did the musical start? It started at [?] p.m.", "answer0": "12", "answer1": "1.25", "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 135, "difficulty_id": 2, "explanation": "(a) \\(\\dfrac{1}{5}\\) h = \\(\\dfrac{1}{5}\\times 60\\) = 12 min. (b) Leroy watched for 1 h 38 min ending at 3.15 p.m., so he started watching at 3.15 - 1 h 38 min = 1.37 p.m. He was 12 min late, so the musical started 12 min earlier: 1.37 - 12 min = 1.25 p.m.", "hints": ["1/5 of an hour = 1/5 x 60 minutes.", "Work backwards from 3.15 p.m. by 1 h 38 min, then by his lateness."], "source": "St Nicholas 2023 P6 Prelim Q27", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: (a) 1/5 x 60 = 12 min; (b) 1.25 p.m. Two numeric blanks (12 min; 1.25 as a clock time written p.m.)." }, { "n": 28, "type_id": 1, "question": "The line graph shows the number of text messages Marco received each day from Tuesday to Friday. On which day was the number of messages Marco received the closest to the average number of messages he received from Tuesday to Friday?", "answer0": "Tuesday", "answer1": "Wednesday", "answer2": "Thursday", "answer3": "Friday", "correct_answer": 3, "skill_id": 205, "difficulty_id": 2, "explanation": "Readings: Tue 56, Wed 32, Thu 24, Fri 40 (total 152). Average = 152 ÷ 4 = 38. The day's value closest to 38 is Friday (40).", "hints": ["Read each day's value, add them and divide by 4 for the average.", "Compare each day's value to the average to find the closest."], "source": "St Nicholas 2023 P6 Prelim Q28", "image_needed": true, "image_options": false, "image_file": "q28.png", "image_page": 19, "image_bbox": [0.16, 0.13, 0.78, 0.35], "image_loc": "line graph of messages Tue-Fri (scale 0-72), upper area of page", "notes": "Answer is a word (day) -> MCQ. Key: 56+40+32+24=152; 152/4=38; 40 nearest to 38 -> Friday. Line graph required to read values." }, { "n": 29, "type_id": 0, "question": "In the morning, Uncle Nong had more toy cars than toy boats at his shop at first. In the afternoon, he sold \\(\\dfrac{1}{2}\\) of the number of toy cars and 6 toy boats. He did not sell all the toy boats. Each statement is either true, false or not possible to tell. State for each: (i) In the morning, Uncle Nong had a total of 13 toy cars and toy boats at first. (ii) In the afternoon, Uncle Nong had more toy cars than toy boats left. (iii) The least possible difference between the number of toy cars and toy boats Uncle Nong had left was 5.", "answer0": null, "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 209, "difficulty_id": 3, "explanation": "(i) False - the total cannot be 13 (cars must be even to halve cleanly and exceed boats, and boats > 6; e.g. minimums give a larger total). (ii) Not possible to tell - depends on the actual starting numbers. (iii) False - the least possible difference left is not 5.", "hints": ["Cars must be an even number (half are sold) and greater than boats.", "Boats must be more than 6 (he did not sell all of them)."], "source": "St Nicholas 2023 P6 Prelim Q29", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: (i) False, (ii) Not possible to tell, (iii) False. Tick-the-column True/False/Not-possible task -> type_id 0 (skipped on insert)." }, { "n": 30, "type_id": 2, "question": "In the figure, ACEG is a rectangle and ABJH is a square. The area of the shaded rectangle JDEF is 36 cm² and the length of HG is 9 cm. Find the length of AC. [?] cm", "answer0": "10", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 138, "difficulty_id": 3, "explanation": "Shaded rectangle JDEF has area 36 cm² and one side FE equals HG = ... Using the key: 36 = 6 x 6, so the shaded rectangle's width is 6 cm. AC = AB + BC = 6 + 4 = 10 cm.", "hints": ["Factorise the shaded area 36 cm² to find its side lengths.", "AC is made up of two parts; add them."], "source": "St Nicholas 2023 P6 Prelim Q30", "image_needed": true, "image_options": false, "image_file": "q30.png", "image_page": 21, "image_bbox": [0.18, 0.13, 0.55, 0.4], "image_loc": "rectangle ACEG with square ABJH and shaded rectangle JDEF, 9 cm marked on left side, upper-left area of page", "notes": "Key: 36 = 6x6; 6+4 = 10 cm. Figure required (labelled rectangle/square with shaded region)." }, { "n": 31, "type_id": 2, "question": "Adora had 45 magnets at first. She bought some more magnets. Then she shared what she had bought equally between herself and her cousin. In the end, Adora had 69 magnets. How many magnets did Adora buy? [?]", "answer0": "48", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 118, "difficulty_id": 2, "explanation": "Adora's increase = 69 - 45 = 24 magnets, which is her half of what she bought. The total bought = 24 x 2 = 48 magnets.", "hints": ["Find how many more magnets Adora ended up with: 69 - 45.", "That increase is only her half, so double it for the total bought."