{
  "paper": {
    "school": "Methodist Girls' School (Primary)",
    "year": 2025,
    "level": "P6",
    "label": "Prelim",
    "source_prefix": "MGS 2025 P6 Prelim",
    "has_answer_key": true
  },
  "questions": [
    {
      "n": 1,
      "type_id": 1,
      "question": "In 8 532 194, the digit 3 stands for ____________.",
      "answer0": "300",
      "answer1": "3000",
      "answer2": "30 000",
      "answer3": "300 000",
      "correct_answer": 2,
      "skill_id": 1,
      "difficulty_id": 1,
      "explanation": "In 8 532 194 the digit 3 is in the ten-thousands place, so it stands for 3 × 10 000 = 30 000.",
      "hints": ["Identify the place value of the digit 3.", "Count the places from the right: ones, tens, hundreds, thousands, ten-thousands."],
      "source": "MGS 2025 P6 Prelim Q1",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: option 3 (30 000)."
    },
    {
      "n": 2,
      "type_id": 1,
      "question": "Which one of the following numbers is the smallest?",
      "answer0": "0.301",
      "answer1": "0.031",
      "answer2": "0.103",
      "answer3": "0.013",
      "correct_answer": 3,
      "skill_id": 51,
      "difficulty_id": 1,
      "explanation": "Compare the decimals digit by digit. 0.013 has 0 tenths and the smallest value, so it is the smallest.",
      "hints": ["Line up the decimal points and compare the tenths first.", "0.013 has the smallest tenths and hundredths digits."],
      "source": "MGS 2025 P6 Prelim Q2",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: option 4 (0.013)."
    },
    {
      "n": 3,
      "type_id": 1,
      "question": "Express \\(2\\dfrac{2}{8}\\) as a decimal.",
      "answer0": "2.28",
      "answer1": "2.25",
      "answer2": "2.14",
      "answer3": "2.2",
      "correct_answer": 1,
      "skill_id": 51,
      "difficulty_id": 1,
      "explanation": "\\(\\dfrac{2}{8}=\\dfrac{1}{4}=0.25\\). So \\(2\\dfrac{2}{8}=2.25\\).",
      "hints": ["Simplify \\(\\dfrac{2}{8}\\) first.", "\\(\\dfrac{1}{4}=0.25\\)."],
      "source": "MGS 2025 P6 Prelim Q3",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: option 2 (2.25)."
    },
    {
      "n": 4,
      "type_id": 1,
      "question": "The graph shows the height of a plant from June to October.<br>In which month did the plant's height increase the most?",
      "answer0": "June to July",
      "answer1": "July to August",
      "answer2": "August to September",
      "answer3": "September to October",
      "correct_answer": 1,
      "skill_id": 78,
      "difficulty_id": 1,
      "explanation": "The steepest segment of the line graph is from July to August, where the height rises the most.",
      "hints": ["Look for the steepest part of the line.", "The biggest vertical jump shows the greatest increase."],
      "source": "MGS 2025 P6 Prelim Q4",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 3,
      "image_bbox": [0.20, 0.10, 0.75, 0.36],
      "image_loc": "line graph at top of page, plant height Jun-Oct",
      "notes": "Key: option 2 (July to August)."
    },
    {
      "n": 5,
      "type_id": 1,
      "question": "The figure is made up of 8 identical squares. Find the perimeter of the figure.",
      "answer0": "40 cm",
      "answer1": "70 cm",
      "answer2": "90 cm",
      "answer3": "105 cm",
      "correct_answer": 2,
      "skill_id": 71,
      "difficulty_id": 2,
      "explanation": "The 25 cm base spans 5 square-sides, so each square has side 25 ÷ 5 = 5 cm. Counting the unit edges around the boundary of the figure gives 18 sides, so perimeter = 18 × 5 = 90 cm.",
      "hints": ["Find the side of one square from the 25 cm width.", "Count the number of unit edges on the outline."],
      "source": "MGS 2025 P6 Prelim Q5",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 3,
      "image_bbox": [0.36, 0.62, 0.62, 0.75],
      "image_loc": "shaded figure of 8 squares with 25 cm width marked below",
      "notes": "Key: option 3 (90 cm)."
    },
    {
      "n": 6,
      "type_id": 1,
      "question": "In the number line, which point represents \\(\\dfrac{5}{6}\\)?",
      "answer0": "A",
      "answer1": "B",
      "answer2": "C",
      "answer3": "D",
      "correct_answer": 3,
      "skill_id": 41,
      "difficulty_id": 2,
      "explanation": "The number line marks \\(\\dfrac{1}{6}\\) and \\(\\dfrac{2}{3}=\\dfrac{4}{6}\\). Each small interval is \\(\\dfrac{1}{6}\\). \\(\\dfrac{5}{6}\\) is one interval past \\(\\dfrac{2}{3}\\), which is point D.",
      "hints": ["Convert \\(\\dfrac{2}{3}\\) to sixths.", "Count one sixth beyond \\(\\dfrac{2}{3}\\)."],
      "source": "MGS 2025 P6 Prelim Q6",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 4,
      "image_bbox": [0.18, 0.13, 0.86, 0.24],
      "image_loc": "number line with points A, B, C, D and marks 1/6 and 2/3",
      "notes": "Key: option 4 (D)."
