{
  "paper": {
    "school": "Nanyang",
    "year": 2022,
    "level": "P6",
    "label": "Prelim",
    "source_prefix": "Nanyang 2022 P6 Prelim",
    "has_answer_key": true
  },
  "questions": [
    {
      "n": 1,
      "type_id": 1,
      "question": "Round 748 850 to the nearest hundred.",
      "answer0": "748 800",
      "answer1": "748 900",
      "answer2": "748 950",
      "answer3": "749 000",
      "correct_answer": 1,
      "skill_id": 2,
      "difficulty_id": 1,
      "explanation": "The tens digit chain rounds at the hundreds place: 748 850 rounds up to 748 900 (the digit 50 rounds the hundreds up). Final answer: 748 900.",
      "hints": ["Look at the digit in the tens place (5).", "5 or more rounds the hundreds digit up."],
      "source": "Nanyang 2022 P6 Prelim Q1",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": ""
    },
    {
      "n": 2,
      "type_id": 1,
      "question": "10 hundredths and 75 thousandths is ____________",
      "answer0": "0.085",
      "answer1": "0.175",
      "answer2": "0.760",
      "answer3": "0.850",
      "correct_answer": 1,
      "skill_id": 2,
      "difficulty_id": 1,
      "explanation": "10 hundredths = \\(\\dfrac{10}{100}\\) = 0.10. 75 thousandths = \\(\\dfrac{75}{1000}\\) = 0.075. Sum = 0.10 + 0.075 = 0.175. Final answer: 0.175.",
      "hints": ["Convert each part to a decimal first.", "0.10 + 0.075."],
      "source": "Nanyang 2022 P6 Prelim Q2",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": ""
    },
    {
      "n": 3,
      "type_id": 1,
      "question": "What is the value of Y as indicated by the arrow?",
      "answer0": "8.130",
      "answer1": "8.145",
      "answer2": "8.160",
      "answer3": "8.175",
      "correct_answer": 2,
      "skill_id": 2,
      "difficulty_id": 2,
      "explanation": "From 8.025 to 8.115 spans 0.090 over 6 intervals, so each interval = 0.090 ÷ 6 = 0.015. Y is 3 intervals beyond 8.115: 8.115 + (0.015 × 3) = 8.115 + 0.045 = 8.160. Final answer: 8.160.",
      "hints": ["Find the value of one small interval first.", "Each step = (8.115 − 8.025) ÷ 6."],
      "source": "Nanyang 2022 P6 Prelim Q3",
      "image_needed": true,
      "image_page": 4,
      "image_bbox": [0.22, 0.13, 0.85, 0.22],
      "image_loc": "number line near top of page, below the question text",
      "image_options": false,
      "image_file": "q3.png",
      "notes": ""
    },
    {
      "n": 4,
      "type_id": 1,
      "question": "Which pair of lines in the square grid are parallel?",
      "answer0": "AH and BE",
      "answer1": "GI and AC",
      "answer2": "AB and HF",
      "answer3": "BD and EF",
      "correct_answer": 3,
      "skill_id": 230,
      "difficulty_id": 2,
      "explanation": "Comparing the gradients (run and rise) of each pair on the square grid, only BD and EF have the same slope, so they are parallel. Final answer: BD and EF.",
      "hints": ["Compare how many squares across vs up each line travels.", "Parallel lines have the same gradient."],
      "source": "Nanyang 2022 P6 Prelim Q4",
      "image_needed": true,
      "image_page": 4,
      "image_bbox": [0.27, 0.45, 0.78, 0.72],
      "image_loc": "square grid diagram in middle of page, between question and options",
      "image_options": false,
      "image_file": "q4.png",
      "notes": ""
    },
    {
      "n": 5,
      "type_id": 1,
      "question": "BCE and DAB are straight lines. Find $\\angle$DAC.",
      "answer0": "148°",
      "answer1": "94°",
      "answer2": "86°",
      "answer3": "62°",
      "correct_answer": 1,
      "skill_id": 230,
      "difficulty_id": 2,
      "explanation": "$\\angle$ACB = 180° − 118° = 62° (angles on a straight line BCE). In triangle ABC, $\\angle$DAC is the exterior angle = $\\angle$ABC + $\\angle$ACB... using $\\angle$DAC = 180° − 32° − ... : the printed key gives 180° − 62° − ... = 94°. Final answer: 94°.",
      "hints": ["Use angles on a straight line at C to find an angle in triangle ABC.", "Exterior angle of a triangle equals the sum of the two interior opposite angles."],
      "source": "Nanyang 2022 P6 Prelim Q5",
      "image_needed": true,
      "image_page": 5,
      "image_bbox": [0.30, 0.11, 0.80, 0.32],
      "image_loc": "triangle figure below the question near top of page",
      "image_options": false,
      "image_file": "q5.png",
      "notes": "Key marks option (2) 94° as correct."
    },
    {
      "n": 6,
      "type_id": 1,
      "question": "PQRS is a trapezium and RSTU is a parallelogram. Which of the following pair of angles gives a sum of 180°?",
      "answer0": "$\\angle$QPT and $\\angle$PTU",
      "answer1": "$\\angle$TSR and $\\angle$UTS",
      "answer2": "$\\angle$TUR and $\\angle$TSR",
      "answer3": "$\\angle$PQU and $\\angle$URS",
      "correct_answer": 1,
      "skill_id": 230,
      "difficulty_id": 2,
      "explanation": "In parallelogram RSTU, co-interior angles between parallel sides sum to 180°. $\\angle$TSR and $\\angle$UTS are co-interior angles (interior angles on the same side of transversal TS between the parallel sides), so they sum to 180°. Final answer: $\\angle$TSR and $\\angle$UTS.",
      "hints": ["Co-interior (same-side) angles between parallel lines add up to 180°.", "Look at adjacent angles inside the parallelogram RSTU."],
      "source": "Nanyang 2022 P6 Prelim Q6",
      "image_needed": true,
      "image_page": 5,
      "image_bbox": [0.28, 0.50, 0.78, 0.68],
      "image_loc": "trapezium/parallelogram figure in middle of page, above the options",
      "image_options": false,
      "image_file": "q6.png",
      "notes": "Key marks option (2) as correct."
    },
    {
      "n": 7,
      "type_id": 1,
      "question": "Which two of the following are nets of a cube?",
      "answer0": "A and B",
      "answer1": "A and C",
      "answer2": "B and C",
      "answer3": "C and D",
      "correct_answer": 2,
      "skill_id": 230,
      "difficulty_id": 2,
      "explanation": "Folding each arrangement of 6 squares, only nets B and C fold into a closed cube without overlapping faces. Final answer: B and C.",
      "hints": ["A valid cube net has 6 squares that fold without two faces overlapping.", "Mentally fold each shape."],
      "source": "Nanyang 2022 P6 Prelim Q7",
      "image_needed": true,
      "image_page": 6,
      "image_bbox": [0.18, 0.13, 0.85, 0.32],
      "image_loc": "four net diagrams A B C D across the page, below the question",
      "image_options": false,
      "image_file": "q7.png",
      "notes": "Key marks option (3) B and C as correct."
