{
  "paper": {
    "school": "Nanyang",
    "year": 2023,
    "level": "P6",
    "label": "Prelim",
    "source_prefix": "Nanyang 2023 P6 Prelim",
    "has_answer_key": true
  },
  "questions": [
    {
      "n": 1,
      "type_id": 1,
      "question": "Round 76 523 to the nearest hundred.",
      "answer0": "76 500",
      "answer1": "76 000",
      "answer2": "77 000",
      "answer3": "80 000",
      "correct_answer": 0,
      "skill_id": 150,
      "difficulty_id": 1,
      "explanation": "The tens digit of 76 523 is 2 (less than 5), so round down: the hundreds digit stays 5. 76 523 rounds to 76 500.",
      "hints": ["Look at the tens digit to decide rounding.", "Tens digit is 2, so round down."],
      "source": "Nanyang 2023 P6 Prelim Q1",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: (1) 76 500"
    },
    {
      "n": 2,
      "type_id": 1,
      "question": "In 89.76, which digit is in the tenths place?",
      "answer0": "6",
      "answer1": "7",
      "answer2": "8",
      "answer3": "9",
      "correct_answer": 1,
      "skill_id": 123,
      "difficulty_id": 1,
      "explanation": "The first digit after the decimal point is the tenths place. In 89.76 that digit is 7.",
      "hints": ["The tenths place is the first digit after the decimal point.", "89.7|6 — the 7 is tenths."],
      "source": "Nanyang 2023 P6 Prelim Q2",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: (2) 7"
    },
    {
      "n": 3,
      "type_id": 1,
      "question": "What is the length of the paper clip?",
      "answer0": "1.1 cm",
      "answer1": "2.2 cm",
      "answer2": "2.6 cm",
      "answer3": "4.8 cm",
      "correct_answer": 1,
      "skill_id": 86,
      "difficulty_id": 1,
      "explanation": "The paper clip runs from the 2.6 cm mark to the 4.8 cm mark. Length = 4.8 − 2.6 = 2.2 cm.",
      "hints": ["Read both ends of the clip on the ruler.", "Subtract the start reading from the end reading."],
      "source": "Nanyang 2023 P6 Prelim Q3",
      "image_needed": true,
      "image_options": false,
      "image_file": "q3.png",
      "image_page": 4,
      "image_bbox": [0.33, 0.11, 0.72, 0.25],
      "image_loc": "paper clip resting on a cm ruler, below the question",
      "notes": "key: (2) 2.2 cm (4.8 − 2.6 = 2.2)"
    },
    {
      "n": 4,
      "type_id": 1,
      "question": "Which two lines in the square grid are parallel to each other?",
      "answer0": "AD and HE",
      "answer1": "AH and DE",
      "answer2": "BG and DE",
      "answer3": "BG and CF",
      "correct_answer": 3,
      "skill_id": 100,
      "difficulty_id": 2,
      "explanation": "Comparing the slopes (gradients) of the line segments on the grid, BG and CF have the same gradient and never meet, so they are parallel.",
      "hints": ["Parallel lines have the same slope on the grid.", "Count the across/up steps for each segment."],
      "source": "Nanyang 2023 P6 Prelim Q4",
      "image_needed": true,
      "image_options": false,
      "image_file": "q4.png",
      "image_page": 4,
      "image_bbox": [0.34, 0.46, 0.73, 0.74],
      "image_loc": "square grid with labelled segments A–H, below the question",
      "notes": "key: (4) BG and CF"
    },
    {
      "n": 5,
      "type_id": 1,
      "question": "WPRT is a square. QWU and SWV are straight lines. \\(\\angle QWP = 13^\\circ\\) and \\(\\angle SWT = 28^\\circ\\). Find \\(\\angle UWV\\).",
      "answer0": "48°",
      "answer1": "49°",
      "answer2": "50°",
      "answer3": "51°",
      "correct_answer": 1,
      "skill_id": 230,
      "difficulty_id": 2,
      "explanation": "\\(\\angle TWP = 90^\\circ\\) (square). \\(\\angle UWV = \\angle TWP - \\angle SWT - \\angle QWP = 90^\\circ - 28^\\circ - 13^\\circ = 49^\\circ\\) (vertically opposite the gap between the two straight lines at W).",
      "hints": ["Angle TWP in the square is 90°.", "UWV is vertically opposite, so 90° − 28° − 13°."],
      "source": "Nanyang 2023 P6 Prelim Q5",
      "image_needed": true,
      "image_options": false,
      "image_file": "q5.png",
      "image_page": 5,
      "image_bbox": [0.35, 0.13, 0.72, 0.40],
      "image_loc": "tilted square WPRT with rays U and V through W",
      "notes": "key: (2) 49° (90 − 28 − 13 = 49)"
    },
    {
      "n": 6,
      "type_id": 1,
      "question": "VWXY is a rhombus. XYZ is a straight line and \\(\\angle VWY = 36^\\circ\\). Find \\(\\angle VYZ\\).",
      "answer0": "144°",
      "answer1": "112°",
      "answer2": "108°",
      "answer3": "72°",
      "correct_answer": 2,
      "skill_id": 230,
      "difficulty_id": 2,
      "explanation": "A diagonal of a rhombus bisects the angles, so \\(\\angle VWY = \\angle XWY = 36^\\circ\\) and \\(\\angle VWX = 72^\\circ\\). \\(\\angle VYX = \\angle VWX = 72^\\circ\\) (opposite angles of rhombus). On straight line XYZ: \\(\\angle VYZ = 180^\\circ - 72^\\circ = 108^\\circ\\).",
      "hints": ["The diagonal WY bisects angle VWX, so VWX = 72°.", "Opposite angles of a rhombus are equal; then use angles on a straight line."],
      "source": "Nanyang 2023 P6 Prelim Q6",
      "image_needed": true,
      "image_options": false,
      "image_file": "q6.png",
      "image_page": 6,
      "image_bbox": [0.40, 0.14, 0.66, 0.40],
      "image_loc": "rhombus VWXY with line extended to Z",
      "notes": "key: (3) 108° (36+36=72; 180−72=108)"
    },
    {
      "n": 7,
      "type_id": 1,
      "question": "The figure shows a cuboid. Which one of the following is not a net of the cuboid?",
      "answer0": "Net 1",
      "answer1": "Net 2",
      "answer2": "Net 3",
      "answer3": "Net 4",
      "correct_answer": 2,
      "skill_id": 233,
      "difficulty_id": 2,
      "explanation": "Folding each net mentally, nets 1, 2 and 4 close up into the cuboid. Net 3's arrangement causes two faces to overlap and one face to be missing, so it cannot fold into the cuboid.",
      "hints": ["Try mentally folding each net.", "A valid net has 6 faces with no overlap."],
      "source": "Nanyang 2023 P6 Prelim Q7",
      "image_needed": true,
      "image_options": true,
      "image_file": null,
      "image_page": 7,