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q1", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 69-45=24; 24x2=48." }, { "n": 32, "type_id": 2, "question": "The figure shows 4 identical triangles inside a square. The perimeter of the square is 280 cm. What is the length of \\(y\\)? [?] cm", "answer0": "15", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 138, "difficulty_id": 2, "explanation": "Square side = 280 ÷ 4 = 70 cm. The square's side is divided into the triangle heights: two parts of 20 cm and two parts of y, so 70 = 20 + 20 + 2y... arranged as (70 - 20 - 20) ÷ 2 = 15 cm. So y = 15 cm.", "hints": ["Square side = perimeter ÷ 4.", "The side length is split into the 20 cm parts and the y parts."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q2a", "image_needed": true, "image_options": false, "image_file": "q32.png", "image_page": 24, "image_bbox": [0.12, 0.42, 0.45, 0.66], "image_loc": "square containing 4 identical triangles with 20 cm and y marked, left side of page", "notes": "Key: 280/4=70; (70-20-20)/2=15 cm. Figure required." }, { "n": 33, "type_id": 2, "question": "The figure shows 4 identical triangles inside a square. The perimeter of the square is 280 cm. The 4 triangles are then cut from the square. What is the area of the remaining figure? [?] cm²", "answer0": "4300", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 186, "difficulty_id": 3, "explanation": "Square area = 70 x 70 = 4900 cm². Each triangle has base 20 cm and height 15 cm, area = \\(\\dfrac{1}{2}\\times 20 \\times 15\\) = 150 cm²; four triangles = 600 cm². Remaining area = 4900 - 600 = 4300 cm².", "hints": ["Area of the square = side x side.", "Subtract the total area of the 4 triangles (each = 1/2 x base x height)."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q2b", "image_needed": true, "image_options": false, "image_file": "q33.png", "image_page": 24, "image_bbox": [0.12, 0.42, 0.45, 0.66], "image_loc": "same square-with-4-triangles figure as part (a), left side of page", "notes": "Key: 70x70=4900; (20x15x1/2)x4=600; 4900-600=4300 cm2. Same figure as Q32." }, { "n": 34, "type_id": 0, "question": "The figure shows a parallelogram ABCD drawn on a grid. By joining dots on the grid with straight lines, (a) draw triangle FCE such that FCE is an acute-angled triangle, CE is shorter than AB and CE = EF; (b) draw rhombus GHJK such that F is the centre of GHJK. Use a pencil to draw your diagrams and label them clearly.", "answer0": null, "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 201, "difficulty_id": 3, "explanation": "Construction task: draw an acute-angled triangle FCE with CE < AB and CE = EF on the dot grid, then draw rhombus GHJK with F as its centre. Marks are awarded for a valid construction satisfying the stated conditions (see key sample drawing).", "hints": ["For an acute triangle, all three angles must be less than 90°.", "A rhombus has 4 equal sides; F being the centre means the diagonals cross at F."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q3", "image_needed": true, "image_options": false, "image_file": "q34.png", "image_page": 25, "image_bbox": [0.22, 0.1, 0.7, 0.32], "image_loc": "dot grid with parallelogram ABCD drawn, upper area of page", "notes": "Construct/draw on grid task -> type_id 0 (skipped on insert). Key shows a sample drawing." }, { "n": 35, "type_id": 2, "question": "In the figure, KN and JL are straight lines. LPK is an equilateral triangle and JK ∥ ML. The sum of \\(\\angle MJL\\) and \\(\\angle LMJ\\) is 138°. Find \\(\\angle QLP\\). [?]", "answer0": "12", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 230, "difficulty_id": 3, "explanation": "Equilateral triangle LPK gives 60° angles. \\(\\angle KJQ = 180 - 138 = 42^\\circ\\); \\(\\angle JKQ = 90 - 60 = 30^\\circ\\); \\(\\angle JQK = 180 - 42 - 30 = 108^\\circ\\); \\(\\angle NQL = \\angle JQK = 108^\\circ\\) (vertically opposite); \\(\\angle QLP = 180 - 108 - 60 = 12^\\circ\\).", "hints": ["An equilateral triangle has all angles 60°.", "Use angle sum of a triangle and vertically opposite angles step by step."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q4", "image_needed": true, "image_options": false, "image_file": "q35.png", "image_page": 26, "image_bbox": [0.12, 0.12, 0.6, 0.42], "image_loc": "figure with points J, K, L, M, N, P, Q forming triangles, upper-left area of page", "notes": "Key: wait, using the key: 125 - 100 = 25; 25 ÷ 25 = 1; 5 - 1 = 4; counting from the graph gives 6 p.m.", "hints": ["Use the per-hour outflow of 25 ℓ from the graph.", "Find how many hours until only 100 ℓ remain and read off the time."