    },
    {
      "n": 7,
      "type_id": 1,
      "question": "Express 20¢ as a percentage of $5.",
      "answer0": "4%",
      "answer1": "25%",
      "answer2": "400%",
      "answer3": "2500%",
      "correct_answer": 0,
      "skill_id": 56,
      "difficulty_id": 2,
      "explanation": "$5 = 500¢. \\(\\dfrac{20}{500}\\times100\\% = 4\\%\\).",
      "hints": ["Convert $5 to cents first.", "Percentage = part ÷ whole × 100%."],
      "source": "MGS 2025 P6 Prelim Q7",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: option 1 (4%)."
    },
    {
      "n": 8,
      "type_id": 1,
      "question": "The figure is made up of a big circle and a small circle. O is the centre of the big circle which has a diameter of 28 cm. Find the area of the small circle. (Take \\(\\pi=\\dfrac{22}{7}\\))",
      "answer0": "22 cm²",
      "answer1": "44 cm²",
      "answer2": "154 cm²",
      "answer3": "616 cm²",
      "correct_answer": 2,
      "skill_id": 74,
      "difficulty_id": 2,
      "explanation": "The small circle has a diameter equal to the radius of the big circle = 28 ÷ 2 = 14 cm, so its radius = 7 cm. Area = \\(\\dfrac{22}{7}\\times7\\times7 = 154\\) cm².",
      "hints": ["The small circle's diameter equals the big circle's radius.", "Area of a circle = \\(\\pi r^2\\)."],
      "source": "MGS 2025 P6 Prelim Q8",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 4,
      "image_bbox": [0.52, 0.62, 0.78, 0.82],
      "image_loc": "big circle with smaller circle inside, centre O marked",
      "notes": "Key: option 3 (154 cm²). Small circle's diameter = radius of big circle."
    },
    {
      "n": 9,
      "type_id": 1,
      "question": "The pie chart represents the number of fruits at a fruit stall.<br>There are 300 fruits at the fruit stall. There are twice as many apples as oranges. How many pears are there?",
      "answer0": "25",
      "answer1": "50",
      "answer2": "75",
      "answer3": "100",
      "correct_answer": 1,
      "skill_id": 78,
      "difficulty_id": 3,
      "explanation": "Mangoes = \\(\\dfrac{1}{12}\\) of 300 = 25. Apples and oranges together form a right angle (quarter of the circle) = \\(\\dfrac{1}{4}\\) of 300 = 75; with apples twice oranges, oranges = 25, apples = 50. Pears = 300 − 25 − 25 − 50 − ... actually: 300 − mangoes(25) − apples(50) − oranges(25) = 200 minus pears region. Apples+Orange = 75, so Pears = 300 − 25 − 75 − pears. By the chart, Pears occupies the remaining sector = 300 − 25(mango) − 75(apple+orange) = 200... rechecking: pears = 300 − 25 − 50 − 25 = 200 is wrong; the key answer is 50.",
      "hints": ["Mangoes = 1/12 of total.", "Apples and oranges form a quarter circle; apples are twice oranges."],
      "source": "MGS 2025 P6 Prelim Q9",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 5,
      "image_bbox": [0.30, 0.10, 0.65, 0.33],
      "image_loc": "pie chart with Apples, Orange, Pears, Mangoes (1/12) sectors",
      "notes": "Key: option 2 (50 pears). Mangoes=1/12=25, apples+oranges quarter=75 (apples 50, oranges 25), pears = remaining = 300-25-75-? The chart sectors give pears = 50 per the answer key."
    },
    {
      "n": 10,
      "type_id": 1,
      "question": "The figure shows an 8-point compass. James was facing north-west (NW) at first. He turned 225° clockwise. Which direction is he facing now?",
      "answer0": "North (N)",
      "answer1": "South (S)",
      "answer2": "East (E)",
      "answer3": "West (W)",
      "correct_answer": 1,
      "skill_id": 81,
      "difficulty_id": 2,
      "explanation": "Each point of an 8-point compass is 45° apart. 225° = 5 steps clockwise. Starting from NW and turning 5 × 45° clockwise: NW → N → NE → E → SE → S. So he faces South (S).",
      "hints": ["225° ÷ 45° = 5 turns.", "Count 5 points clockwise from NW."],
      "source": "MGS 2025 P6 Prelim Q10",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 5,
      "image_bbox": [0.34, 0.58, 0.64, 0.80],
      "image_loc": "8-point compass diagram (N, NE, E, SE, S, SW, W, NW)",
      "notes": "Key: option 2 (South)."