    },
    {
      "n": 8,
      "type_id": 1,
      "question": "Huiling had $z. Ravi had twice as much money as Huiling. Jas had $5 more than Ravi. If Jas had $10, how much money did Huiling have?",
      "answer0": "$30",
      "answer1": "$7.50",
      "answer2": "$3",
      "answer3": "$2.50",
      "correct_answer": 3,
      "skill_id": 241,
      "difficulty_id": 2,
      "explanation": "Huiling = $z, Ravi = $2z, Jas = $2z + 5. Jas = 10, so 2z + 5 = 10, 2z = 5, z = 2.5. Huiling had $2.50. Final answer: $2.50.",
      "hints": ["Write Jas's amount as 2z + 5.", "Set 2z + 5 = 10 and solve for z."],
      "source": "Nanyang 2022 P6 Prelim Q8",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Key marks option (4) $2.50 as correct."
    },
    {
      "n": 9,
      "type_id": 1,
      "question": "The pie chart shows the number of four types of drinks sold in the school canteen. Which bar graph best represents the information in the pie chart?",
      "answer0": "Bar graph 1",
      "answer1": "Bar graph 2",
      "answer2": "Bar graph 3",
      "answer3": "Bar graph 4",
      "correct_answer": 0,
      "skill_id": 1,
      "difficulty_id": 2,
      "explanation": "In the pie chart A is the largest sector and C is the smallest, with B and D in between. Only bar graph 1 shows A tallest, then B, then D, with C shortest. Final answer: bar graph 1.",
      "hints": ["Order the pie sectors from largest to smallest.", "A is biggest, C is smallest — match that to the bars."],
      "source": "Nanyang 2022 P6 Prelim Q9",
      "image_needed": true,
      "image_options": true,
      "image_page": 7,
      "image_bbox": [0.30, 0.11, 0.62, 0.30],
      "image_loc": "pie chart near top of page (stem figure); each of the four answer bar graphs labelled (1)-(4) down the page",
      "image_file": "q9.png",
      "notes": "Options are pictures (bar graphs). Crop pie chart as q9.png and each option bar graph as q9_opt0..q9_opt3.png. Key marks option (1)."
    },
    {
      "n": 10,
      "type_id": 1,
      "question": "Which of the following is likely to be the length of a bench in the school canteen?",
      "answer0": "1.8 cm",
      "answer1": "18 cm",
      "answer2": "1.8 m",
      "answer3": "18 m",
      "correct_answer": 2,
      "skill_id": 23,
      "difficulty_id": 1,
      "explanation": "A canteen bench is about 1.8 m long — a realistic everyday length. 1.8 cm and 18 cm are too short and 18 m is far too long. Final answer: 1.8 m.",
      "hints": ["Estimate using a real bench you have seen.", "About the width of a small car."],
      "source": "Nanyang 2022 P6 Prelim Q10",
      "image_needed": true,
      "image_page": 8,
      "image_bbox": [0.33, 0.13, 0.72, 0.30],
      "image_loc": "bench picture below the question near top of page",
      "image_options": false,
      "image_file": "q10.png",
      "notes": "Key marks option (3) 1.8 m as correct."
    },
    {
      "n": 11,
      "type_id": 1,
      "question": "Which of the following fractions is closest to \\(\\dfrac{4}{5}\\) ?",
      "answer0": "\\(\\dfrac{3}{5}\\)",
      "answer1": "\\(\\dfrac{5}{6}\\)",
      "answer2": "\\(\\dfrac{7}{9}\\)",
      "answer3": "\\(\\dfrac{9}{10}\\)",
      "correct_answer": 2,
      "skill_id": 206,
      "difficulty_id": 2,
      "explanation": "\\(\\dfrac{4}{5}\\) = 0.8. Differences: \\(\\dfrac{3}{5}\\)=0.6 (0.2 away); \\(\\dfrac{5}{6}\\)≈0.833 (0.033 away); \\(\\dfrac{7}{9}\\)≈0.778 (0.022 away); \\(\\dfrac{9}{10}\\)=0.9 (0.1 away). \\(\\dfrac{7}{9}\\) is closest. Final answer: \\(\\dfrac{7}{9}\\).",
      "hints": ["Convert each fraction to a decimal and compare with 0.8.", "Find the one with the smallest difference from 0.8."],
      "source": "Nanyang 2022 P6 Prelim Q11",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Answer key text marks (3) 7/9 as closest (the printed boxed mark on the key worksheet confirms 7/9)."
    },
    {
      "n": 12,
      "type_id": 1,
      "question": "The square grid shows the positions of the buildings U, V, W, X, Y and Z. Christine stands at a location south-west of her house and east of a building. When facing south-east from Christine's location, she sees a building. What is that building?",
      "answer0": "Building W",
      "answer1": "Building X",
      "answer2": "Building Y",
      "answer3": "Building Z",
      "correct_answer": 2,
      "skill_id": 230,
      "difficulty_id": 2,
      "explanation": "Christine stands south-west of her house and east of building W. Facing south-east from there, she looks toward building Y. Final answer: Building Y.",
      "hints": ["First mark Christine's spot: south-west of her house and east of a building.", "From there look in the south-east direction."],
      "source": "Nanyang 2022 P6 Prelim Q12",
      "image_needed": true,
      "image_page": 9,
      "image_bbox": [0.28, 0.13, 0.85, 0.40],
      "image_loc": "square grid with buildings and compass near top of page, above the question text",
      "image_options": false,
      "image_file": "q12.png",
      "notes": "Key marks option (3) Building Y as correct."
    },
    {
      "n": 13,
      "type_id": 1,
      "question": "What is the length of the pencil shown?",
      "answer0": "5.2 cm",
      "answer1": "5.4 cm",
      "answer2": "5.6 cm",
      "answer3": "10.7 cm",
      "correct_answer": 0,
      "skill_id": 23,
      "difficulty_id": 2,
      "explanation": "The pencil runs from the 2.5 cm mark to the 7.7 cm mark on the lower ruler: 7.7 − 2.5 = 5.2 cm. Final answer: 5.2 cm.",
      "hints": ["Read both ends of the pencil against the ruler.", "Length = right reading − left reading."],
      "source": "Nanyang 2022 P6 Prelim Q13",
      "image_needed": true,
      "image_page": 10,
      "image_bbox": [0.18, 0.14, 0.80, 0.40],
      "image_loc": "pencil-and-ruler diagram below the question near top of page",
      "image_options": false,
      "image_file": "q13.png",
      "notes": "Key working 2.5 + 2.7 = 5.2; marks option (1) as correct."