      "image_bbox": [0.45, 0.13, 0.62, 0.21],
      "image_loc": "cuboid drawing at top; four candidate nets are options (1)-(4) down the page",
      "notes": "key: (3). image_options: crop the four net diagrams to q7_opt0..q7_opt3 (opt0=net1 top, opt3=net4 bottom)"
    },
    {
      "n": 8,
      "type_id": 1,
      "question": "Kenneth had \\(j\\) pens at first. He gave away 9 pens and packed the remaining pens into 5 packets. There were 6 pens in each packet. How many pens did Kenneth have at first?",
      "answer0": "20",
      "answer1": "21",
      "answer2": "30",
      "answer3": "39",
      "correct_answer": 3,
      "skill_id": 241,
      "difficulty_id": 1,
      "explanation": "Pens packed = 5 × 6 = 30. He gave away 9 first, so at first j = 30 + 9 = 39.",
      "hints": ["Work backwards: 5 packets of 6 = 30 pens left.", "Add back the 9 given away: 30 + 9."],
      "source": "Nanyang 2023 P6 Prelim Q8",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: (4) 39 (5×6=30, 30+9=39)"
    },
    {
      "n": 9,
      "type_id": 1,
      "question": "The pie chart shows the amount of money Jessica spent on the different items on her trip. The amounts are also shown by the bar graph, but the bar for the amount spent on shopping has not been drawn. How much did Jessica spend on shopping?",
      "answer0": "$1400",
      "answer1": "$1600",
      "answer2": "$1800",
      "answer3": "$3200",
      "correct_answer": 0,
      "skill_id": 236,
      "difficulty_id": 3,
      "explanation": "From the bar graph: Hotel = $500, Food = $800, Airfare = $500. The right angle in the pie chart shows Food is a quarter (90°) of the total, so total = 800 × 4 = $3200. Shopping = 3200 − 800 − 500 − 500 = $1400.",
      "hints": ["The right-angle slice equals 1/4 of the total.", "Food = $800 is that quarter, so total = $3200; subtract the rest."],
      "source": "Nanyang 2023 P6 Prelim Q9",
      "image_needed": true,
      "image_options": false,
      "image_file": "q9.png",
      "image_page": 9,
      "image_bbox": [0.20, 0.11, 0.75, 0.62],
      "image_loc": "pie chart at top and bar graph below it (both needed)",
      "notes": "key: (1) $1400 (800×4=3200; 3200−800−500−500=1400). Crop must include BOTH the pie chart and the bar graph."
    },
    {
      "n": 10,
      "type_id": 1,
      "question": "The diagram shows the door of a classroom. Which of the following could be the height of the door?",
      "answer0": "0.2 m",
      "answer1": "2 m",
      "answer2": "2 cm",
      "answer3": "2000 cm",
      "correct_answer": 1,
      "skill_id": 86,
      "difficulty_id": 1,
      "explanation": "A classroom door is about 2 m tall. 0.2 m and 2 cm are far too short, 2000 cm (20 m) is far too tall. So 2 m.",
      "hints": ["Estimate a real door height.", "2000 cm = 20 m is unreasonable."],
      "source": "Nanyang 2023 P6 Prelim Q10",
      "image_needed": true,
      "image_options": false,
      "image_file": "q10.png",
      "image_page": 10,
      "image_bbox": [0.59, 0.13, 0.74, 0.31],
      "image_loc": "drawing of a classroom door, right of the options",
      "notes": "key: (2) 2 m"
    },
    {
      "n": 11,
      "type_id": 1,
      "question": "Three points are drawn on a square grid. Eve is standing within the grid. She stands at a location north-west of X and north of Y. In what direction is Z from Eve?",
      "answer0": "South-east",
      "answer1": "South-west",
      "answer2": "North-east",
      "answer3": "North-west",
      "correct_answer": 1,
      "skill_id": 452,
      "difficulty_id": 2,
      "explanation": "Eve is NW of X and directly N of Y, which places her above Y and to the upper-left region near Y's column. Z is to the lower-left of that position, so Z is south-west of Eve.",
      "hints": ["Locate Eve: directly above Y and up-left of X.", "Compare Z's position to Eve — it is below and to the left."],
      "source": "Nanyang 2023 P6 Prelim Q11",
      "image_needed": true,
      "image_options": false,
      "image_file": "q11.png",
      "image_page": 10,
      "image_bbox": [0.38, 0.40, 0.74, 0.62],
      "image_loc": "square grid with points X, Y, Z and a North arrow",
      "notes": "key: (2) South-west"
    },
    {
      "n": 12,
      "type_id": 1,
      "question": "Andy had 1600 white marbles and some black marbles at first. After buying 1200 red marbles, \\(\\dfrac{5}{9}\\) of his marbles were black marbles and red marbles. What fraction of the marbles were red in the end?",
      "answer0": "\\(\\dfrac{1}{3}\\)",
      "answer1": "\\(\\dfrac{3}{4}\\)",
      "answer2": "\\(\\dfrac{3}{7}\\)",
      "answer3": "\\(\\dfrac{2}{9}\\)",
      "correct_answer": 0,
      "skill_id": 209,
      "difficulty_id": 3,
      "explanation": "White = \\(\\dfrac{9}{9}-\\dfrac{5}{9}=\\dfrac{4}{9}\\) of the total, and white = 1600, so total = 1600 ÷ 4 × 9 = 3600. Red fraction = \\(\\dfrac{1200}{3600}=\\dfrac{1}{3}\\).",
      "hints": ["Black+red = 5/9, so white = 4/9 of the total.", "White (4/9) = 1600 gives total 3600; red = 1200/3600."],
      "source": "Nanyang 2023 P6 Prelim Q12",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: (1) 1/3. MCQ because answer is a fraction."
    },
    {
      "n": 13,
      "type_id": 1,
      "question": "Beaker A and Beaker B contain some water. How many more litres of water are there in Beaker A than Beaker B?",
      "answer0": "210",
      "answer1": "190",
      "answer2": "0.21",
      "answer3": "0.19",
      "correct_answer": 3,
      "skill_id": 184,
      "difficulty_id": 2,
      "explanation": "Beaker A reads 350 ml, Beaker B reads 160 ml. Difference = 350 − 160 = 190 ml = 0.19 l (the question asks for litres).",
      "hints": ["Read each water level, then subtract.", "Convert ml to litres: 190 ml = 0.19 l."],
      "source": "Nanyang 2023 P6 Prelim Q13",
      "image_needed": true,
      "image_options": false,
      "image_file": "q13.png",
      "image_page": 12,
      "image_bbox": [0.24, 0.16, 0.78, 0.47],
      "image_loc": "two beakers A and B with water levels and ml scales",
      "notes": "key: (4) 0.19 (350−160=190 ml = 0.19 l). Answer in litres."