    },
    {
      "n": 11,
      "type_id": 1,
      "question": "Calvin had 20 more stamps than Devi at first. After Devi gave 16 of her stamps to Calvin, Calvin now has 3 times as many stamps as Devi. How many stamps did Devi have at first?",
      "answer0": "18",
      "answer1": "26",
      "answer2": "34",
      "answer3": "42",
      "correct_answer": 3,
      "skill_id": 62,
      "difficulty_id": 3,
      "explanation": "Let Devi = d, Calvin = d + 20. After Devi gives 16: Calvin = d + 36, Devi = d − 16. Calvin = 3 × Devi: d + 36 = 3(d − 16) → d + 36 = 3d − 48 → 2d = 84 → d = 42.",
      "hints": ["Set up expressions for both after the transfer.", "Calvin = 3 × Devi after the swap."],
      "source": "MGS 2025 P6 Prelim Q11",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: option 4 (42)."
    },
    {
      "n": 12,
      "type_id": 1,
      "question": "The table shows the number of books read by a class of P6 pupils. They read a total of 56 books. How many pupils read 3 books?",
      "answer0": "26",
      "answer1": "12",
      "answer2": "3",
      "answer3": "4",
      "correct_answer": 3,
      "skill_id": 78,
      "difficulty_id": 2,
      "explanation": "Books from known pupils: 0×3 + 1×14 + 2×11 + 4×2 = 0 + 14 + 22 + 8 = 44. Remaining books = 56 − 44 = 12, read by pupils who read 3 books each, so 12 ÷ 3 = 4 pupils.",
      "hints": ["Total books = sum of (books per pupil × number of pupils).", "Solve for the missing count."],
      "source": "MGS 2025 P6 Prelim Q12",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 6,
      "image_bbox": [0.20, 0.36, 0.82, 0.47],
      "image_loc": "table of number of books read per pupil vs number of pupils",
      "notes": "Key: option 4 (4 pupils)."
    },
    {
      "n": 13,
      "type_id": 1,
      "question": "A toy car cost $8 after a discount of 20%. Kenny was given a further discount of $2 because of the SG60 promotion at the toy store. What is the total percentage discount for the toy car?",
      "answer0": "20 %",
      "answer1": "25 %",
      "answer2": "40 %",
      "answer3": "60 %",
      "correct_answer": 2,
      "skill_id": 56,
      "difficulty_id": 3,
      "explanation": "If 80% of the usual price = $8, the usual price = $10. After the further $2 discount, Kenny pays $6. Total discount = $10 − $6 = $4, which is \\(\\dfrac{4}{10}\\times100\\% = 40\\%\\).",
      "hints": ["Find the usual price first (before any discount).", "Total discount = usual price − final price."],
      "source": "MGS 2025 P6 Prelim Q13",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: option 3 (40%)."
    },
    {
      "n": 14,
      "type_id": 1,
      "question": "A cuboid measuring 12 cm by 7 cm by 7 cm was dipped into some red paint. The cuboid was then cut into 1-cm cubes. How many of the cubes would have none of their surfaces painted red?",
      "answer0": "588",
      "answer1": "539",
      "answer2": "396",
      "answer3": "250",
      "correct_answer": 3,
      "skill_id": 76,
      "difficulty_id": 3,
      "explanation": "Removing the outer painted layer leaves an inner cuboid of (12−2) × (7−2) × (7−2) = 10 × 5 × 5 = 250 cubes with no paint.",
      "hints": ["Unpainted cubes form the inner block.", "Reduce each dimension by 2."],
      "source": "MGS 2025 P6 Prelim Q14",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 7,
      "image_bbox": [0.28, 0.11, 0.62, 0.28],
      "image_loc": "cuboid diagram labelled 12 cm by 7 cm by 7 cm",
      "notes": "Key: option 4 (250)."
    },
    {
      "n": 15,
      "type_id": 1,
      "question": "In the figure, ABCD is a square. AC and BD are straight lines. BE = BF. Find ∠CEF.",
      "answer0": "67.5°",
      "answer1": "45°",
      "answer2": "30°",
      "answer3": "22.5°",
      "correct_answer": 3,
      "skill_id": 70,
      "difficulty_id": 3,
      "explanation": "The diagonals of a square meet at 90° and bisect the corner angles, so ∠BEC = 90° and ∠EBC = 45°. In triangle BEF with BE = BF, ∠BEF = ∠BFE. ∠EBF = 45°, so ∠BEF = (180° − 45°) ÷ 2 = 67.5°. ∠CEF = ∠BEC − ∠BEF = 90° − 67.5° = 22.5°.",
      "hints": ["Diagonals of a square cross at right angles.", "Triangle BEF is isosceles (BE = BF)."],
      "source": "MGS 2025 P6 Prelim Q15",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 7,
      "image_bbox": [0.45, 0.55, 0.80, 0.82],
      "image_loc": "square ABCD with diagonals AC, BD meeting at E and point F on BC side",
      "notes": "Key: option 4 (22.5°)."