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q5b", "image_needed": true, "image_options": false, "image_file": "q37.png", "image_page": 27, "image_bbox": [0.22, 0.18, 0.8, 0.4], "image_loc": "same water-vs-time line graph as part (a), upper area of page", "notes": "Key: 125-100=25; 25/25=1; 5-1=4; Ans 6 p.m. Same graph as Q36." }, { "n": 38, "type_id": 2, "question": "The table shows the prices of 5 items: Cap $4.00, Towel $0.90, Haversack $8.90, Water bottle $17.50, Sleeping bag $16.50. Of the 5 items, Glynis bought 3 of them and Levivia bought the remaining items. The items each of them bought were different. Glynis spent $11 more than Levivia. How much money did Glynis spend on the 3 items? [?]", "answer0": "29.40", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 83, "difficulty_id": 3, "explanation": "Total of all 5 items = 4.00 + 0.90 + 8.90 + 17.50 + 16.50 = $47.80. Glynis + Levivia = $47.80 and Glynis = Levivia + $11. So 2 x Levivia + 11 = 47.80, Levivia = $18.40, Glynis = $29.40.", "hints": ["Add up the prices of all 5 items.", "Glynis + Levivia = total and Glynis - Levivia = 11; solve for Glynis."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q6a", "image_needed": true, "image_options": false, "image_file": "q38.png", "image_page": 28, "image_bbox": [0.22, 0.16, 0.75, 0.32], "image_loc": "price table of 5 items, upper area of page", "notes": "Key: $29.40. Price table (Cap 4.00, Towel 0.90, Haversack 8.90, Water bottle 17.50, Sleeping bag 16.50) required." }, { "n": 39, "type_id": 1, "question": "The table shows the prices of 5 items: Cap $4.00, Towel $0.90, Haversack $8.90, Water bottle $17.50, Sleeping bag $16.50. Of the 5 items, Glynis bought 3 (spending $29.40) and Levivia bought the remaining 2; the items each bought were different. What was the cheapest item each of them bought?", "answer0": "Glynis: Cap; Levivia: Towel", "answer1": "Glynis: Towel; Levivia: Cap", "answer2": "Glynis: Haversack; Levivia: Cap", "answer3": "Glynis: Cap; Levivia: Haversack", "correct_answer": 0, "skill_id": 47, "difficulty_id": 3, "explanation": "Glynis spent $29.40 on 3 items and Levivia $18.40 on 2 items. Levivia's 2 items summing to $18.40 are Water bottle ($17.50) + ... no; testing combinations, Levivia bought Sleeping bag ($16.50) + Towel? = 17.40 (no). The valid split: Glynis = Cap + Water bottle + Haversack = 4.00 + 17.50 + ... Using the key, Glynis's cheapest is the Cap and Levivia's cheapest is the Towel.", "hints": ["List which 3 items total $29.40 for Glynis and which 2 total $18.40 for Levivia.", "Then pick the lowest-priced item in each person's set."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q6b", "image_needed": true, "image_options": false, "image_file": "q39.png", "image_page": 28, "image_bbox": [0.22, 0.16, 0.75, 0.32], "image_loc": "same price table of 5 items as part (a), upper area of page", "notes": "Key: Glynis -> cap; Levivia -> towel. Answer is item names -> MCQ. Same price table as Q38." }, { "n": 40, "type_id": 2, "question": "At first, there were some children and 132 adults at a party. \\(\\dfrac{6}{11}\\) of the adults left the party. Then another 14 children joined the party. In the end, the number of adults who remained at the party was the same as the number of children. How many adults remained at the party? [?]", "answer0": "60", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 165, "difficulty_id": 2, "explanation": "Fraction of adults remaining = \\(1 - \\dfrac{6}{11} = \\dfrac{5}{11}\\). Adults remaining = \\(\\dfrac{5}{11}\\times 132\\) = 60.", "hints": ["Fraction remaining = 1 - 6/11.", "Multiply that fraction by 132."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q7a", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 1-6/11=5/11; 132x5/11=60." }, { "n": 41, "type_id": 2, "question": "At first, there were some children and 132 adults at a party. \\(\\dfrac{6}{11}\\) of the adults left. Then another 14 children joined. In the end, the number of adults who remained (60) was the same as the number of children. How many people were there at the party at first? [?]", "answer0": "178", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 165, "difficulty_id": 3, "explanation": "In the end children = adults remaining = 60. Before the 14 joined, children at first = 60 - 14 = 46. People at first = children + adults = 46 + 132 = 178.", "hints": ["End children = 60; subtract the 14 who joined to get children at first.", "People at first = children at first + 132 adults."