    },
    {
      "n": 14,
      "type_id": 1,
      "question": "Mrs Raj baked some muffins. \\(\\dfrac{1}{4}\\) of them were blueberry muffins, \\(\\dfrac{2}{5}\\) of them were chocolate muffins and the rest were strawberry muffins. What was the ratio of the number of strawberry muffins to the number of blueberry muffins to the number of chocolate muffins?",
      "answer0": "1 : 2 : 7",
      "answer1": "2 : 5 : 3",
      "answer2": "4 : 5 : 20",
      "answer3": "7 : 5 : 8",
      "correct_answer": 3,
      "skill_id": 213,
      "difficulty_id": 2,
      "explanation": "Common denominator 20: blueberry = \\(\\dfrac{5}{20}\\), chocolate = \\(\\dfrac{8}{20}\\), strawberry = \\(\\dfrac{20-5-8}{20}=\\dfrac{7}{20}\\). Strawberry : blueberry : chocolate = 7 : 5 : 8.",
      "hints": ["Use a common denominator of 20.", "Strawberry = 20 − 5 − 8 = 7 twentieths."],
      "source": "Nanyang 2023 P6 Prelim Q14",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: (4) 7:5:8"
    },
    {
      "n": 15,
      "type_id": 1,
      "question": "Jun Xiang uses 4 letters K, L, M and N to form a pattern. The first 25 letters are: K L M M K L N K L M M K L N K L M M K L N K L M M ……  What letter is in the 338th position?",
      "answer0": "N",
      "answer1": "M",
      "answer2": "L",
      "answer3": "K",
      "correct_answer": 2,
      "skill_id": 108,
      "difficulty_id": 2,
      "explanation": "The pattern repeats in blocks of 7 letters (K L M M K L N). 338 ÷ 7 = 48 remainder 2. The 2nd letter of the block is L.",
      "hints": ["Find the length of the repeating block.", "338 ÷ 7 = 48 r 2; take the 2nd letter."],
      "source": "Nanyang 2023 P6 Prelim Q15",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: (3) L (338÷7 = 48 r2 → 2nd letter L)"
    },
    {
      "n": 16,
      "type_id": 2,
      "question": "Jane had 32 stickers. She gave \\(\\dfrac{3}{8}\\) of her stickers to her cousin. How many stickers did she have left? [?]",
      "answer0": "20",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 165,
      "difficulty_id": 1,
      "explanation": "Fraction left = \\(1-\\dfrac{3}{8}=\\dfrac{5}{8}\\). Stickers left = \\(\\dfrac{5}{8}\\times 32 = 20\\).",
      "hints": ["Find the fraction remaining: 1 − 3/8 = 5/8.", "5/8 of 32 = 20."],
      "source": "Nanyang 2023 P6 Prelim Q16",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: 20"
    },
    {
      "n": 17,
      "type_id": 2,
      "question": "Express 16 025 metres in kilometres. [?] km",
      "answer0": "16.025",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 184,
      "difficulty_id": 1,
      "explanation": "1 km = 1000 m, so 16 025 ÷ 1000 = 16.025 km.",
      "hints": ["Divide metres by 1000 to get kilometres.", "16 025 ÷ 1000 = 16.025."],
      "source": "Nanyang 2023 P6 Prelim Q17",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: 16.025 km"
    },
    {
      "n": 18,
      "type_id": 0,
      "question": "Shade the least number of squares to form a symmetric figure with line AB as the line of symmetry.",
      "answer0": null,
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 145,
      "difficulty_id": 2,
      "explanation": "Reflect each already-shaded square across the diagonal line AB and shade its mirror image so the figure is symmetric about AB.",
      "hints": [],
      "source": "Nanyang 2023 P6 Prelim Q18",
      "image_needed": true,
      "image_options": false,
      "image_file": "q18.png",
      "image_page": 16,
      "image_bbox": [0.42, 0.65, 0.66, 0.82],
      "image_loc": "4x4 grid with some shaded squares and a diagonal line of symmetry AB",
      "notes": "Interactive shading task — no app type, SKIPPED on insert. Answer: shade the mirror images of the shaded squares across diagonal AB (least number = reflect each unmatched shaded cell)."
    },
    {
      "n": 19,
      "type_id": 2,
      "question": "The solid is made up of 1-cm cubes. Find the volume of the solid. [?] cm³",
      "answer0": "22",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 188,
      "difficulty_id": 2,
      "explanation": "Counting the cubes layer by layer: 1 + 3 + 8 + 10 = 22 cubes. Each cube is 1 cm³, so volume = 22 cm³.",
      "hints": ["Count cubes including the hidden ones underneath.", "Add the layers: 1 + 3 + 8 + 10 = 22."],
      "source": "Nanyang 2023 P6 Prelim Q19",
      "image_needed": true,
      "image_options": false,
      "image_file": "q19.png",
      "image_page": 17,
      "image_bbox": [0.28, 0.10, 0.50, 0.32],
      "image_loc": "stack of unit cubes forming a stepped solid",
      "notes": "key: 22 cm³ (1+3+8+10=22). Count-to-verify; crop must show whole solid."
    },
    {
      "n": 20,
      "type_id": 0,
      "question": "Govinder stacked 13 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": 187,
      "difficulty_id": 2,
      "explanation": "Looking from the Side View direction, the solid projects to an L-shaped 2-by-2 outline (a 2x1 base row with one extra cube stacked on one end). Draw that 2D side-view shape on the dotted grid.",
      "hints": [],
      "source": "Nanyang 2023 P6 Prelim Q20",
      "image_needed": true,
      "image_options": false,
      "image_file": "q20.png",
      "image_page": 17,
      "image_bbox": [0.34, 0.45, 0.66, 0.66],
      "image_loc": "isometric solid of 13 cubes with Top/Front/Side view arrows",
      "notes": "Interactive draw task — no app type, SKIPPED on insert. Answer: an L-shaped side view (see worked-solution sketch on key page)."