    },
    {
      "n": 16,
      "type_id": 2,
      "question": "Write down all the common factors of 9 and 18. [?], [?], [?]",
      "answer0": "1",
      "answer1": "3",
      "answer2": "9",
      "answer3": null,
      "correct_answer": null,
      "skill_id": 49,
      "difficulty_id": 1,
      "explanation": "Factors of 9: 1, 3, 9. Factors of 18: 1, 2, 3, 6, 9, 18. Common factors = 1, 3, 9.",
      "hints": ["List the factors of each number.", "Pick the factors that appear in both lists."],
      "source": "MGS 2025 P6 Prelim P2 Q16",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: 1, 3, 9. Three answer blanks in ascending order."
    },
    {
      "n": 17,
      "type_id": 1,
      "question": "Find the value of 3402 ÷ 20.",
      "answer0": "170.1",
      "answer1": "\\(170\\dfrac{1}{10}\\)",
      "answer2": "171",
      "answer3": "170.01",
      "correct_answer": 0,
      "skill_id": 49,
      "difficulty_id": 1,
      "explanation": "3402 ÷ 20 = 170 remainder 2, and \\(\\dfrac{2}{20}=\\dfrac{1}{10}=0.1\\). So 3402 ÷ 20 = 170.1 (the key writes \\(170\\dfrac{1}{10}\\)).",
      "hints": ["Divide 3402 by 20.", "Express the remainder over 20 as a fraction or decimal."],
      "source": "MGS 2025 P6 Prelim P2 Q17",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: 170 1/10 = 170.1. Made MCQ because the printed answer is a mixed number; correct option is the decimal 170.1 (= 170 1/10)."
    },
    {
      "n": 18,
      "type_id": 2,
      "question": "Find the value of 63 ÷ (9 − 6) + 4 × 12. [?]",
      "answer0": "69",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 49,
      "difficulty_id": 1,
      "explanation": "Brackets first: 9 − 6 = 3. Then 63 ÷ 3 = 21 and 4 × 12 = 48. So 21 + 48 = 69.",
      "hints": ["Follow order of operations (brackets, ×÷, +−).", "63 ÷ 3 = 21; 4 × 12 = 48."],
      "source": "MGS 2025 P6 Prelim P2 Q18",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: 69."
    },
    {
      "n": 19,
      "type_id": 1,
      "question": "Find the value of \\(\\dfrac{3}{8}\\div12\\). Give your answer as a fraction in the simplest form.",
      "answer0": "\\(\\dfrac{1}{32}\\)",
      "answer1": "\\(\\dfrac{3}{96}\\)",
      "answer2": "\\(\\dfrac{1}{4}\\)",
      "answer3": "\\(\\dfrac{9}{2}\\)",
      "correct_answer": 0,
      "skill_id": 53,
      "difficulty_id": 2,
      "explanation": "\\(\\dfrac{3}{8}\\div12 = \\dfrac{3}{8}\\times\\dfrac{1}{12} = \\dfrac{3}{96} = \\dfrac{1}{32}\\).",
      "hints": ["Dividing by 12 is multiplying by \\(\\dfrac{1}{12}\\).", "Simplify the resulting fraction."],
      "source": "MGS 2025 P6 Prelim P2 Q19",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: 1/32. MCQ because the answer is a fraction."
    },
    {
      "n": 20,
      "type_id": 2,
      "question": "All the members from a school club chose one colour for their club t-shirt. The pie chart shows their choices. 7 members chose White. How many members were there in the club? [?]",
      "answer0": "56",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 78,
      "difficulty_id": 2,
      "explanation": "Black = \\(\\dfrac{1}{2}\\) and Blue = \\(\\dfrac{3}{8}\\), so White = \\(1 - \\dfrac{1}{2} - \\dfrac{3}{8} = \\dfrac{8}{8}-\\dfrac{4}{8}-\\dfrac{3}{8} = \\dfrac{1}{8}\\). If \\(\\dfrac{1}{8}\\) = 7 members, the total = 7 × 8 = 56.",
      "hints": ["Find the fraction for White.", "If 1/8 = 7, multiply by 8 for the whole."],
      "source": "MGS 2025 P6 Prelim P2 Q20",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 11,
      "image_bbox": [0.58, 0.40, 0.82, 0.56],
      "image_loc": "pie chart with Black 1/2, Blue 3/8, White sectors",
      "notes": "Key: 56."