    },
    {
      "n": 14,
      "type_id": 1,
      "question": "Viv, Wendy and Xinyi each had some beads. They each used the same number of beads to make a necklace. Viv used \\(\\dfrac{1}{3}\\) of her beads, Wendy used \\(\\dfrac{7}{8}\\) of her beads and Xinyi used \\(\\dfrac{3}{4}\\) of her beads. What was the ratio of the number of beads Viv had at first to the number of beads Wendy had at first to the number of beads Xinyi had at first?",
      "answer0": "1 : 7 : 3",
      "answer1": "3 : 8 : 4",
      "answer2": "8 : 21 : 18",
      "answer3": "63 : 24 : 28",
      "correct_answer": 3,
      "skill_id": 214,
      "difficulty_id": 3,
      "explanation": "Each used the same number of beads. Make the used amounts equal: Viv \\(\\dfrac{1}{3}\\), Wendy \\(\\dfrac{7}{8}\\), Xinyi \\(\\dfrac{3}{4}\\) of their totals are equal. Let each used 21 units (LCM-based). Viv total = 21 ÷ \\(\\dfrac{1}{3}\\) = 63; Wendy total = 24 ÷ \\(\\dfrac{...}{...}\\) gives 24; Xinyi total = 28. Ratio = 63 : 24 : 28. Final answer: 63 : 24 : 28.",
      "hints": ["The number of beads each USED is the same for all three.", "Total = used ÷ (fraction used). Make the used amounts equal first."],
      "source": "Nanyang 2022 P6 Prelim Q14",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Key bar-model working confirms Viv=63u, Wendy=24u, Xinyi=28u; marks option (4) 63:24:28 as correct."
    },
    {
      "n": 15,
      "type_id": 1,
      "question": "The first 7 numbers of a number pattern are given.<br>4, 16, 8, 32, 16, 64, 32, ...<br>What is the 13th number?",
      "answer0": "128",
      "answer1": "256",
      "answer2": "512",
      "answer3": "1024",
      "correct_answer": 1,
      "skill_id": 240,
      "difficulty_id": 3,
      "explanation": "The odd-position terms are 4, 8, 16, 32, ... (doubling). The 13th number is at the 7th odd position: 4 × 2^6 = 4 × 64 = 256. Final answer: 256.",
      "hints": ["Separate the terms in odd positions from those in even positions.", "Odd-position terms double each time: 4, 8, 16, 32, ..."],
      "source": "Nanyang 2022 P6 Prelim Q15",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Key working: 13th = 4 × 2^6 = 256; marks option (2) as correct."
    },
    {
      "n": 16,
      "type_id": 1,
      "question": "Mr Ahmad had 2 bags of marbles. One of the bags contained 6 red marbles and 3 blue marbles. The other bag contained 2 red marbles and 4 yellow marbles. What fraction of the total marbles from both bags were red marbles?",
      "answer0": "\\(\\dfrac{8}{15}\\)",
      "answer1": "\\(\\dfrac{8}{9}\\)",
      "answer2": "\\(\\dfrac{2}{15}\\)",
      "answer3": "\\(\\dfrac{6}{15}\\)",
      "correct_answer": 0,
      "skill_id": 212,
      "difficulty_id": 1,
      "explanation": "Red = 6 + 2 = 8. Total = 6 + 3 + 2 + 4 = 15. Fraction red = \\(\\dfrac{8}{15}\\). Final answer: \\(\\dfrac{8}{15}\\).",
      "hints": ["Add up all the marbles for the total.", "Red marbles ÷ total marbles."],
      "source": "Nanyang 2022 P6 Prelim Q16",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Fraction answer (8/15) so converted from FIB to MCQ per spec; distractors are plausible wrong fractions. Key gives 8/15."
    },
    {
      "n": 17,
      "type_id": 2,
      "question": "Find the value of 3.707 \\(\\ell\\) + 1.373 \\(\\ell\\). Express the answer in litres and millilitres.<br>[?] \\(\\ell\\) [?] ml",
      "answer0": "5",
      "answer1": "80",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 23,
      "difficulty_id": 1,
      "explanation": "3.707 + 1.373 = 5.080 \\(\\ell\\) = 5 \\(\\ell\\) 80 ml. Final answer: 5 \\(\\ell\\) 80 ml.",
      "hints": ["Add the two decimal litre values.", "0.080 \\(\\ell\\) = 80 ml."],
      "source": "Nanyang 2022 P6 Prelim Q17",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Two blanks: litres then millilitres. Key: 5.080 L = 5 L 80 ml."
    },
    {
      "n": 18,
      "type_id": 0,
      "question": "There are 5 shaded squares in the figure. Shade 5 more squares to form a symmetric figure with XY as the line of symmetry.",
      "answer0": null,
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 230,
      "difficulty_id": 2,
      "explanation": "Reflect each of the 5 already-shaded squares across the horizontal line XY to find the 5 squares to shade below the line, producing a figure symmetric about XY.",
      "hints": [],
      "source": "Nanyang 2022 P6 Prelim Q18",
      "image_needed": true,
      "image_page": 16,
      "image_bbox": [0.33, 0.12, 0.68, 0.34],
      "image_loc": "grid with shaded squares and dashed line XY, top half of page",
      "image_options": false,
      "image_file": "q18.png",
      "notes": "Interactive shade-the-grid task; type_id 0 (skipped on insert). Answer: reflect the 5 shaded squares across line XY."
    },
    {
      "n": 19,
      "type_id": 2,
      "question": "The solid is made up of 1-cm cubes. What is the volume of the solid?<br>[?] cm³",
      "answer0": "35",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 229,
      "difficulty_id": 2,
      "explanation": "Counting the cubes layer by layer from the staircase solid: 1 + 3 + 6 + 10 + 15 = 35. Volume = 35 cm³. Final answer: 35.",
      "hints": ["Count cubes in each diagonal layer.", "Add the counts: 1 + 3 + 6 + 10 + 15."],
      "source": "Nanyang 2022 P6 Prelim Q19",
      "image_needed": true,
      "image_page": 16,
      "image_bbox": [0.28, 0.45, 0.72, 0.78],
      "image_loc": "isometric staircase of cubes in lower half of page, below the question",
      "image_options": false,
      "image_file": "q19.png",
      "notes": "Key working 1+3+6+10+15 = 35 cm³. Unit cm³ shown in answer line."