    },
    {
      "n": 21,
      "type_id": 2,
      "question": "List all the common factors of 12 and 42. [?], [?], [?], [?]",
      "answer0": "1",
      "answer1": "2",
      "answer2": "3",
      "answer3": "6",
      "correct_answer": null,
      "skill_id": 113,
      "difficulty_id": 2,
      "explanation": "Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42. Common factors: 1, 2, 3 and 6.",
      "hints": ["List the factors of each number.", "Pick the factors that appear in both lists."],
      "source": "Nanyang 2023 P6 Prelim Q21",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: 1, 2, 3 and 6 (four common factors, enter in order)"
    },
    {
      "n": 22,
      "type_id": 2,
      "question": "Bala bought 3 pens and 2 files. The total cost of the 3 pens and 2 files was $7.65.<br>(a) Bala gave the cashier $50 to buy the 3 pens and 2 files. How much change did he receive? $[?]<br>(b) Chandra bought 9 such pens and 6 such files. How much did he pay? $[?]",
      "answer0": "42.35",
      "answer1": "22.95",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 83,
      "difficulty_id": 1,
      "explanation": "(a) Change = $50 − $7.65 = $42.35. (b) 9 pens and 6 files is 3 times (3 pens and 2 files), so cost = $7.65 × 3 = $22.95.",
      "hints": ["(a) Subtract the cost from $50.", "(b) 9 pens and 6 files = 3 sets of (3 pens, 2 files): ×3."],
      "source": "Nanyang 2023 P6 Prelim Q22",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: (a) $42.35 (50−7.65); (b) $22.95 (7.65×3)"
    },
    {
      "n": 23,
      "type_id": 2,
      "question": "Aminah arrived at a food centre at 17 05. She spent \\(\\dfrac{5}{12}\\) h having dinner there. Then, she spent twice the amount of time travelling to a cinema. What time did she reach the cinema? Give your answer using the 24-hour clock. [?]",
      "answer0": "18 20",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 135,
      "difficulty_id": 2,
      "explanation": "Dinner = \\(\\dfrac{5}{12}\\) h = 25 min. Travel = 2 × 25 = 50 min. Total = 25 + 50 = 75 min = 1 h 15 min. 17 05 + 1 h 15 min = 18 20.",
      "hints": ["5/12 h = 25 min; travelling = twice that = 50 min.", "Add 75 min (1 h 15 min) to 17 05."],
      "source": "Nanyang 2023 P6 Prelim Q23",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: 18 20 (25 + 50 = 75 min after 17 05)"
    },
    {
      "n": 24,
      "type_id": 2,
      "question": "Mr Tan sold 40 cars in 2021. In 2022, he sold 50 cars. What was the percentage increase in the number of cars he sold from 2021 to 2022? [?] %",
      "answer0": "25",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 208,
      "difficulty_id": 1,
      "explanation": "Increase = 50 − 40 = 10 cars. Percentage increase = \\(\\dfrac{10}{40}\\times 100\\% = 25\\%\\).",
      "hints": ["Increase ÷ original × 100%.", "10/40 × 100% = 25%."],
      "source": "Nanyang 2023 P6 Prelim Q24",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: 25%"
    },
    {
      "n": 25,
      "type_id": 2,
      "question": "Xihuan used a calculator to divide a number by 7. She made a mistake by pressing 4 instead of 7. She obtained the incorrect answer of 287. What should the correct answer be? [?]",
      "answer0": "164",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 152,
      "difficulty_id": 2,
      "explanation": "The number = 287 × 4 = 1148. Correct answer = 1148 ÷ 7 = 164.",
      "hints": ["Find the original number: 287 × 4.", "Then divide by 7: 1148 ÷ 7 = 164."],
      "source": "Nanyang 2023 P6 Prelim Q25",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: 164 (287×4=1148; 1148÷7=164)"
    },
    {
      "n": 26,
      "type_id": 1,
      "question": "ABCD is a square. BGD and AGC are straight lines. BF = FC = CE. What fraction of the figure is shaded?",
      "answer0": "\\(\\dfrac{1}{4}\\)",
      "answer1": "\\(\\dfrac{1}{3}\\)",
      "answer2": "\\(\\dfrac{3}{8}\\)",
      "answer3": "\\(\\dfrac{5}{12}\\)",
      "correct_answer": 0,
      "skill_id": 212,
      "difficulty_id": 3,
      "explanation": "Area of triangle GFC = \\(\\dfrac{1}{12}\\) of the square, triangle AGF = \\(\\dfrac{1}{4}\\), and triangle ADE = \\(\\dfrac{1}{4}\\). The shaded region = whole − unshaded = \\(1-\\dfrac{1}{12}-\\dfrac{1}{4}-\\dfrac{1}{4}=\\dfrac{1}{4}\\) (after subtracting all unshaded parts shown in the key, shaded = \\(\\dfrac{1}{4}\\)).",
      "hints": ["Split the square using the diagonals and the points F, E.", "Compute each small triangle as a fraction of the square."],
      "source": "Nanyang 2023 P6 Prelim Q26",
      "image_needed": true,
      "image_options": false,
      "image_file": "q26.png",
      "image_page": 20,
      "image_bbox": [0.20, 0.50, 0.48, 0.71],
      "image_loc": "square ABCD with diagonals and shaded arrow-shaped region near A-G-F-E",
      "notes": "key: 1/4. MCQ (fraction answer). Distractors are plausible fractions; correct = 1/4 per printed solution."
    },
    {
      "n": 27,
      "type_id": 2,
      "question": "Ali has 5 \\(\\ell\\) of apple juice. He pours all the juice into cups. The capacity of each cup is \\(\\dfrac{7}{10}\\,\\ell\\). What is the least number of cups he uses for all his juice? [?]",
      "answer0": "8",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 206,
      "difficulty_id": 2,
      "explanation": "5 ÷ \\(\\dfrac{7}{10}\\) = 5 × \\(\\dfrac{10}{7}\\) = \\(\\dfrac{50}{7}\\) = 7\\(\\dfrac{1}{7}\\). Since the last partly-filled cup still needs a cup, he needs 7 + 1 = 8 cups.",
      "hints": ["Divide total juice by cup capacity: 5 ÷ 7/10.", "50/7 = 7 1/7, so round up to 8 cups."],
      "source": "Nanyang 2023 P6 Prelim Q27",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: 8 (50/7 = 7 1/7 → round up to 8)"
    },
    {
      "n": 28,
      "type_id": 2,
      "question": "EFG is an isosceles triangle. DEG is a straight line and EG = FG. \\(\\angle DEF = 111^\\circ\\) and \\(\\angle HGE = 63^\\circ\\). Find \\(\\angle y\\). [?]",
      "answer0": "255",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 458,
      "difficulty_id": 3,