    },
    {
      "n": 21,
      "type_id": 2,
      "question": "The table shows the hourly rate of work done at a call center. Weekdays (Monday to Friday): $8. Weekend (Saturday and Sunday): $10.<br>Leonard works at the call center for 10 hours daily on weekdays and 5 hours on Saturdays. How much does he earn in a week? $[?]",
      "answer0": "450",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 56,
      "difficulty_id": 2,
      "explanation": "Weekday hours = 10 × 5 = 50, earning 50 × $8 = $400. Saturday = 5 × $10 = $50. Total = $400 + $50 = $450.",
      "hints": ["Weekdays are Monday to Friday (5 days).", "Use the weekend rate for Saturday."],
      "source": "MGS 2025 P6 Prelim P2 Q21",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: $450. Answer entered as a number (dollar sign in stem)."
    },
    {
      "n": 22,
      "type_id": 2,
      "question": "PR and QS are straight lines. Find ∠y. [?]°",
      "answer0": "78",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 70,
      "difficulty_id": 2,
      "explanation": "Angles on the straight line PR at the meeting point: 98° + ∠y + 20°... using vertically opposite / angles on a line, ∠y = 98° − 20° = 78°.",
      "hints": ["PR and QS are straight lines crossing.", "∠y = 98° − 20°."],
      "source": "MGS 2025 P6 Prelim P2 Q22",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 12,
      "image_bbox": [0.30, 0.58, 0.78, 0.84],
      "image_loc": "two crossing straight lines PR and QS with angles 98°, 20° and y marked",
      "notes": "Key: 78°."
    },
    {
      "n": 23,
      "type_id": 2,
      "question": "12 identical cubes are joined together to form a cuboid. The shaded face of the cuboid is 16 cm². Find the volume of the cuboid. [?] cm³",
      "answer0": "96",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 76,
      "difficulty_id": 3,
      "explanation": "The shaded face shows 4 cube-faces, so one cube face = 16 ÷ 4 = 4 cm², giving a cube edge of \\(\\sqrt{4}=2\\) cm. The cuboid is 2 cubes by 2 cubes by 3 cubes = (2×2) cm by (2×2) cm... dimensions 4 cm × 4 cm × 6 cm. Volume = 4 × 4 × 6 = 96 cm³.",
      "hints": ["Find one cube's edge from the shaded face.", "Multiply the three cuboid dimensions."],
      "source": "MGS 2025 P6 Prelim P2 Q23",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 13,
      "image_bbox": [0.20, 0.12, 0.45, 0.28],
      "image_loc": "cuboid made of 12 small cubes with one end face shaded",
      "notes": "Key: 96 cm³."
    },
    {
      "n": 24,
      "type_id": 2,
      "question": "The figure is made up of a circle and a parallelogram. O is the centre of the circle and ∠QPS = 54°. Find ∠PQR. [?]°",
      "answer0": "72",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 70,
      "difficulty_id": 3,
      "explanation": "OP and OS are radii, so triangle OPS is isosceles. ∠QPS = ∠OPS = 54° (∠OSP = 54°), so ∠POS = 180° − 54° − 54° = 72°. In the parallelogram ∠PQR = ∠POS-related... the printed key gives 180° − 54° − 54° = 72°.",
      "hints": ["OP and OS are radii (equal), so the triangle is isosceles.", "Base angles are equal; angles in a triangle sum to 180°."],
      "source": "MGS 2025 P6 Prelim P2 Q24",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 13,
      "image_bbox": [0.16, 0.50, 0.55, 0.66],
      "image_loc": "circle centred at O with parallelogram PQRS, 54° marked at P",
      "notes": "Key: 72°."
    },
    {
      "n": 25,
      "type_id": 1,
      "question": "A bag contains marbles of three different colours. \\(\\dfrac{1}{3}\\) of the marbles are blue. The ratio of the number of red marbles to green marbles is 2 : 3. Find the ratio of the number of blue marbles to that of the green marbles.",
      "answer0": "5 : 6",
      "answer1": "1 : 3",
      "answer2": "2 : 3",
      "answer3": "1 : 2",
      "correct_answer": 0,
      "skill_id": 60,
      "difficulty_id": 3,
      "explanation": "Blue = \\(\\dfrac{1}{3}\\) of total, so red + green = \\(\\dfrac{2}{3}\\). Red : green = 2 : 3, total 5 parts = \\(\\dfrac{2}{3}\\). Blue = \\(\\dfrac{1}{3}\\) = 5 parts of those same units... taking total as 15: blue = 5, red+green = 10 with red = 4, green = 6. Blue : green = 5 : 6.",
      "hints": ["Use a common total (e.g. 15) so blue is a whole number.", "Split the remaining two-thirds as red : green = 2 : 3."],
      "source": "MGS 2025 P6 Prelim P2 Q25",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: 5 : 6. MCQ because the answer is a ratio."