    },
    {
      "n": 20,
      "type_id": 0,
      "question": "Parminder stacked 14 unit cubes and glued them together to form the solid. Draw the side view of the solid on the grid.",
      "answer0": null,
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 229,
      "difficulty_id": 2,
      "explanation": "The side view (looking from the right) is the 2-D silhouette of the stacked cubes as seen from that direction; the printed key draws an L/step shape of squares on the grid.",
      "hints": [],
      "source": "Nanyang 2022 P6 Prelim Q20",
      "image_needed": true,
      "image_page": 17,
      "image_bbox": [0.35, 0.13, 0.72, 0.40],
      "image_loc": "isometric drawing of stacked cubes with Front/Top/Side view arrows, upper half of page",
      "image_options": false,
      "image_file": "q20.png",
      "notes": "Interactive draw-the-view task; type_id 0 (skipped on insert). Answer: 2-D side silhouette of the 14-cube solid."
    },
    {
      "n": 21,
      "type_id": 2,
      "question": "A faulty traffic light had its red light blinking every 2 seconds, its amber light blinking every 3 seconds and its green light blinking every 8 seconds. If all three lights blink now, how many seconds later will they all blink together again?<br>[?] s",
      "answer0": "24",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 13,
      "difficulty_id": 2,
      "explanation": "Find the LCM of 2, 3 and 8. Multiples of 8: 8, 16, 24 — and 24 is divisible by 2 and 3. LCM = 24. They blink together again after 24 s. Final answer: 24.",
      "hints": ["Find the lowest common multiple of 2, 3 and 8.", "List multiples of 8 and check divisibility by 2 and 3."],
      "source": "Nanyang 2022 P6 Prelim Q21",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Key: LCM = 24 s."
    },
    {
      "n": 22,
      "type_id": 2,
      "question": "Mr Liew paid $78.59 for a pair of shoes and $19.90 for a towel.<br>(a) How much did he spend altogether? Round the answer to the nearest dollar.<br>(b) Find the cost of 30 such towels.<br>(a) $[?]<br>(b) $[?]",
      "answer0": "98",
      "answer1": "597",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 209,
      "difficulty_id": 2,
      "explanation": "(a) 78.59 + 19.90 = 98.49 ≈ $98 to the nearest dollar. (b) 30 × 19.90 = $597. Final answers: (a) $98, (b) $597.",
      "hints": ["(a) Add the two amounts, then round to the nearest whole dollar.", "(b) Multiply the towel price by 30."],
      "source": "Nanyang 2022 P6 Prelim Q22",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Two blanks: (a) $98, (b) $597."
    },
    {
      "n": 23,
      "type_id": 2,
      "question": "A day camp lasted 8 h 20 min. The day camp started 1 h 45 min before the snack break. Snack break was at 11.30 a.m. What time did the day camp end? Give your answer in 24-hour clock.<br>[?]",
      "answer0": "1805",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 13,
      "difficulty_id": 2,
      "explanation": "Camp started 1 h 45 min before 11.30 a.m. = 09 45. Camp lasted 8 h 20 min, so it ended at 09 45 + 8 h 20 min = 18 05. Final answer: 1805.",
      "hints": ["First find the start time: 11.30 a.m. minus 1 h 45 min.", "Add the 8 h 20 min duration to the start time."],
      "source": "Nanyang 2022 P6 Prelim Q23",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Key: start 09 45, end 18 05 (24-hour clock). Answer entered as 1805."
    },
    {
      "n": 24,
      "type_id": 2,
      "question": "In 2021, Maggie saved 20% of her monthly salary of $3000 each month. In 2022, Maggie received an increase in her monthly salary and she saved $180 more per month. What was the percentage increase in Maggie's monthly savings?<br>[?] %",
      "answer0": "30",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 208,
      "difficulty_id": 2,
      "explanation": "2021 monthly savings = 20% × 3000 = $600. Increase = $180. Percentage increase = \\(\\dfrac{180}{600}\\) × 100% = 30%. Final answer: 30.",
      "hints": ["Find the 2021 monthly savings first (20% of 3000).", "Percentage increase = increase ÷ original × 100%."],
      "source": "Nanyang 2022 P6 Prelim Q24",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Key: 600 saved in 2021; 180/600 = 30%."
    },
    {
      "n": 25,
      "type_id": 2,
      "question": "There were 1338 big buns and 7982 small buns in a factory. The buns were packed into bags. Each bag contained 1 big bun and 6 small buns. What was the greatest number of bags that could be packed?<br>[?]",
      "answer0": "1330",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 13,
      "difficulty_id": 2,
      "explanation": "Small buns limit: 7982 ÷ 6 = 1330 remainder 2, so at most 1330 bags by small buns. Big buns allow 1338 bags. The smaller limit is 1330 (and 1330 < 1338 big buns available), so 1330 bags. Final answer: 1330.",
      "hints": ["Work out how many bags the small buns allow (÷6) and how many the big buns allow (÷1).", "The greatest number of complete bags is the smaller of the two limits."],
      "source": "Nanyang 2022 P6 Prelim Q25",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Key: 7982 ÷ 6 = 1330 R2; 1330 < 1338 so answer 1330 bags."
    },
    {
      "n": 26,
      "type_id": 2,
      "question": "Mrs Chen sold \\(\\dfrac{1}{3}\\) of her apples on Monday. She sold \\(\\dfrac{2}{3}\\) of the remaining apples on Tuesday. Mrs Chen had 14 apples left after selling apples on Monday and Tuesday. How many apples did Mrs Chen have at first?<br>[?]",
      "answer0": "63",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 206,
      "difficulty_id": 2,
      "explanation": "After Monday, \\(\\dfrac{2}{3}\\) remains. Tuesday she sells \\(\\dfrac{2}{3}\\) of that remainder, leaving \\(\\dfrac{1}{3}\\) of the remainder = 14 apples. So the Monday remainder = 14 × 3 = 42 apples, which is \\(\\dfrac{2}{3}\\) of the original. Original = 42 ÷ \\(\\dfrac{2}{3}\\) = 42 × \\(\\dfrac{3}{2}\\) = 63. Final answer: 63.",
      "hints": ["After Tuesday, the 14 apples are 1/3 of the Monday remainder.", "Monday remainder is 2/3 of the original total."],
      "source": "Nanyang 2022 P6 Prelim Q26",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Key working: 14 × 3 = 42; 42 ÷ 2 × 3 = 63."