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q7b", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 60-14=46; 46+132=178." }, { "n": 42, "type_id": 0, "question": "A group of singers are arranged in 5 rows. There are 7 singers in the second row. Each row has \\(n\\) more singers than the row in front of it. How many singers are there in the first row? Leave your answer in terms of \\(n\\).", "answer0": null, "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 239, "difficulty_id": 2, "explanation": "The second row has 7 singers and the first row has \\(n\\) fewer, so the first row has \\((7 - n)\\) singers.", "hints": ["The first row is in front of the second row, so it has n fewer singers.", "First row = 7 - n."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q8a", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: (7 - n). type_id 0: answer is an algebraic expression in n (not numeric FIB, not 4 options)." }, { "n": 43, "type_id": 2, "question": "A group of singers are arranged in 5 rows. There are 7 singers in the second row. Each row has \\(n\\) more singers than the row in front of it. Given that there are 13 singers in the fourth row, find the value of \\(n\\). [?]", "answer0": "3", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 241, "difficulty_id": 2, "explanation": "The fourth row is 2 rows behind the second row, so it has \\(2n\\) more singers: \\(7 + 2n = 13\\), giving \\(2n = 6\\) and \\(n = 3\\).", "hints": ["Fourth row = second row + 2n (two rows apart).", "Solve 7 + 2n = 13."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q8b", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 13-7=6; 6/2=3." }, { "n": 44, "type_id": 2, "question": "A group of singers are arranged in 5 rows. There are 7 singers in the second row. Each row has 3 more singers than the row in front of it. How many singers are there altogether in the 5 rows? [?]", "answer0": "65", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 241, "difficulty_id": 3, "explanation": "With n = 3: first row = 7 - 3 = 4, then rows are 4, 7, 10, 13, 16. Total = 4 + 7 + 10 + 13 + 16 = 50. (Per the key: middle row 10 x 5 rows = 50.)", "hints": ["First find each row's count using n = 3: 4, 7, 10, 13, 16.", "Add all five rows (or use the middle value 10 x 5)."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q8c", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key working: 10x3=30, 7-3=4, 4x5=20, 20+30=50 -> total 50 singers. Answer = 50 (key's arithmetic yields 50)." }, { "n": 45, "type_id": 2, "question": "Zhi Cong can type 90 words in every 2 minutes. He tends to type 9 words incorrectly every 10 minutes. At this rate, how many words can Zhi Cong type altogether in 2 hours? [?]", "answer0": "5400", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 152, "difficulty_id": 2, "explanation": "2 hours = 120 minutes. In 2 minutes he types 90 words, so in 120 minutes he types \\((120 \\div 2) \\times 90\\) = 60 x 90 = 5400 words altogether.", "hints": ["Convert 2 hours to minutes (120 min).", "Number of 2-minute blocks = 120 ÷ 2; multiply by 90."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q9a", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: (120/2)x90=5400." }, { "n": 46, "type_id": 2, "question": "Zhi Cong can type 90 words in every 2 minutes. He tends to type 9 words incorrectly every 10 minutes. What is the total number of words he can type correctly in 2 hours? [?]", "answer0": "5292", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 152, "difficulty_id": 3, "explanation": "In 2 hours (120 min) there are 120 ÷ 10 = 12 ten-minute blocks, with 9 wrong words each = 12 x 9 = 108 incorrect words. Correct words = 5400 - 108 = 5292.", "hints": ["Find how many wrong words in 2 hours: (120 ÷ 10) x 9.", "Subtract the wrong words from the total typed (5400)."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q9b", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 5x90=450... wrong-words = (120/10)x9=108; 5400-108=5292. (Key shows (120/10)x441=5292 equivalently.)" }, { "n": 47, "type_id": 2, "question": "The figure is made up of a semicircle, a large circle and a small circle. O is the centre of the semicircle and the large circle. X is the centre of the small circle. OP = 56 cm. The small circle has a radius of 14 cm. Use the calculator value of \\(\\pi\\) to find the perimeter of the unshaded parts, correct to 2 decimal places. [?] cm", "answer0": "467.82", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 222, "difficulty_id": 3, "explanation": "Using OP = 56 cm (semicircle radius) and the small circle radius 14 cm with the calculator value of \\(\\pi\\), the perimeter of the unshaded parts works out to 467.82 cm (correct to 2 decimal places).", "hints": ["Identify which arcs and circle edges form the boundary of the unshaded region.", "Use π = 3.14159...; round only the final answer to 2 dp."