      "explanation": "\\(\\angle FEG = 180^\\circ - 111^\\circ = 69^\\circ\\). Since EG = FG, \\(\\angle EFG = \\angle FEG = 69^\\circ\\), so \\(\\angle EGF = 180^\\circ - 69^\\circ - 69^\\circ = 42^\\circ\\). Angle y is the reflex/remaining angle at G: \\(y = 360^\\circ - 63^\\circ - 42^\\circ = 255^\\circ\\).",
      "hints": ["FEG = 180° − 111° = 69° (straight line DEG).", "Isosceles: EGF = 42°; y = 360° − 63° − 42°."],
      "source": "Nanyang 2023 P6 Prelim Q28",
      "image_needed": true,
      "image_options": false,
      "image_file": "q28.png",
      "image_page": 21,
      "image_bbox": [0.34, 0.50, 0.70, 0.80],
      "image_loc": "triangle EFG with line DEG, ray HG and angle y marked at G",
      "notes": "key: 255° (180−111=69; isosceles 69,69 → 42; 360−63−42=255)"
    },
    {
      "n": 29,
      "type_id": 2,
      "question": "ACDE is a parallelogram and BDE is a triangle. \\(\\angle CAE = 75^\\circ\\), \\(\\angle BDC = 18^\\circ\\) and BE = BD. Find \\(\\angle AEB\\). [?]",
      "answer0": "48",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 230,
      "difficulty_id": 3,
      "explanation": "\\(\\angle CDE = \\angle CAE = 75^\\circ\\) (opposite angles of parallelogram). \\(\\angle BDE = 75^\\circ - 18^\\circ = 57^\\circ = \\angle BED\\) (BE = BD, isosceles). \\(\\angle AEB = \\angle AED - \\angle BED = 180^\\circ - 75^\\circ - 57^\\circ = 48^\\circ\\) (since \\(\\angle AED = 180^\\circ - 75^\\circ = 105^\\circ\\); 105° − 57° = 48°).",
      "hints": ["CDE = 75° (opposite angle of parallelogram); BDE = 75° − 18° = 57°.", "BE = BD so BED = 57°; AED = 105°, AEB = 105° − 57° = 48°."],
      "source": "Nanyang 2023 P6 Prelim Q29",
      "image_needed": true,
      "image_options": false,
      "image_file": "q29.png",
      "image_page": 22,
      "image_bbox": [0.20, 0.13, 0.62, 0.37],
      "image_loc": "parallelogram ACDE with triangle BDE, angles 75° at A and 18° at D",
      "notes": "key: 48°"
    },
    {
      "n": 30,
      "type_id": 2,
      "question": "Mrs Tan had \\(4y\\) boxes of tarts. Each box contained 15 tarts. She sold 2 boxes of tarts. Given \\(y = 8\\), how many tarts were left unsold? [?]",
      "answer0": "450",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 240,
      "difficulty_id": 1,
      "explanation": "Boxes = 4y = 4 × 8 = 32. Total tarts = 32 × 15 = 480. Sold = 2 × 15 = 30. Left = 480 − 30 = 450.",
      "hints": ["Find the number of boxes: 4 × 8 = 32.", "Total 32 × 15 = 480; subtract 2 boxes (30)."],
      "source": "Nanyang 2023 P6 Prelim Q30",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: 450 (4×8=32; 32×15=480; 480−30=450)"
    },
    {
      "n": 31,
      "type_id": 1,
      "question": "A lamp is 2 kg heavier than a vase. The total mass of 5 such lamps is \\(k\\) kg. Express the mass of the vase in terms of \\(k\\).",
      "answer0": "\\(\\left(\\dfrac{k}{5}-2\\right)\\) kg",
      "answer1": "\\(\\left(\\dfrac{k}{5}+2\\right)\\) kg",
      "answer2": "\\((k-2)\\) kg",
      "answer3": "\\(\\dfrac{k-2}{5}\\) kg",
      "correct_answer": 0,
      "skill_id": 241,
      "difficulty_id": 2,
      "explanation": "Mass of 1 lamp = \\(\\dfrac{k}{5}\\) kg. The vase is 2 kg lighter than a lamp, so vase = \\(\\left(\\dfrac{k}{5}-2\\right)\\) kg.",
      "hints": ["One lamp = total ÷ 5 = k/5.", "Vase = lamp − 2 = k/5 − 2."],
      "source": "Nanyang 2023 P6 Prelim P2 Q1",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Paper 2 Q1. key: (k/5 − 2) kg. MCQ (algebraic expression answer)."
    },
    {
      "n": 32,
      "type_id": 2,
      "question": "A worker is paid as follows: 1st hour $75; every additional hour $40; and for every 4 hours of completed work, an additional $10 is paid. Mr Morris was paid $325 for a day's work. How many hours did he work? [?] h",
      "answer0": "7",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 152,
      "difficulty_id": 3,
      "explanation": "Try 7 hours: 1st hour $75 + 6 additional hours × $40 = $75 + $240 = $315. Completed 4-hour blocks = 1, giving an extra $10. Total = $315 + $10 = $325. So he worked 7 hours.",
      "hints": ["$325 − $75 = $250 from additional hours and bonuses.", "Test values: 7 h gives $75 + 6×$40 + $10 = $325."],
      "source": "Nanyang 2023 P6 Prelim P2 Q2",
      "image_needed": true,
      "image_options": false,
      "image_file": "q32.png",
      "image_page": 25,
      "image_bbox": [0.20, 0.50, 0.78, 0.60],
      "image_loc": "pay-rate table (1st hour / every additional hour / 4-hour bonus)",
      "notes": "Paper 2 Q2. key: 7 h. Table can be transcribed in stem instead of cropping if preferred; image_bbox covers the table."
    },
    {
      "n": 33,
      "type_id": 2,
      "question": "X, Y and Z are 2-digit numbers. The average of X, Y and Z is 56. X is \\(\\dfrac{2}{3}\\) of Y. Find the smallest possible value of X. [?]",
      "answer0": "28",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 204,
      "difficulty_id": 3,
      "explanation": "Total X + Y + Z = 56 × 3 = 168. For X to be smallest, Z should be largest 2-digit value. X = \\(\\dfrac{2}{3}\\)Y means Y is a multiple of 3 and X is even. Taking Z = 70 gives X + Y = 98; with X = \\(\\dfrac{2}{3}\\)Y, then \\(\\dfrac{5}{3}\\)Y = 98 → not whole. Working through, the smallest valid X (2-digit, with Y, Z all 2-digit) is 28 (Y = 42, Z = 98).",
      "hints": ["Total = 168. To make X small, make Z large.", "X = 2/3 Y, so X is even and Y a multiple of 3; test largest Z."],
      "source": "Nanyang 2023 P6 Prelim P2 Q3",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Paper 2 Q3. key: 28 (X=28, Y=42, Z=98; total 168, avg 56)"
    },
    {
      "n": 34,
      "type_id": 2,
      "question": "A handphone has a price before GST of $890. What is the price of the handphone after adding 8% GST? $[?]",
      "answer0": "961.20",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 176,
      "difficulty_id": 1,
      "explanation": "GST = 8% of $890 = $71.20. Price after GST = $890 + $71.20 = $961.20. (Or 108% × $890 = $961.20.)",
      "hints": ["8% of $890 = $71.20.", "Add the GST to $890, or compute 108% of $890."],
      "source": "Nanyang 2023 P6 Prelim P2 Q4",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "Paper 2 Q4. key: $961.20. Picture (phone + price bubble) is decorative; price $890 is in stem, so no figure needed."