    },
    {
      "n": 26,
      "type_id": 2,
      "question": "A total of 93 Lego bricks are used to build a tower. There are at least 5 red bricks between any 2 blue bricks. What is the greatest number of blue bricks used? [?]",
      "answer0": "16",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 48,
      "difficulty_id": 3,
      "explanation": "Each repeating set is 1 blue + 5 red = 6 bricks. 93 ÷ 6 = 15 remainder 3. The remaining 3 bricks allow one more blue at the top, so 15 + 1 = 16 blue bricks.",
      "hints": ["Group as 1 blue followed by 5 red = 6 bricks.", "Account for the remainder by allowing a final blue brick."],
      "source": "MGS 2025 P6 Prelim P2 Q26",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: 16. Tower diagram is illustrative only; not required to answer."
    },
    {
      "n": 27,
      "type_id": 1,
      "question": "June had $5a. After buying some cakes at $5 each, she had $2a left. How many cakes did she buy? Give your answer in terms of a.",
      "answer0": "\\(\\dfrac{3a}{5}\\)",
      "answer1": "3a",
      "answer2": "\\(\\dfrac{5a}{3}\\)",
      "answer3": "\\(\\dfrac{2a}{5}\\)",
      "correct_answer": 0,
      "skill_id": 62,
      "difficulty_id": 2,
      "explanation": "Money spent = $5a − $2a = $3a. Each cake costs $5, so number of cakes = \\(\\dfrac{3a}{5}\\).",
      "hints": ["Find the amount spent on cakes.", "Divide the amount spent by the price per cake."],
      "source": "MGS 2025 P6 Prelim P2 Q27",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: 3a/5. MCQ because the answer is an algebraic fraction."
    },
    {
      "n": 28,
      "type_id": 2,
      "question": "In the figure, QRUV and PRST are rectangles. U is the midpoint of PT. RS = 16 cm, ST = 6 cm and UR = 10 cm. Find the length of QR. [?] cm",
      "answer0": "4.8",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 71,
      "difficulty_id": 3,
      "explanation": "U is the midpoint of PT, so PU = UT = 8 cm. Triangle area using base 8 cm and height 6 cm: \\(\\dfrac{1}{2}\\times8\\times6 = 24\\) cm². The rectangle QRUV relates: 24 × 2 = 48, and 48 ÷ 10 = 4.8 cm = QR.",
      "hints": ["U is the midpoint, so PU = UT = 8 cm.", "Use equal areas to find QR (the key: 1/2 × 8 × 6 = 24; 24 × 2 ÷ 10 = 4.8)."],
      "source": "MGS 2025 P6 Prelim P2 Q28",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 15,
      "image_bbox": [0.18, 0.38, 0.65, 0.58],
      "image_loc": "rectangles QRUV and PRST with R, S, T, U, P, Q, V labelled; 16 cm, 10 cm, 6 cm marked",
      "notes": "Key: 4.8 cm."
    },
    {
      "n": 29,
      "type_id": 1,
      "question": "A table with 4 columns is filled with numbers in a pattern. Column A: 3, 10, 11, 18, ... Column B: 4, 9, 12, 17, ... Column C: 5, 8, 13, 16, ... Column D: 6, 7, 14, 15, ...<br>In which column will the number 272 appear?",
      "answer0": "Column A",
      "answer1": "Column B",
      "answer2": "Column C",
      "answer3": "Column D",
      "correct_answer": 2,
      "skill_id": 48,
      "difficulty_id": 3,
      "explanation": "Each row holds 4 consecutive numbers in a boustrophedon (left-right then right-left) pattern. 272 − 2 = 270; 270 ÷ 8 = 33 remainder 6. The remainder places 272 in Column C.",
      "hints": ["Notice the numbers snake left-to-right then right-to-left.", "Use the position of 272 within its block of 8."],
      "source": "MGS 2025 P6 Prelim P2 Q29",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: Column C. MCQ (answer is a column label, not a numeric value)."
    },
    {
      "n": 30,
      "type_id": 0,
      "question": "The dot grid shows a trapezium ABCD and a right-angled triangle CDE. By joining the dots on the grid with straight lines, (a) draw and label another trapezium BCDX, such that DX is twice as long as BC. (b) draw and label another triangle CDY, such that it has the same area as triangle CDE. Triangle CDY must not overlap with trapezium ABCD.",
      "answer0": null,
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 70,
      "difficulty_id": 3,
      "explanation": "This is a drawing task on a dot grid. (a) Draw trapezium BCDX with DX parallel to BC and DX = 2 × BC. (b) Draw triangle CDY (point Y on the grid) with the same base/height product as triangle CDE so its area equals CDE, placed so it does not overlap trapezium ABCD. See answer key page for the drawn solution.",
      "hints": ["For DX twice BC, extend a line parallel to BC at double the length.", "Same area means same base × height ÷ 2."],
      "source": "MGS 2025 P6 Prelim P2 Q30",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 17,
      "image_bbox": [0.10, 0.13, 0.92, 0.55],
      "image_loc": "dot grid with trapezium ABCD and triangle CDE",
      "notes": "type_id 0: interactive draw-on-grid task, no value/sequence answer. Answer key shows the drawn figure (page 40 of PDF)."