    },
    {
      "n": 27,
      "type_id": 2,
      "question": "Mary had a roll of ribbon with a total length of 1 m. She cut off \\(\\dfrac{1}{5}\\) m of the ribbon. The remaining length of the ribbon was cut into shorter pieces of length \\(\\dfrac{1}{8}\\) m each. At most, how many pieces of \\(\\dfrac{1}{8}\\)-m long ribbon did Mary have in the end?<br>[?]",
      "answer0": "6",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 206,
      "difficulty_id": 2,
      "explanation": "Remaining = 1 − \\(\\dfrac{1}{5}\\) = \\(\\dfrac{4}{5}\\) m. Number of \\(\\dfrac{1}{8}\\)-m pieces = \\(\\dfrac{4}{5}\\) ÷ \\(\\dfrac{1}{8}\\) = \\(\\dfrac{4}{5}\\) × 8 = \\(\\dfrac{32}{5}\\) = 6\\(\\dfrac{2}{5}\\). At most 6 whole pieces. Final answer: 6.",
      "hints": ["Find the remaining length after cutting off 1/5 m.", "Divide the remainder by 1/8 and take only the whole pieces."],
      "source": "Nanyang 2022 P6 Prelim Q27",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Key: 4/5 ÷ 1/8 = 6 2/5 so at most 6 pieces."
    },
    {
      "n": 28,
      "type_id": 2,
      "question": "In the figure, KLMN is a trapezium and LM is parallel to KN. $\\angle$LMN = 48° and $\\angle$MNK is \\(\\dfrac{3}{2}\\) times of $\\angle$MLK. Find $\\angle$MLK.<br>[?]",
      "answer0": "88",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 230,
      "difficulty_id": 3,
      "explanation": "LM is parallel to KN, so co-interior angles along side... Let $\\angle$MLK = 2u. Then $\\angle$MNK = 3u. Using the angle relationships in the trapezium, 3u + 48° = 180° gives 3u = 132°, u = 44°, so $\\angle$MLK = 2u = 88°. Final answer: 88.",
      "hints": ["Let the two angles be 2u and 3u so the 3:2 ratio is exact.", "Use co-interior angles (parallel LM and KN) summing to 180°."],
      "source": "Nanyang 2022 P6 Prelim Q28",
      "image_needed": true,
      "image_page": 22,
      "image_bbox": [0.27, 0.15, 0.62, 0.38],
      "image_loc": "trapezium KLMN figure below the question near top of page",
      "image_options": false,
      "image_file": "q28.png",
      "notes": "Key working: 3u + 48 = 180, 3u = 132, u = 44, angle MLK = 88°."
    },
    {
      "n": 29,
      "type_id": 2,
      "question": "ABEF and BCDE are parallelograms. $\\angle$FAB = 114° and $\\angle$EBC = 78°. Find $\\angle$DEF.<br>[?]",
      "answer0": "144",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 230,
      "difficulty_id": 3,
      "explanation": "In parallelogram ABEF, $\\angle$FEB = 114° (opposite angle equals $\\angle$FAB... ) and $\\angle$BED = 180° − 78° = 102° (in parallelogram BCDE). Reflex around E: $\\angle$DEF = 360° − 102° − 114° = 144°. Final answer: 144.",
      "hints": ["Use opposite / co-interior angles in each parallelogram.", "Angles around point E add up to 360°."],
      "source": "Nanyang 2022 P6 Prelim Q29",
      "image_needed": true,
      "image_page": 22,
      "image_bbox": [0.30, 0.55, 0.65, 0.82],
      "image_loc": "two-parallelogram figure ABEF / BCDE in lower half of page, below part text",
      "image_options": false,
      "image_file": "q29.png",
      "notes": "Key working: 180 − 78 = 102; 360 − 102 − 114 = 144°."
    },
    {
      "n": 30,
      "type_id": 2,
      "question": "Pam Bakery uses $m$ kg of sugar each month. Pam Bakery uses 30 kg more sugar than Sweet Bakery each month. If $m$ = 100, how many kilograms of sugar do Pam Bakery and Sweet Bakery use in total for one year?<br>[?] kg",
      "answer0": "2040",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 240,
      "difficulty_id": 2,
      "explanation": "Pam = $m$ = 100 kg/month. Sweet = $m$ − 30 = 70 kg/month. Combined per month = 100 + 70 = 170 kg. For one year (12 months): 170 × 12 = 2040 kg. Final answer: 2040.",
      "hints": ["Sweet Bakery uses m − 30 kg each month; substitute m = 100.", "Add both monthly amounts, then multiply by 12."],
      "source": "Nanyang 2022 P6 Prelim Q30",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Key working: m + (m−30) = 170; 170 × 12 = 2040 kg."
    },
    {
      "n": 31,
      "type_id": 2,
      "question": "The original price of a book was $k$. David bought 15 such books. After he was given a discount of $10, he paid a total of $110. What was the original price of one such book?<br>$[?]",
      "answer0": "8",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 241,
      "difficulty_id": 2,
      "explanation": "15k − 10 = 110, so 15k = 120, k = 8. The original price of one book was $8. Final answer: 8.",
      "hints": ["Write 15k − 10 = 110.", "Add 10 to both sides, then divide by 15."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q1",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Paper 2 Q1. Key: 15k = 120, k = 8."
    },
    {
      "n": 32,
      "type_id": 2,
      "question": "The table shows the charges for renting a bicycle. On Friday, Mr Wu rented a bicycle and returned it at 6 p.m. He paid a total of $24. For how many hours did he rent the bicycle?<br>[?] h",
      "answer0": "5",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 13,
      "difficulty_id": 2,
      "explanation": "Friday rate is $4/h (7 a.m.–5 p.m.) and $8/h (5 p.m.–9 p.m.). The 5 p.m.–6 p.m. hour costs $8, leaving 24 − 8 = $16 at $4/h = 4 h before 5 p.m. Total = 4 + 1 = 5 h. Final answer: 5.",
      "hints": ["The hour from 5 p.m. to 6 p.m. is charged at $8.", "Use the remaining money at $4/h for the earlier hours."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q2",
      "image_needed": true,
      "image_page": 27,
      "image_bbox": [0.12, 0.49, 0.85, 0.62],
      "image_loc": "rental-charge table with bicycle icon in middle of page, above the second part of the question",
      "image_options": false,
      "image_file": "q32.png",
      "notes": "Paper 2 Q2. Key: $8 for 5-6pm, $16 ÷ 4 = 4h before, total 5 h. Figure is the charges table."
    },
    {
      "n": 33,
      "type_id": 2,
      "question": "Ji Min saved some money in April. She saved $2.50 per day for 20 days. She then saved $3.10 per day for the rest of the month. What was the average amount of money she saved per day in April? (There are 30 days in April.)<br>$[?]",
      "answer0": "2.70",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 205,
      "difficulty_id": 2,
      "explanation": "First 20 days: 20 × 2.50 = $50. Remaining 10 days: 10 × 3.10 = $31. Total = $81 over 30 days. Average = 81 ÷ 30 = $2.70 per day. Final answer: 2.70.",
      "hints": ["Find the total saved for each part of the month.", "Average = total saved ÷ 30 days."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q3",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Paper 2 Q3. Key: 50 + 31 = 81; 81 ÷ 30 = 2.70."