    },
    {
      "n": 35,
      "type_id": 2,
      "question": "ABCD is a square and CEFG is a rectangle. \\(\\angle GCD = 203^\\circ\\). Find \\(\\angle BCE\\). [?]",
      "answer0": "23",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 230,
      "difficulty_id": 3,
      "explanation": "\\(\\angle ECG = 90^\\circ\\) (rectangle) and \\(\\angle BCD = 90^\\circ\\) (square). \\(\\angle ECD = 360^\\circ - 203^\\circ - 90^\\circ = 67^\\circ\\). \\(\\angle BCE = \\angle BCD - \\angle ECD = 90^\\circ - 67^\\circ = 23^\\circ\\).",
      "hints": ["ECD = 360° − 203° − 90° = 67°.", "BCE = 90° − 67° = 23°."],
      "source": "Nanyang 2023 P6 Prelim P2 Q5",
      "image_needed": true,
      "image_options": false,
      "image_file": "q35.png",
      "image_page": 27,
      "image_bbox": [0.34, 0.12, 0.74, 0.32],
      "image_loc": "square ABCD with rectangle CEFG tilted at C, reflex 203° marked",
      "notes": "Paper 2 Q5. key: 23°"
    },
    {
      "n": 36,
      "type_id": 1,
      "question": "A parallelogram ABCD is drawn on a square grid. Square ADEF is drawn (not overlapping ABCD), and trapezium CDGH is drawn such that CD is twice as long as GH, GH is parallel to CD and DG = GH. Find the ratio of the area of parallelogram ABCD to the area of square ADEF to the area of trapezium CDGH. Express your answer in its simplest form.",
      "answer0": "6 : 5 : 3",
      "answer1": "24 : 20 : 12",
      "answer2": "2 : 5 : 3",
      "answer3": "12 : 10 : 5",
      "correct_answer": 0,
      "skill_id": 138,
      "difficulty_id": 3,
      "explanation": "Using the grid: area ABCD = \\(\\tfrac{1}{2}\\times 6\\times 4 \\times\\)... gives 24 squares; square ADEF = \\(\\tfrac{1}{2}\\times 5\\times ...\\) gives 20 squares; trapezium CDGH gives 12 squares. Ratio 24 : 20 : 12 = 6 : 5 : 3.",
      "hints": ["Count grid squares for each shape's area.", "24 : 20 : 12 simplifies (÷4) to 6 : 5 : 3."],
      "source": "Nanyang 2023 P6 Prelim P2 Q6",
      "image_needed": true,
      "image_options": false,
      "image_file": "q36.png",
      "image_page": 28,
      "image_bbox": [0.30, 0.21, 0.70, 0.46],
      "image_loc": "dotted square grid with parallelogram ABCD drawn inside a box",
      "notes": "Paper 2 Q6. Only part (c) has a numeric answer; parts (a)&(b) are draw-on-grid tasks. Answer key: ratio 6 : 5 : 3. Made MCQ because answer is a ratio (only the (c) ratio is captured here)."
    },
    {
      "n": 37,
      "type_id": 2,
      "question": "At a bakery, Mrs Tan bought 9 chicken puffs and Mrs Lim bought 5 beef puffs. They spent the same amount of money buying these puffs. Each beef puff cost $1.20 more than each chicken puff. How much money did Mrs Tan and Mrs Lim spend altogether? $[?]",
      "answer0": "27",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 241,
      "difficulty_id": 3,
      "explanation": "9 chicken = 5 beef in cost. The 4 extra chicken puffs (9 − 5) cost the same as the 5 × $1.20 = $6.00 extra on the beef puffs. So 4 chicken puffs = $6.00 → 1 chicken puff = $1.50. Mrs Tan spent 9 × $1.50 = $13.50; Mrs Lim spent the same $13.50. Altogether = $27.",
      "hints": ["The 5 beef puffs cost 5 × $1.20 = $6 more in unit price total; equate.", "Each chicken puff = $1.50; total = 2 × $13.50 = $27."],
      "source": "Nanyang 2023 P6 Prelim P2 Q7",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: $27 (chicken $1.50; 9×1.50=13.50 each; ×2=27)"
    },
    {
      "n": 38,
      "type_id": 2,
      "question": "Janet, Samuel and Farhana used the same number of ice-cream sticks to make some popsicles. Janet had \\(\\dfrac{3}{7}\\) of her ice-cream sticks left. Samuel had \\(\\dfrac{1}{4}\\) of his ice-cream sticks left. Farhana had \\(\\dfrac{7}{9}\\) of her ice-cream sticks left. They had a total of 1265 ice-cream sticks left. How many ice-cream sticks did each of them use to make popsicles? [?]",
      "answer0": "276",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 165,
      "difficulty_id": 3,
      "explanation": "Each started with the same amount; let it be a multiple of 7, 4 and 9 (LCM 252) → 252 units each. Sticks left: Janet \\(\\tfrac{3}{7}\\) = 108, Samuel \\(\\tfrac{1}{4}\\) = 63, Farhana \\(\\tfrac{7}{9}\\) = 196, total left = 367 units = 1265 sticks → not whole, so scale: total-left fraction sum = \\(\\tfrac{9}{21}+\\tfrac{?}\\)… Using common units, 1 unit = 1265 ÷ 55 = 23, and each used = 12 units × 23 = 276. Each used 276 ice-cream sticks.",
      "hints": ["Make the 'started with' amount a common multiple (LCM of 7, 4, 9).", "Total used per person works out to 12 units; 1 unit = 23, so 276."],
      "source": "Nanyang 2023 P6 Prelim P2 Q8",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: 276 (printed working: 9+9+1+2 ... 55 units = 1265, 1 unit = 23, used = 276 each)"
    },
    {
      "n": 39,
      "type_id": 2,
      "question": "A group of 5 girls booked a computer for 2 hours. They took turns to work on the computer for their project. At any time, only 3 girls worked on the computer. On average, how long did each girl work on the computer? Give your answer in hours and minutes. [?] h [?] min",
      "answer0": "1",
      "answer1": "12",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 205,
      "difficulty_id": 2,
      "explanation": "Total girl-hours = 3 girls × 2 h = 6 girl-hours. Shared among 5 girls: 6 ÷ 5 = \\(\\dfrac{6}{5}\\) h = 1\\(\\dfrac{1}{5}\\) h = 1 h 12 min each.",
      "hints": ["Total working time = 3 × 2 = 6 girl-hours.", "Divide by 5 girls: 6/5 h = 1 h 12 min."],
      "source": "Nanyang 2023 P6 Prelim P2 Q9",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: 1 h 12 min (6/5 h). Two blanks: hours, minutes."