    },
    {
      "n": 31,
      "type_id": 2,
      "question": "The usual price of a tennis racket is $120. During a sale, Joe bought the racket at a discount of 20%. How much did Joe pay for the racket? $[?]",
      "answer0": "96",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 56,
      "difficulty_id": 1,
      "explanation": "100% = $120, so 80% = $120 ÷ 100 × 80 = $96. Joe paid $96.",
      "hints": ["A 20% discount means paying 80% of the price.", "80% of $120."],
      "source": "MGS 2025 P6 Prelim P2 Q1",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 20,
      "image_bbox": [0.20, 0.24, 0.66, 0.40],
      "image_loc": "tennis racket image with 'Usual price $120' and 'Sale 20% Discount' tags",
      "notes": "Key: $96. Image is decorative (price tags); not required to solve."
    },
    {
      "n": 32,
      "type_id": 2,
      "question": "Judy had a 1-m long metal wire. She used some of it to form a square of sides (4s + 3) cm and had 32 cm of wire left. What is the value of s? [?]",
      "answer0": "3.5",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 62,
      "difficulty_id": 2,
      "explanation": "Wire used for the square = 100 − 32 = 68 cm. The square has 4 equal sides: 4(4s + 3) = 68 → 16s + 12 = 68 → 16s = 56 → s = 3.5.",
      "hints": ["1 m = 100 cm.", "Perimeter of the square = 4 × (4s + 3)."],
      "source": "MGS 2025 P6 Prelim P2 Q2",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: s = 3.5. Square diagram is illustrative; not needed."
    },
    {
      "n": 33,
      "type_id": 2,
      "question": "The sum of three different 3-digit numbers is 405. Of the three numbers, find the largest possible number. [?]",
      "answer0": "204",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 48,
      "difficulty_id": 2,
      "explanation": "To maximise one number, make the other two as small as possible: the two smallest different 3-digit numbers are 100 and 101, summing to 201. So the largest = 405 − 201 = 204.",
      "hints": ["Make the other two numbers as small as possible.", "Smallest two different 3-digit numbers are 100 and 101."],
      "source": "MGS 2025 P6 Prelim P2 Q3",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: 204."
    },
    {
      "n": 34,
      "type_id": 2,
      "question": "Participants of a shooting contest must obtain at least a certain score in the first round to qualify for the second round. There were 150 participants in the first round and the table shows the number of participants for each score: score 0 → 28, score 1 → 32, score 2 → 30, score 3 → 23, score 4 → 25, score 5 or more → 12.<br>60% of the participants did not qualify for the second round. From the table, what was the lowest score of a participant who qualified for the second round? [?]",
      "answer0": "3",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 56,
      "difficulty_id": 3,
      "explanation": "60% of 150 = 90 participants did not qualify. The lowest scores sum to 90: 28 + 32 + 30 = 90 (scores 0, 1, 2). So those scoring 0, 1, 2 did not qualify; the lowest qualifying score is 3.",
      "hints": ["Find 60% of 150 to get the non-qualifiers.", "Add the lowest score counts until you reach that number."],
      "source": "MGS 2025 P6 Prelim P2 Q4",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 21,
      "image_bbox": [0.28, 0.46, 0.70, 0.62],
      "image_loc": "table of score vs number of participants",
      "notes": "Key: 3. Table can be transcribed in the stem; image optional."
    },
    {
      "n": 35,
      "type_id": 0,
      "question": "The figure is made up of squares. Shade 3 more squares so that the figure has a line of symmetry.",
      "answer0": null,
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 80,
      "difficulty_id": 2,
      "explanation": "Shading task: add 3 squares so the overall pattern is symmetric about a line. The answer key shows the three squares to shade (page 41 of the PDF).",
      "hints": ["Find a possible line of symmetry.", "Mirror the existing shaded squares across that line."],
      "source": "MGS 2025 P6 Prelim P2 Q5",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 22,
      "image_bbox": [0.22, 0.13, 0.66, 0.46],
      "image_loc": "10x9 grid of squares with some shaded",
      "notes": "type_id 0: interactive shading task, no value answer. Answer key shows the squares to shade."
    },
    {
      "n": 36,
      "type_id": 2,
      "question": "Mrs Tan needs 21 pieces of ribbon, each of length 70 cm, to tie some presents. Ribbon is sold in rolls of 4.5 m each. What is the least number of rolls of ribbons that Mrs Tan needs to buy? [?]",
      "answer0": "4",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 65,
      "difficulty_id": 2,
      "explanation": "Each roll is 4.5 m = 450 cm; 450 ÷ 70 = 6 remainder 30, so each roll gives 6 usable pieces. 21 ÷ 6 = 3 remainder 3, so 3 rolls give 18 pieces and a 4th roll is needed for the remaining 3. Least rolls = 4.",
      "hints": ["Find how many 70 cm pieces fit in one 450 cm roll.", "Divide the 21 pieces needed by pieces per roll, then round up."],
      "source": "MGS 2025 P6 Prelim P2 Q6",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: 4 rolls. Ribbon image is decorative."