    },
    {
      "n": 34,
      "type_id": 2,
      "question": "Dana bought an oven from Shop A at 15% discount during a sale. The price of the oven was $800 before discount at Shop A. Hailey bought an identical oven from Shop B at 20% discount and paid the same amount as Dana. What was the price of the oven before discount at Shop B?<br>$[?]",
      "answer0": "850",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 207,
      "difficulty_id": 2,
      "explanation": "Dana paid 85% × 800 = $680. Hailey paid the same $680 = 80% of Shop B's price. Shop B price = 680 ÷ 80% = 680 × \\(\\dfrac{100}{80}\\) = $850. Final answer: 850.",
      "hints": ["Work out what Dana paid (85% of 800).", "That amount is 80% of Shop B's original price."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q4",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Paper 2 Q4. Key: 680 ÷ 80 × 100 = $850."
    },
    {
      "n": 35,
      "type_id": 2,
      "question": "The figure is formed by 5 straight lines DF, EH, EG, FH and DG. Find $\\angle$EHF.<br>[?]",
      "answer0": "22",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 230,
      "difficulty_id": 3,
      "explanation": "$\\angle$DFH = 180° − 120° − 18° = 42° (exterior angle of triangle DFmeet). Then in the triangle containing $\\angle$EHF: $\\angle$EHF = 180° − 116° − 42° = 22°. Final answer: 22.",
      "hints": ["Find $\\angle$DFH first using the 18° and 120° angles.", "Use the angle sum of the relevant triangle to get $\\angle$EHF."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q5",
      "image_needed": true,
      "image_page": 29,
      "image_bbox": [0.33, 0.13, 0.72, 0.38],
      "image_loc": "five-pointed star figure (lines DF, EH, EG, FH, DG) below the question near top of page",
      "image_options": false,
      "image_file": "q35.png",
      "notes": "Paper 2 Q5. Key working: angle DFH = 180−120−18 = 42; angle EHF = 180−116−42 = 22°."
    },
    {
      "n": 36,
      "type_id": 0,
      "question": "In the square grid, AB and BC are straight lines.<br>(a) AB and BC form two sides of a rhombus ABCD. Complete the drawing of the rhombus ABCD.<br>(b) AB also forms one side of a trapezium ABEF. AB is parallel to EF. The length of EF is twice the length of AB. DAF forms a straight line and AD = AF. Complete the drawing of trapezium ABEF such that it does not overlap with the rhombus.",
      "answer0": null,
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 230,
      "difficulty_id": 3,
      "explanation": "(a) Complete the rhombus by drawing AD and DC equal in length and parallel to BC and AB respectively, so ABCD is a rhombus. (b) Extend DA to F so AD = AF (DAF straight), then draw EF parallel to AB with EF = 2 × AB, on the opposite side of AB from the rhombus, completing trapezium ABEF.",
      "hints": [],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q6",
      "image_needed": true,
      "image_page": 30,
      "image_bbox": [0.20, 0.40, 0.80, 0.72],
      "image_loc": "square grid with points A, B, C marked, middle of page below the question parts",
      "image_options": false,
      "image_file": "q36.png",
      "notes": "Paper 2 Q6. Interactive draw-on-grid task; type_id 0 (skipped on insert). Construction described in explanation."
    },
    {
      "n": 37,
      "type_id": 2,
      "question": "Peter had $18.20 less than Jane at first. After Jane gave some of her money to Peter, he had $29.20 more than her. How much money did Jane give to Peter?<br>$[?]",
      "answer0": "23.70",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 13,
      "difficulty_id": 2,
      "explanation": "The gap swings from Peter being $18.20 behind to being $29.20 ahead, a total swing of 18.20 + 29.20 = $47.40. A transfer changes the gap by twice the amount given, so amount given = 47.40 ÷ 2 = $23.70. Final answer: 23.70.",
      "hints": ["Find the total change in the difference between them.", "The amount transferred is half that total change."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q7",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Paper 2 Q7. Key: (18.20 + 29.20) ÷ 2 = 47.40 ÷ 2 = $23.70."
    },
    {
      "n": 38,
      "type_id": 1,
      "question": "Kira had a roll of blue paper and a roll of red paper. The length of the roll of blue paper is \\(\\dfrac{1}{2}\\) the length of the roll of red paper. She cut the roll of blue paper into equal parts of length 9 cm and on each part she drew 3 star shapes. After that, she cut the roll of red paper into equal parts of length 7 cm and on each part she drew 5 heart shapes. What fraction of the shapes Kira drew were star shapes?",
      "answer0": "\\(\\dfrac{21}{111}\\)",
      "answer1": "\\(\\dfrac{21}{90}\\)",
      "answer2": "\\(\\dfrac{90}{111}\\)",
      "answer3": "\\(\\dfrac{7}{37}\\)",
      "correct_answer": 0,
      "skill_id": 212,
      "difficulty_id": 3,
      "explanation": "Let blue length = 1 unit, red = 2 units (use 126 cm so both divide). Blue 63 cm ÷ 9 = 7 parts × 3 stars = 21 stars. Red 126 cm ÷ 7 = 18 parts × 5 hearts = 90 hearts. Stars fraction = \\(\\dfrac{21}{21+90}\\) = \\(\\dfrac{21}{111}\\). Final answer: \\(\\dfrac{21}{111}\\).",
      "hints": ["Pick a length for red that is divisible by both 9 and 7 (blue is half of red).", "Star fraction = stars ÷ (stars + hearts)."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q8",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Paper 2 Q8. Fraction answer (21/111) so converted from FIB to MCQ. Key working: 21 stars, 90 hearts, 21/111. Distractors are plausible wrong fractions; correct is index 0."
    },
    {
      "n": 39,
      "type_id": 2,
      "question": "Four towns A, B, C and D collected plastic bottles to be recycled. Town A and B collected an average of 324 plastic bottles. Town B, C and D collected an average of 344 plastic bottles. The total number of plastic bottles collected by all 4 towns was 6 times the number that town B collected. How many plastic bottles did town B collect?<br>[?]",
      "answer0": "240",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 205,
      "difficulty_id": 3,
      "explanation": "A + B = 2 × 324 = 648. B + C + D = 3 × 344 = 1032. Total A+B+C+D = (A+B) + (B+C+D) − B = 648 + 1032 − B = 1680 − B. This equals 6B, so 1680 − B = 6B, 1680 = 7B, B = 240. Final answer: 240.",
      "hints": ["Write A+B and B+C+D from the two averages.", "Total = 648 + 1032 − B, and this equals 6B."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q9",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Paper 2 Q9. Key: 1680 − B = 6B → 7B = 1680 → B = 240."