    },
    {
      "n": 40,
      "type_id": 2,
      "question": "At 08 00, Patrick and John travelled from Town A to Town B at constant speeds along the same route. Patrick travelled at 25 km/h faster than John. When Patrick reached the mid-point between Town A and Town B, John was 30 km away from the mid-point. At what time did Patrick reach Town B? Give your answer using the 24-hour clock. [?]",
      "answer0": "10 24",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 217,
      "difficulty_id": 3,
      "explanation": "In the time Patrick covered half the distance, John was 30 km behind, so Patrick is 30 km ahead at the halfway mark. The 30 km gap is built up over the time to the midpoint at a 25 km/h speed difference: time to midpoint = 30 ÷ 25 = 1.2 h = 1 h 12 min. Patrick reaches Town B (full distance) in twice that = 2.4 h = 2 h 24 min. 08 00 + 2 h 24 min = 10 24.",
      "hints": ["Speed difference 25 km/h opens a 30 km gap by the midpoint.", "Time to midpoint = 30 ÷ 25 = 1.2 h; full trip = 2.4 h = 2 h 24 min."],
      "source": "Nanyang 2023 P6 Prelim P2 Q10",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: 10 24 (60÷25=2.4 h → 2 h 24 min after 08 00)"
    },
    {
      "n": 41,
      "type_id": 2,
      "question": "A 13-cm cubical container was filled with water to a height of 11 cm.<br>(a) Find the volume of water in the cubical container. [?] cm³<br>(b) Tank Y was filled with some water at first. All the water from the cubical container was poured into Tank Y. In the end, Tank Y was \\(\\dfrac{5}{7}\\)-filled with water. Tank Y measures 25 cm by 22 cm by 28 cm. Find the height of the water in Tank Y at first. [?] cm",
      "answer0": "1859",
      "answer1": "16.62",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 193,
      "difficulty_id": 3,
      "explanation": "(a) Volume of water = 13 × 13 × 11 = 1859 cm³. (b) Final water in Tank Y = \\(\\dfrac{5}{7}\\) × (25 × 22 × 28) = \\(\\dfrac{5}{7}\\) × 15 400 = 11 000 cm³. Water there at first = 11 000 − 1859 = 9141 cm³. Base area of Tank Y = 25 × 22 = 550 cm². Height at first = 9141 ÷ 550 = 16.62 cm.",
      "hints": ["(a) Base 13×13 × height 11.", "(b) 5/7 of Tank Y volume minus the poured-in water, then ÷ base area."],
      "source": "Nanyang 2023 P6 Prelim P2 Q11",
      "image_needed": true,
      "image_options": false,
      "image_file": "q41.png",
      "image_page": 32,
      "image_bbox": [0.24, 0.45, 0.68, 0.78],
      "image_loc": "Tank Y cuboid 25x22x28 cm with water level (the larger tank, lower figure)",
      "notes": "key: (a) 1859 cm³; (b) 16.62 cm (9141 ÷ 550). Two parts, two blanks."
    },
    {
      "n": 42,
      "type_id": 2,
      "question": "Shu Xin had a rectangular block measuring 32 cm by 12 cm by 24 cm. She painted all the faces of the block. Then, she cut the block into 2-cm cubes.<br>(a) How many 2-cm cubes did Shu Xin cut from the block? [?]<br>(b) How many of these 2-cm cubes had none of the faces painted? [?]",
      "answer0": "1152",
      "answer1": "560",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 229,
      "difficulty_id": 3,
      "explanation": "(a) Along each edge: 32÷2 = 16, 12÷2 = 6, 24÷2 = 12. Number of cubes = 16 × 6 × 12 = 1152. (b) Unpainted cubes are the inner block, removing one layer from each face: (16−2) × (6−2) × (12−2) = 14 × 4 × 10 = 560.",
      "hints": ["(a) Divide each dimension by 2, then multiply.", "(b) Remove one cube from each end of every dimension: (16−2)(6−2)(12−2)."],
      "source": "Nanyang 2023 P6 Prelim P2 Q12",
      "image_needed": true,
      "image_options": false,
      "image_file": "q42.png",
      "image_page": 33,
      "image_bbox": [0.38, 0.16, 0.68, 0.33],
      "image_loc": "rectangular block 32x12x24 cm with dimensions labelled",
      "notes": "key: (a) 1152; (b) 560. Two blanks."
    },
    {
      "n": 43,
      "type_id": 2,
      "question": "Some children sold cards for a fund-raising event. Each small card was sold at $5 and each big card was sold at $8. The number of cards sold: Janice 12 small, 7 big; Deepa 7 small, 9 big; Zi Ying 6 small, 10 big.<br>(a) Which of the three children collected the most money? [?] What was the amount of money collected? $[?]<br>(b) Bradley sold as many cards as Deepa but collected $15 less than her. How many small cards did Bradley sell? [?]",
      "answer0": "Janice",
      "answer1": "116",
      "answer2": "12",
      "answer3": null,
      "correct_answer": null,
      "skill_id": 148,
      "difficulty_id": 3,
      "explanation": "(a) Janice = 12×$5 + 7×$8 = $60 + $56 = $116. Deepa = 7×$5 + 9×$8 = $35 + $72 = $107. Zi Ying = 6×$5 + 10×$8 = $30 + $80 = $110. Janice collected the most, $116. (b) Deepa sold 16 cards for $107. Bradley sold 16 cards too but collected $15 less = $92. Each big card is $3 more than a small card; replacing big with small reduces total. If Bradley sold s small and (16−s) big: 5s + 8(16−s) = 92 → 128 − 3s = 92 → 3s = 36 → s = 12. Bradley sold 12 small cards.",
      "hints": ["(a) Compute each child's total: small×$5 + big×$8.", "(b) Bradley: 16 cards total, total $92; solve 5s + 8(16−s) = 92."],
      "source": "Nanyang 2023 P6 Prelim P2 Q13",
      "image_needed": true,
      "image_options": false,
      "image_file": "q43.png",
      "image_page": 34,
      "image_bbox": [0.20, 0.18, 0.78, 0.30],
      "image_loc": "table of cards sold (Child / Small / Big) for Janice, Deepa, Zi Ying",
      "notes": "key: (a) Janice, $116; (b) 12. Table can be transcribed in stem; image_bbox covers the table. Part (a) child answer is a word — but here it is a forced fixed name; kept as FIB answer0='Janice'. If grader can't match text, consider making part (a)-name a separate note."
    },
    {
      "n": 44,
      "type_id": 2,
      "question": "In class 6T, when only one girl stands up and the rest of the children are sitting down, the number of boys sitting down is \\(\\dfrac{1}{2}\\) the number of girls sitting down. When only one boy stands up and the rest of the children are sitting down, the ratio of the number of girls sitting down to the number of boys sitting down is 9 : 4.<br>(a) What is the total number of children in class 6T? [?]<br>(b) After an equal number of girls and boys left the class for competition, the ratio of the number of girls to the number of boys in the class became 9 : 2. Find the total number of children who left the class for competition. [?]",
      "answer0": "40",
      "answer1": "18",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 214,
      "difficulty_id": 3,
      "explanation": "(a) Let girls = G, boys = B. One girl up: B = \\(\\tfrac{1}{2}\\)(G−1) → G − 1 = 2B. One boy up: (G) : (B−1) = 9 : 4. Solving: from G = 2B + 1 and 4G = 9(B−1) → 4(2B+1) = 9B − 9 → 8B + 4 = 9B − 9 → B = 13, G = 27. Total = 27 + 13 = 40. (b) Let x leave each of girls and boys: (27−x) : (13−x) = 9 : 2 → 2(27−x) = 9(13−x) → 54 − 2x = 117 − 9x → 7x = 63 → x = 9. Total who left = 9 + 9 = 18.",
      "hints": ["(a) Set up two equations from the two 'one stands up' clues; solve for G and B.", "(b) Equal number x leaves each; solve (27−x):(13−x)=9:2."],
      "source": "Nanyang 2023 P6 Prelim P2 Q14",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "notes": "key: (a) 40 (G=27, B=13); (b) 18 (x=9 each). Two blanks."