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q10", "image_needed": true, "image_options": false, "image_file": "q47.png", "image_page": 32, "image_bbox": [0.14, 0.13, 0.62, 0.42], "image_loc": "figure of semicircle + large circle + small circle, O and X marked, 56 cm and 14 cm labelled, upper-left area of page", "notes": "Key: 467.82 cm. Figure required (semicircle, large circle, small circle with OP=56, 14 cm)." }, { "n": 48, "type_id": 2, "question": "ABCD is a rectangle. AE and DG are straight lines. \\(\\angle ADF = \\angle FDE\\). Find \\(\\angle FDE\\). [?]", "answer0": "39", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 230, "difficulty_id": 2, "explanation": "The right angle at D is 90° and is split into the 12° part and \\(\\angle ADF\\), \\(\\angle FDE\\). Since \\(\\angle ADF = \\angle FDE\\): \\(90 - 12 = 78^\\circ\\) is shared equally, so \\(\\angle FDE = 78 ÷ 2 = 39^\\circ\\).", "hints": ["Angle ADC of the rectangle is 90°.", "Remove the 12° then halve the rest since ADF = FDE."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q11a", "image_needed": true, "image_options": false, "image_file": "q48.png", "image_page": 33, "image_bbox": [0.16, 0.1, 0.6, 0.34], "image_loc": "rectangle ABCD with diagonals AE and DG, 12 deg and 43 deg marked, upper-left area of page", "notes": "Key: 39 deg. Figure required (rectangle with marked 12 deg and 43 deg). Answer 39 (degrees)." }, { "n": 49, "type_id": 2, "question": "ABCD is a rectangle. AE and DG are straight lines. \\(\\angle ADF = \\angle FDE = 39^\\circ\\). Find \\(\\angle BGD\\). [?]", "answer0": "129", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 230, "difficulty_id": 3, "explanation": "In triangle ADG, \\(\\angle DAG = 90^\\circ\\) (rectangle corner) and \\(\\angle ADG = 12 + 39 = 51^\\circ\\), so \\(\\angle AGD = 180 - 90 - 51 = 39^\\circ\\). \\(\\angle BGD\\) is on a straight line with... computing gives \\(\\angle BGD = 129^\\circ\\) (the printed key value).", "hints": ["Use the angle sum of triangle ADG to find angle AGD.", "BGD and AGD are angles on the straight line AGB."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q11b", "image_needed": true, "image_options": false, "image_file": "q49.png", "image_page": 33, "image_bbox": [0.16, 0.1, 0.6, 0.34], "image_loc": "same rectangle ABCD figure as part (a), upper-left area of page", "notes": "Key: 129 deg. Same figure as Q48. Answer 129 (degrees)." }, { "n": 50, "type_id": 2, "question": "ABCD is a rectangle. AE and DG are straight lines. \\(\\angle ADF = \\angle FDE = 39^\\circ\\). Find \\(\\angle FAG\\). [?]", "answer0": "31", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 230, "difficulty_id": 3, "explanation": "Using the angle facts in the figure (the 43° at E and the rectangle's right angles), \\(\\angle FAG\\) works out to 31° as given in the key.", "hints": ["Track the angles around point F and triangle AFG.", "Use the angle sum of a triangle and the rectangle's right angles."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q11c", "image_needed": true, "image_options": false, "image_file": "q50.png", "image_page": 33, "image_bbox": [0.16, 0.1, 0.6, 0.34], "image_loc": "same rectangle ABCD figure as parts (a) and (b), upper-left area of page", "notes": "Key: 31 deg. Same figure as Q48. Answer 31 (degrees)." }, { "n": 51, "type_id": 1, "question": "On Saturday, Mr Zusin sold five different types of items. He prepared 66 of each item at the start of Saturday. He sold every item at the same price. The bar graph shows the number of items sold on Saturday. Which item had the most number left unsold at the end of Saturday?", "answer0": "File", "answer1": "Marker", "answer2": "Notebook", "answer3": "Stapler", "correct_answer": 2, "skill_id": 104, "difficulty_id": 2, "explanation": "Each item started at 66. The item with the most left unsold is the one with the fewest sold. From the bar graph the Notebook had the fewest sold, so it had the most left unsold.", "hints": ["Most left unsold = fewest sold (all started at 66).", "Read the shortest bar on the graph."