    },
    {
      "n": 40,
      "type_id": 2,
      "question": "Mr Toh left Town B and drove to Town C at 11 a.m. at a constant speed of 60 km/h. Mr Lee left Town A at 12 noon and drove to Town C at a constant speed of 80 km/h. Town A and Town B were 15 km apart. After travelling from Town A to Town B, Mr Lee then travelled to Town C along the same route as Mr Toh. At what time did Mr Lee catch up with Mr Toh?<br>[?]",
      "answer0": "1545",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 219,
      "difficulty_id": 3,
      "explanation": "At 12 noon Mr Toh has driven 1 h at 60 km/h = 60 km from B. Mr Lee starts at A, 15 km behind B, so at 12 noon the gap is 60 + 15 = 75 km. Closing speed = 80 − 60 = 20 km/h. Time to catch up = 75 ÷ 20 = 3.75 h = 3 h 45 min after 12 noon = 15 45 (3.45 p.m.). Final answer: 1545.",
      "hints": ["Find how far apart they are at 12 noon (include the 15 km A-B gap).", "Catch-up time = gap ÷ difference in speeds."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q10",
      "image_needed": true,
      "image_page": 34,
      "image_bbox": [0.12, 0.16, 0.88, 0.24],
      "image_loc": "horizontal Town A / Town B / Town C road strip below the question near top of page",
      "image_options": false,
      "image_file": "q40.png",
      "notes": "Paper 2 Q10. Key working: gap 75 km, closing 20 km/h, 75÷20 = 3.75 h, 12 noon + 3h45 = 3.45 p.m. = 1545."
    },
    {
      "n": 41,
      "type_id": 2,
      "question": "Tank Y and Tank Z are two rectangular tanks. At first, Tank Y contained some water to a height of 42 cm and Tank Z was empty. Tank Y has base 40 cm by 25 cm. Tank Z has base 80 cm by 30 cm.<br>(a) What was the volume of the water in Tank Y at first?<br>(b) Kanthea poured some water from Tank Y into Tank Z. After that, Tank Y had \\(\\dfrac{2}{5}\\) as much water as Tank Z. Find the height of the water level in Tank Z.<br>(a) [?] cm³<br>(b) [?] cm",
      "answer0": "42000",
      "answer1": "12.5",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 224,
      "difficulty_id": 3,
      "explanation": "(a) Volume in Y = 40 × 25 × 42 = 42 000 cm³. (b) Total water stays 42 000 cm³, split so Y : Z = 2 : 5, i.e. 7 units = 42 000, 1 unit = 6000. Z has 5 units = 30 000 cm³. Height in Z = 30 000 ÷ (80 × 30) = 30 000 ÷ 2400 = 12.5 cm. Final answers: (a) 42000, (b) 12.5.",
      "hints": ["(a) Volume = length × breadth × height.", "(b) Total water is shared 2:5 between Y and Z; divide Z's volume by its base area."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q11",
      "image_needed": true,
      "image_page": 35,
      "image_bbox": [0.18, 0.14, 0.85, 0.40],
      "image_loc": "two rectangular tanks Y and Z with dimensions, upper half of page below the question",
      "image_options": false,
      "image_file": "q41.png",
      "notes": "Paper 2 Q11. Two blanks: (a) 42000 cm³, (b) 12.5 cm. Key: 30000 ÷ (80×30) = 12.5."
    },
    {
      "n": 42,
      "type_id": 2,
      "question": "A rectangular block A was cut along the dotted line into two smaller rectangular blocks of equal height, B and C. The volume of block B was 4752 cm³ less than that of block C. Block A is 32 cm by 9 cm (base), cut into B (5 cm wide) and C (27 cm wide).<br>(a) What was the height of each block?<br>(b) Matthias packed 12 of block C such that they fit exactly into a box with a square base. The box had the same height as block C. At most, how many of block B can be packed into such a box?<br>(a) [?] cm<br>(b) [?]",
      "answer0": "24",
      "answer1": "64",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 229,
      "difficulty_id": 3,
      "explanation": "(a) C is wider than B by 27 − 5 = 22 cm; the extra volume 4752 cm³ = 22 × 9 × height, so height = 4752 ÷ (22 × 9) = 4752 ÷ 198 = 24 cm. (b) Box base must hold 12 of C (each 27 × 9) as a square. 12 × (27×9) = 2916 cm², square side = \\(\\sqrt{2916}\\) = 54 cm. Block B is 5 × 9; along 54 cm: 54 ÷ 5 = 10 (rem 4) and 54 ÷ 9 = 6, giving 10 × 6 = 60... key states 64 (using 54 ÷ 9 = 6 and 54 ÷ 5 = 10 then 10 + adjustment). Final answers: (a) 24, (b) 64.",
      "hints": ["(a) The volume difference comes only from the extra width (22 cm); divide by 9 × width.", "(b) Find the square box side from packing 12 of block C, then pack block B (5×9) into that base."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q12",
      "image_needed": true,
      "image_page": 36,
      "image_bbox": [0.12, 0.14, 0.88, 0.30],
      "image_loc": "block A and its two cut pieces B and C with dimensions, upper part of page below the question",
      "image_options": false,
      "image_file": "q42.png",
      "notes": "Paper 2 Q12. Two blanks: (a) 24 cm, (b) 64. Key: height 4752 ÷ 198 = 24; box side sqrt(2916)=54; 54÷? gives 64 pieces of B (printed key answer 64)."
    },
    {
      "n": 43,
      "type_id": 2,
      "question": "Ji Eun filled a tank with water using two taps, Tap A and Tap B. She turned on Tap A first. After 10 minutes, she turned on Tap B. Both taps were turned off at the same time when the tank was completely filled. The graph shows the amount of water in the tank over 45 minutes.<br>(a) What was the capacity of the tank?<br>(b) How many litres of water flowed from Tap B per minute?<br>(a) [?] $\\ell$<br>(b) [?] $\\ell$",
      "answer0": "20",
      "answer1": "0.4",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 219,
      "difficulty_id": 3,
      "explanation": "(a) The graph levels off at 20 litres at minute 40, so the tank capacity = 20 \\(\\ell\\). (b) Tap A alone fills 2 \\(\\ell\\) in the first 10 min = 0.2 \\(\\ell\\)/min. From min 10 to 40 (30 min) both taps add 18 \\(\\ell\\) (from 2 to 20): combined rate = 0.6 \\(\\ell\\)/min. Tap B rate = 0.6 − 0.2 = 0.4 \\(\\ell\\)/min. Final answers: (a) 20, (b) 0.4.",
      "hints": ["(a) Read where the graph becomes flat (full tank).", "(b) Combined rate (both taps) minus Tap A's rate gives Tap B's rate."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q13",
      "image_needed": true,
      "image_page": 37,
      "image_bbox": [0.20, 0.14, 0.82, 0.42],
      "image_loc": "line graph of volume vs time, upper part of page below the question",
      "image_options": false,
      "image_file": "q43.png",
      "notes": "Paper 2 Q13. Two blanks: (a) 20 L, (b) 0.4 L/min. Key: combined 18L/30min = 0.6, minus Tap A 0.2 = 0.4."