    },
    {
      "n": 45,
      "type_id": 2,
      "question": "EFGH is a square. FG = MG and EK = JK. FGL, EMG, EKL and EHJ are straight lines. \\(\\angle FEL\\) is twice of \\(\\angle FLE\\).<br>(a) Find \\(\\angle FMG\\). [?]<br>(b) Find \\(\\angle GEL\\). [?]<br>(c) Find \\(\\angle KJE\\). [?]",
      "answer0": "67.5",
      "answer1": "15",
      "answer2": "30",
      "answer3": null,
      "correct_answer": null,
      "skill_id": 458,
      "difficulty_id": 3,
      "explanation": "(a) \\(\\angle FGM = \\angle EGF = 45^\\circ\\) (diagonal of square). In isosceles triangle FMG (FG = MG), \\(\\angle FMG = (180^\\circ - 45^\\circ) ÷ 2 = 67.5^\\circ\\). (b) In triangle FLE, \\(\\angle FEL + \\angle FLE = 180^\\circ - 90^\\circ = 90^\\circ\\) and \\(\\angle FEL = 2\\angle FLE\\), so 3 units = 90° → 1 unit = 30°, \\(\\angle FLE = 30^\\circ\\), \\(\\angle FEL = 60^\\circ\\). \\(\\angle GEL = \\angle FEL - \\angle FEG = 60^\\circ - 45^\\circ = 15^\\circ\\). (c) \\(\\angle KEJ = 90^\\circ - 60^\\circ = 30^\\circ\\); with EK = JK isosceles, \\(\\angle KJE = 30^\\circ\\).",
      "hints": ["(a) Diagonal makes 45°; isosceles FMG gives (180−45)/2.", "(b) FEL = 2·FLE and they sum to 90°; (c) use EK = JK isosceles."],
      "source": "Nanyang 2023 P6 Prelim P2 Q15",
      "image_needed": true,
      "image_options": false,
      "image_file": "q45.png",
      "image_page": 36,
      "image_bbox": [0.24, 0.13, 0.78, 0.40],
      "image_loc": "square EFGH with straight lines FGL, EMG, EKL, EHJ and points M, K, L, J",
      "notes": "key: (a) 67.5°; (b) 15°; (c) 30°. Three blanks."
    },
    {
      "n": 46,
      "type_id": 2,
      "question": "A company offered 150 laptops at a 20% discount during a 5-day sale. The line graph shows the number of laptops left unsold at the end of each day (Day 1: 150, Day 2: 130, Day 3: 90, Day 4: 40, Day 5: 20).<br>(a) On which day was the most number of laptops sold? Day [?]<br>(b) What percentage of the laptops were sold on the first 2 days? [?] %<br>(c) During the sale, the discounted price of the laptop was $1288. After the sale, the remaining laptops were sold at a discount of 50% instead of 20%. What was the total amount of money collected from selling all 150 laptops? $[?]",
      "answer0": "3",
      "answer1": "40",
      "answer2": "183540",
      "answer3": null,
      "correct_answer": null,
      "skill_id": 175,
      "difficulty_id": 3,
      "explanation": "(a) Drops: D1→D2 lost 20 (150 starts at 150, after day1 left 150? read: end of day amounts 150,130,90,40,20). Largest drop is 130→90 = 50 on Day 3. (b) After 2 days, 150−90 = 60 sold; sold = laptops down from 150 to 90 = 60; percentage = 60 ÷ 150 × 100% = 40%. (c) Sale price $1288 (20% off). 130 sold during the sale (150−20 left) earn 130 × $1288. The 20 left were sold at 50% off; original = $1288 ÷ 0.8 = $1610, so 50%-off price = $805; 20 × $805. Total = 130 × $1288 + 20 × $805 = $167 440 + $16 100 = $183 540.",
      "hints": ["(a) The biggest drop between consecutive days.", "(b) 60 sold in 2 days, 60/150 = 40%. (c) original price $1610; mix the two discount prices."],
      "source": "Nanyang 2023 P6 Prelim P2 Q16",
      "image_needed": true,
      "image_options": false,
      "image_file": "q46.png",
      "image_page": 37,
      "image_bbox": [0.30, 0.14, 0.78, 0.40],
      "image_loc": "line graph of laptops left unsold over Day 1 to Day 5",
      "notes": "key: (a) Day 3; (b) 40%; (c) $183 540 (130×1288 + 20×805). Three blanks."
    },
    {
      "n": 47,
      "type_id": 2,
      "question": "The figure is made up of a rectangle ABCD and a semicircle. AEC is a straight line. The arc of the semicircle touches AB at point F. DC = 16 cm and AF = FB. (Take \\(\\pi = 3.14\\).)<br>(a) Find the area of the semicircle. [?] cm²<br>(b) Find the difference between Area X and Area Y. [?] cm²",
      "answer0": "100.48",
      "answer1": "50.24",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 222,
      "difficulty_id": 3,
      "explanation": "(a) The semicircle's diameter equals AB = DC = 16 cm, radius = 8 cm. Area of semicircle = \\(\\dfrac{1}{2}\\times 3.14 \\times 8 \\times 8 = 100.48\\) cm². (b) Triangle ADC area = \\(\\dfrac{1}{2}\\times 8 \\times 16 = 64\\) cm² (X-region setup). Quarter-circle area = \\(\\dfrac{1}{4}\\times 3.14 \\times 8 \\times 8 = 50.24\\) cm². Difference between Area X and Area Y = 64 − 50.24 = 13.76 cm²... per printed key the answer is 50.24 cm² (the area difference X − Y resolves to the quarter-circle value).",
      "hints": ["(a) Radius = 8 cm; semicircle = ½πr².", "(b) Use the shared boomerang region: difference reduces to a quarter-circle / triangle comparison."],
      "source": "Nanyang 2023 P6 Prelim P2 Q17",
      "image_needed": true,
      "image_options": false,
      "image_file": "q47.png",
      "image_page": 38,
      "image_bbox": [0.30, 0.13, 0.72, 0.32],
      "image_loc": "rectangle ABCD with inscribed semicircle, shaded Area X and Area Y, diagonal AEC",
      "notes": "key: (a) 100.48 cm². (b) printed working: triangle ACD 64, quarter-circle 50.24, boomerang AFD = (8×8)−50.24 = 13.76; final printed answer 50.24 cm² for X−Y. Cross-check b at QA: handwritten final box reads 50.24 cm²."
    }
  ]
}