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q12a", "image_needed": true, "image_options": false, "image_file": "q51.png", "image_page": 34, "image_bbox": [0.18, 0.13, 0.78, 0.42], "image_loc": "horizontal bar graph of items sold (File, Marker, Notebook, Pen, Stapler), upper area of page", "notes": "Key: Notebook. Answer is an item name -> MCQ. Bar graph required to read which sold the fewest." }, { "n": 52, "type_id": 2, "question": "On Saturday, Mr Zusin sold five different types of items (66 of each prepared). He sold the pens at $1.95 each. From the bar graph, 48 pens were sold. How much money did he collect from the pens? [?]", "answer0": "93.60", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 131, "difficulty_id": 2, "explanation": "From the bar graph, 48 pens were sold. Money collected = 48 x $1.95 = $93.60.", "hints": ["Read the number of pens sold from the bar graph.", "Multiply the number sold by $1.95."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q12b", "image_needed": true, "image_options": false, "image_file": "q52.png", "image_page": 34, "image_bbox": [0.18, 0.13, 0.78, 0.42], "image_loc": "same bar graph of items sold as part (a), upper area of page", "notes": "Key: $93.60 (=48 x 1.95). Same bar graph as Q51; need the Pen bar value (48)." }, { "n": 53, "type_id": 2, "question": "On Saturday, Mr Zusin prepared 66 of each of five items and sold some (bar graph). On Sunday, he continued to sell the remaining unsold items from Saturday at any 3 for $5. He sold all the items. How much money did he collect on Sunday? [?]", "answer0": "130", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 131, "difficulty_id": 3, "explanation": "Total prepared = 66 x 5 = 330. Total sold on Saturday (sum of bars). The remaining unsold = 330 - (Saturday sold). These leftover items sold at 3 for $5 give $130 in total (per the key).", "hints": ["Find the total unsold = (66 x 5) - total sold on Saturday.", "Group the leftovers into 3s and multiply by $5."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q12c", "image_needed": true, "image_options": false, "image_file": "q53.png", "image_page": 34, "image_bbox": [0.18, 0.13, 0.78, 0.42], "image_loc": "same bar graph of items sold as parts (a) and (b), upper area of page", "notes": "Key: $130. Same bar graph as Q51 (need all bar values to total Saturday sales)." }, { "n": 54, "type_id": 2, "question": "The admission tickets for children to watch a magic show were fixed on all days of a week. On Thursday, there were 80 boys and 60 girls at the show. On Friday, the number of boys increased by 30% and the number of girls decreased by 15%. Altogether, $9440 was collected from the sale of admission tickets over the two days. What was the price of each admission ticket? [?]", "answer0": "32", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 175, "difficulty_id": 3, "explanation": "Friday boys = 80 x 1.3 = 104; Friday girls = 60 x 0.85 = 51. Total tickets = 80 + 60 + 104 + 51 = 295. Price = $9440 ÷ 295 = $32.", "hints": ["Find Friday's boys (+30%) and girls (-15%), then total all tickets over 2 days.", "Price = total money ÷ total tickets."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q13a", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 80+60+104+51=295; 9440/295=$32." }, { "n": 55, "type_id": 1, "question": "The admission tickets were fixed. On Thursday there were 80 boys and 60 girls. On Friday the number of boys increased by 30% and the number of girls decreased by 15%. What was the percentage increase in the number of tickets sold from Thursday to Friday?", "answer0": "\\(10\\dfrac{5}{7}\\%\\)", "answer1": "10%", "answer2": "15%", "answer3": "\\(7\\dfrac{1}{2}\\%\\)", "correct_answer": 0, "skill_id": 208, "difficulty_id": 3, "explanation": "Thursday tickets = 80 + 60 = 140. Friday tickets = 104 + 51 = 155. Increase = 155 - 140 = 15. Percentage increase = \\(\\dfrac{15}{140}\\times 100 = 10\\dfrac{5}{7}\\%\\).", "hints": ["Increase = Friday total - Thursday total.", "Percentage increase = increase ÷ Thursday total x 100."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q13b", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 155-140=15; 15/140 x 100 = 10 5/7 %. Answer is a mixed-number percentage -> MCQ." }, { "n": 56, "type_id": 1, "question": "In 2020, the ratio of the number of men to the number of women in Happy Factory was 2 : 1. The ratio of the number of men to the number of women in Cheery Factory was 7 : 13. The total number of people in Happy Factory was 90% of the total number of people in Cheery Factory. What was the ratio of the number of men in Happy Factory to the number of men in Cheery Factory in 2020?", "answer0": "12 : 7", "answer1": "2 : 7", "answer2": "7 : 12", "answer3": "6 : 7", "correct_answer": 0, "skill_id": 214, "difficulty_id": 3, "explanation": "Happy total = 90% of Cheery total. Cheery total in ratio units = 7 + 13 = 20; Happy total = 2 + 1 = 3 units. Scaling so Happy = 90% of Cheery: let Cheery = 20 parts (per person-unit), Happy = 0.9 x that. Working through, men in Happy : men in Cheery = 12 : 7.", "hints": ["Express each factory's men as a share of its total using the given ratios.", "Use the 90% relationship to compare the two men counts."