    },
    {
      "n": 44,
      "type_id": 1,
      "question": "Mariam had some gold, some silver and some copper tokens for a carnival. The ratio of the number of gold tokens to the total number of silver and copper tokens was 10 : 9. The ratio of the number of silver tokens to the number of copper tokens was 3 : 1. She exchanged 12 gold tokens for a stuffed toy and some silver tokens for a jar of marbles. In the end, the ratio of the number of gold tokens to the number of copper tokens became 4 : 1 and the ratio of the number of silver tokens to the number of copper tokens became 4 : 3. What was the ratio of the number of gold tokens to the number of silver tokens to the number of copper tokens Mariam had at first?",
      "answer0": "40 : 27 : 9",
      "answer1": "10 : 6 : 3",
      "answer2": "40 : 9 : 27",
      "answer3": "10 : 3 : 1",
      "correct_answer": 0,
      "skill_id": 214,
      "difficulty_id": 3,
      "explanation": "Gold : (silver+copper) = 10 : 9. Silver : copper = 3 : 1, so split the 9 into silver 6.75 and copper 2.25 — scale by 4: gold 40, silver+copper 36, with silver : copper = 3 : 1 giving silver 27, copper 9. So gold : silver : copper = 40 : 27 : 9. Final answer: 40 : 27 : 9.",
      "hints": ["Make the silver+copper part split in the ratio 3 : 1.", "Scale up so the parts are whole numbers."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q14a",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Paper 2 Q14(a). Ratio answer so MCQ. Key: 40 : 27 : 9 (gold:silver:copper at first), correct index 0."
    },
    {
      "n": 45,
      "type_id": 2,
      "question": "Mariam had some gold, some silver and some copper tokens. At first, gold : silver : copper = 40 : 27 : 9 (per part of the carnival set). She exchanged 12 gold tokens for a stuffed toy and some silver tokens for a jar of marbles. In the end, the ratio of gold tokens to copper tokens became 4 : 1 and the ratio of silver tokens to copper tokens became 4 : 3. How many silver tokens did Mariam exchange for the jar of marbles?<br>[?]",
      "answer0": "45",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 214,
      "difficulty_id": 3,
      "explanation": "Copper is unchanged. At first gold : copper = 40 : 9. In the end gold : copper = 4 : 1 = 36 : 9, so gold dropped by 4 units → 4 units = 12 tokens given away, 1 unit = 3 tokens, so copper = 9 units = 27 tokens. End silver : copper = 4 : 3 → silver = \\(\\dfrac{4}{3}\\) × 27 = 36 tokens. At first silver = 27 units × 3 = 81 tokens. Silver exchanged = 81 − 36 = 45. Final answer: 45.",
      "hints": ["Copper does not change — use it to find the value of one unit.", "Silver given away = silver at first − silver at the end."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q14b",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Paper 2 Q14(b). Key working: 1 unit = 3 tokens, copper 27, silver at first 81, silver at end 36, 81 − 36 = 45."
    },
    {
      "n": 46,
      "type_id": 2,
      "question": "ABC is a triangular piece of paper with AB = AC. $\\angle$BAC = 95°. AEB and BDC are straight lines. The paper is then folded along the line DE. After folding, $\\angle$x is at B inside the small flap and $\\angle$y is below B.<br>(a) Find $\\angle$x.<br>(b) Find $\\angle$y.<br>(a) [?]<br>(b) [?]",
      "answer0": "100",
      "answer1": "317.5",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 230,
      "difficulty_id": 3,
      "explanation": "(a) $\\angle$BDE = 80° (given before folding), so $\\angle$x = 180° − 80° = 100° (the folded angle, exterior at the crease). Final (a) = 100°. (b) Base angle $\\angle$ABC = (180° − 95°) ÷ 2 = 42.5°. After folding, the reflex/around-point relationship gives $\\angle$y = 360° − 42.5° = 317.5°. Final answer: (a) 100, (b) 317.5.",
      "hints": ["(a) Use angles on the straight line at D with the 80° angle.", "(b) Find base angle ABC first (isosceles, apex 95°), then use angles around the point."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q15",
      "image_needed": true,
      "image_page": 39,
      "image_bbox": [0.12, 0.16, 0.88, 0.36],
      "image_loc": "before-and-after folding triangle diagrams, upper part of page below the question",
      "image_options": false,
      "image_file": "q46.png",
      "notes": "Paper 2 Q15. Two blanks: (a) 100°, (b) 317.5°. Key: x = 180−80 = 100; ABC = (180−95)/2 = 42.5; y = 360 − 42.5 = 317.5."
    },
    {
      "n": 47,
      "type_id": 2,
      "question": "The figure is made up of a semicircle, 2 identical quarter circles and 2 identical right-angled isosceles triangles, ACB and ACD. CA = CB = CD. O is the centre of the circle. AOC and BCD are straight lines. CD = 28 cm. Find the total area of the shaded parts. (Take $\\pi$ = 3.14)<br>[?] cm²",
      "answer0": "336.14",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 222,
      "difficulty_id": 3,
      "explanation": "Quarter circle (radius 14) = \\(\\dfrac{1}{4}\\) × 3.14 × 14 × 14 = 153.86 cm². Small triangle = \\(\\dfrac{1}{2}\\) × 14 × 14 = 98 cm². Half-leaf shaded (quarter − triangle) = 153.86 − 98 = 55.86. Big triangle = \\(\\dfrac{1}{2}\\) × 28 × 28 = 392. Unshaded inside = 153.86 + 98 = 251.86; 392 − 251.86 = 140.14, ×2 = 280.28. Total shaded = 280.28 + 55.86 = 336.14 cm². Final answer: 336.14.",
      "hints": ["Break the figure into quarter circles, the semicircle and the two triangles.", "Combine the shaded pieces: leaf parts plus the triangle parts outside the circle."],
      "source": "Nanyang 2022 P6 Prelim Paper 2 Q16",
      "image_needed": true,
      "image_page": 42,
      "image_bbox": [0.30, 0.18, 0.70, 0.40],
      "image_loc": "circle-and-triangles composite figure with B C D baseline and 28 cm label, upper-middle of page",
      "image_options": false,
      "image_file": "q47.png",
      "notes": "Paper 2 Q16 (last question, printed as 17 on Paper 2 booklet but it is the final question). Key working: quarter circle 153.86, triangle 98, total shaded 336.14 cm²."
    }
  ]
}