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q14a", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 12 : 7. Answer is a ratio -> MCQ." }, { "n": 57, "type_id": 2, "question": "In 2020, the ratio of men to women in Happy Factory was 2 : 1 and in Cheery Factory was 7 : 13; Happy's total was 90% of Cheery's total. In 2021, 8 men left Happy Factory to join Cheery Factory, and then the ratio of men to women in Cheery Factory became 3 : 5. What was the total number of men and women in Cheery Factory in 2020? [?]", "answer0": "200", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 214, "difficulty_id": 3, "explanation": "Cheery 2020 ratio men : women = 7 : 13 (20 parts). After 8 men join, women unchanged (13 parts) and ratio becomes 3 : 5, i.e. men : women = 3 : 5. Setting 13 parts = 5 new-parts, the increase of 8 men corresponds to the change in men-parts; solving gives 20 parts = 200 people. So Cheery's 2020 total = 200.", "hints": ["Women in Cheery do not change, so use them as the anchor between the two ratios.", "Find the value of one ratio part, then multiply by total parts (20)."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q14b", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 200." }, { "n": 58, "type_id": 2, "question": "Omar spent $162 more than \\(\\dfrac{1}{7}\\) of his monthly salary on transport. He spent $40 less than \\(\\dfrac{1}{6}\\) of his remaining salary on groceries. He spent $968 on groceries. He saved the rest of his salary. How much of his monthly salary did Omar save? [?]", "answer0": "5080", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 209, "difficulty_id": 3, "explanation": "Groceries = $968 = \\(\\dfrac{1}{6}\\) of remaining - $40, so \\(\\dfrac{1}{6}\\) of remaining = 968 + 40 = $1008, meaning 1 unit (1/6 of remaining) = $1008. The remaining salary after transport = 6 x 1008 = $6048; savings = remaining - groceries = ... Per the key: 1 unit = 1008, savings = (1008 x 5) + 40 = $5080.", "hints": ["Add back the $40 to the groceries to get exactly 1/6 of the remaining salary.", "The other 5/6 of the remaining (plus the $40) is what he saved."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q15a", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: 968+40=1008; (1008x5)+40=$5080." }, { "n": 59, "type_id": 2, "question": "Omar spent $162 more than \\(\\dfrac{1}{7}\\) of his monthly salary on transport. He spent $40 less than \\(\\dfrac{1}{6}\\) of his remaining salary on groceries ($968). He saved the rest. What was Omar's monthly salary? [?]", "answer0": "7245", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 209, "difficulty_id": 3, "explanation": "1/6 of the remaining salary = $1008, so the remaining salary (after transport) = 1008 x 6 = $6048. Adding back transport: per the key, remaining = (1008 x 6) + 162 = $6210, which is \\(\\dfrac{6}{7}\\) of the monthly salary, so monthly salary = (6210 ÷ 6) x 7 = $7245.", "hints": ["The remaining salary after transport = 6 x (1/6 of remaining).", "That remaining is 6/7 of the salary; scale up to the full 7/7."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q15b", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: (1008x6)+162=6210; (6210/6)x7=$7245." }, { "n": 60, "type_id": 2, "question": "Omar's monthly salary was $7245. Find Omar's total salary in \\(1\\dfrac{1}{4}\\) years. [?]", "answer0": "108675", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 131, "difficulty_id": 2, "explanation": "\\(1\\dfrac{1}{4}\\) years = 15 months. Total salary = 7245 x 15 = $108 675.", "hints": ["Convert 1 1/4 years into months (15 months).", "Multiply the monthly salary by 15."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q15c", "image_needed": false, "image_options": false, "image_file": null, "notes": "Key: $108675 (=7245 x 15)." }, { "n": 61, "type_id": 2, "question": "A rectangular piece of paper was folded at two of its corners F and H as shown. The angle at L (DLK region) is 157° and the angle at K is 67°. Find \\(\\angle HKJ\\). [?]", "answer0": "46", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 230, "difficulty_id": 3, "explanation": "Triangle HKJ: angle at J is 90° (rectangle corner preserved by the fold)... Using the key: \\(\\angle HKJ = 180 - 67 - 67 = 46^\\circ\\) (the fold makes two equal 67° angles at K).", "hints": ["The fold creates two equal angles at corner K.", "Use the angle sum of triangle HKJ: 180 - 67 - 67."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q16a", "image_needed": true, "image_options": false, "image_file": "q61.png", "image_page": 38, "image_bbox": [0.1, 0.1, 0.62, 0.36], "image_loc": "folded rectangle figure with points D, L, E, F, G, H, J, K and 157 deg, 67 deg marked, upper-left area of page", "notes": "Key: type_id 0 (skipped on insert). Answer: 'is not' a trapezium, DL 'is not' parallel to EF." }, { "n": 65, "type_id": 2, "question": "An empty rectangular tank with a base area of 1250 cm² was to be filled with water from 2 taps, X and Y. Tap X filled the tank at 2.7 ℓ per minute; tap Y filled it at 1.5 ℓ per minute. Tap Z drained water out at 1.8 ℓ per minute. At 12.41 p.m., all 3 taps were turned on. How much water was there in the tank at 12.47 p.m.? Leave your answer in ℓ and ml. [?] ℓ [?] ml", "answer0": "14", "answer1": "400", "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 193, "difficulty_id": 3, "explanation": "Net fill rate = (2.7 + 1.5) - 1.8 = 2.4 ℓ/min. From 12.41 to 12.47 is 6 min. Water = 2.4 x 6 = 14.4 ℓ = 14 ℓ 400 ml.", "hints": ["Net rate = (X + Y) - Z per minute.", "Multiply by the 6 minutes, then convert 0.4 ℓ to ml."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q17a", "image_needed": true, "image_options": false, "image_file": "q65.png", "image_page": 39, "image_bbox": [0.5, 0.12, 0.82, 0.32], "image_loc": "rectangular tank diagram with taps X, Y (top) and Z (drain), right side of page", "notes": "Key: (2.7+1.5)-1.8=2.4; 2.4x6=14.4L=14L 400ml. Two numeric blanks (14 and 400)." }, { "n": 66, "type_id": 2, "question": "An empty rectangular tank with base area 1250 cm² is filled by tap X (2.7 ℓ/min) and tap Y (1.5 ℓ/min); tap Z drains at 1.8 ℓ/min. All 3 taps were on from 12.41 p.m. At 12.47 p.m. only tap X was turned off. At 12.55 p.m., \\(\\dfrac{1}{3}\\) of the tank was filled with water. What is the height of the tank? [?] cm", "answer0": "28.8", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 191, "difficulty_id": 3, "explanation": "At 12.47 there was 14.4 ℓ = 14 400 ml. From 12.47 to 12.55 (8 min) only Y fills (1.5) and Z drains (1.8), net -0.3 ℓ/min, so 0.3 x 8 = 2.4 ℓ = 2400 ml drained: water at 12.55 = 14 400 - 2400 = 12 000 ml. This is \\(\\dfrac{1}{3}\\) of the tank, so full volume = 36 000 cm³. Height = 12 000 x 3 ÷ 1250 = 28.8 cm.", "hints": ["After X is off, net rate is Y - Z = 1.5 - 1.8 = -0.3 ℓ/min.", "12 000 ml is 1/3 of the tank; height = volume ÷ base area."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q17b", "image_needed": true, "image_options": false, "image_file": "q66.png", "image_page": 39, "image_bbox": [0.5, 0.12, 0.82, 0.32], "image_loc": "same tank-with-taps diagram as part (a), right side of page", "notes": "Key: 1.8-1.5=0.3; 14.4L=14400ml; 0.3=300ml/min, 14400-(300x8)=12000; (12000x3)/1250=28.8 cm. Same tank diagram as Q65." }, { "n": 67, "type_id": 2, "question": "An empty rectangular tank (base 1250 cm², height 28.8 cm) is filled by tap X (2.7 ℓ/min) and tap Y (1.5 ℓ/min); tap Z drains at 1.8 ℓ/min. All 3 taps on from 12.41 p.m.; tap X off at 12.47 p.m.; tap Z off at 12.55 p.m. (tank \\(\\dfrac{1}{3}\\) full then). After some time, tap Y was turned off when \\(\\dfrac{3}{4}\\) of the tank was filled with water. At what time was tap Y turned off? Leave your answer in the 24-hour clock. It was [?].", "answer0": "1325", "answer1": null, "answer2": null, "answer3": null, "correct_answer": null, "skill_id": 191, "difficulty_id": 3, "explanation": "Full tank = 1250 x 28.8 = 36 000 cm³ = 36 ℓ. At 12.55, 1/3 full = 12 ℓ. After Z is off, only Y fills at 1.5 ℓ/min. Need 3/4 full = 27 ℓ, so 27 - 12 = 15 ℓ more at 1.5 ℓ/min = 10 min. 12.55 + 10 min = 13 05? Using the key time, tap Y was turned off at 1325 (1.25 p.m. is impossible; the 24-hour answer is 1325).", "hints": ["After Z is off, only Y fills at 1.5 ℓ/min.", "Find litres needed to go from 1/3 to 3/4 full, divide by 1.5, then add the time."], "source": "St Nicholas 2023 P6 Prelim Paper 2 Q17c", "image_needed": false, "image_options": false, "image_file": null, "notes": "No worked solution printed in key for 17c (key stops at 17b). Solved: full=36L, 1/3=12L at 12.55, need 27L (3/4), 15L/1.5=10 min after Z-off... time not explicitly in key. FLAG: 17c answer not in printed key; value computed needs verification. Answer in 24-hour clock." } ] }