    },
    {
      "n": 37,
      "type_id": 2,
      "question": "The figure is made up of 5 identical rectangles. Find the area of the figure. [?] cm²",
      "answer0": "105",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 71,
      "difficulty_id": 3,
      "explanation": "Breadth of each rectangle = (17 − 2 × 4) ÷ 3 = 3 cm. Length = 3 + 4 = 7 cm. Area of one rectangle = 3 × 7 = 21 cm². Total = 21 × 5 = 105 cm².",
      "hints": ["Use the 17 cm height and 4 cm width to find one rectangle's breadth.", "Multiply one rectangle's area by 5."],
      "source": "MGS 2025 P6 Prelim P2 Q7",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 23,
      "image_bbox": [0.50, 0.55, 0.72, 0.78],
      "image_loc": "L-shaped figure of 5 identical rectangles, 17 cm and 4 cm marked",
      "notes": "Key: 105 cm²."
    },
    {
      "n": 38,
      "type_id": 2,
      "question": "The graph shows the number of books borrowed by the pupils in a school from January to May.<br>(a) What was the average number of books borrowed per month from January to May? [?]<br>(b) What was the percentage increase in the number of books borrowed from February to March? [?]%",
      "answer0": "525",
      "answer1": "60",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 78,
      "difficulty_id": 3,
      "explanation": "(a) Monthly values from the graph: Jan 300, Feb 375, Mar 600, Apr 525, May 825. Average = (300 + 375 + 600 + 525 + 825) ÷ 5 = 2625 ÷ 5 = 525. (b) Increase from Feb to Mar = 600 − 375 = 225; percentage = \\(\\dfrac{225}{375}\\times100\\% = 60\\%\\).",
      "hints": ["Read each month's value off the graph and average them.", "Percentage increase = increase ÷ original × 100%."],
      "source": "MGS 2025 P6 Prelim P2 Q8",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 24,
      "image_bbox": [0.18, 0.13, 0.80, 0.36],
      "image_loc": "line graph of number of books borrowed Jan-May",
      "notes": "Key: (a) 525, (b) 60%. Two answer blanks. The graph y-axis grid lines read Jan 300, Feb 375, Mar 600, Apr 525, May 825."
    },
    {
      "n": 39,
      "type_id": 2,
      "question": "In the figure, ACDE is a trapezium and AC // ED. AC and FD are straight lines. BG = FG, ∠GDE = 68°, ∠ABD = 71° and ∠BCD = 36°.<br>(a) Find ∠BDC. [?]°<br>(b) Find ∠BFD. [?]°",
      "answer0": "35",
      "answer1": "34",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 70,
      "difficulty_id": 3,
      "explanation": "(a) ∠BDC = 180° − 109° − 36° = 35° (angle sum of triangle BCD, where the angle at B inside the triangle is 109°). (b) Using BG = FG (isosceles) and the angle of 112° at the apex: ∠BFD = (180° − 112°) ÷ 2 = 34°.",
      "hints": ["For (a) use the angle sum of triangle BCD.", "For (b) the triangle with BG = FG is isosceles."],
      "source": "MGS 2025 P6 Prelim P2 Q9",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 25,
      "image_bbox": [0.18, 0.12, 0.72, 0.40],
      "image_loc": "trapezium ACDE with F, B, G, D, E and angles 68°, 71°, 36° marked",
      "notes": "Key: (a) 35°, (b) 34°. Two answer blanks."
    },
    {
      "n": 40,
      "type_id": 2,
      "question": "The bar graph shows the number of four different brands of pens sold at a shop. 560 pens were sold altogether. Part of the graph was torn off. The prices per pen are: A $2.80, B $2.20, C $2.50, D $1.60. The number of pens sold from Brand C was \\(\\dfrac{3}{5}\\) the number of pens sold from Brand D. The total amount collected from the sales of pens from Brand C and Brand D was $620.<br>(a) How many Brand C pens were sold? [?]<br>(b) How many Brand B pens were sold? [?]",
      "answer0": "120",
      "answer1": "140",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 78,
      "difficulty_id": 3,
      "explanation": "(a) Let one 'set' be 3 Brand C pens and 5 Brand D pens (since C : D = 3 : 5). Cost of one set = 3 × $2.50 + 5 × $1.60 = $7.50 + $8.00 = $15.50. Number of sets = $620 ÷ $15.50 = 40. Brand C pens = 40 × 3 = 120. (b) Brand D = 40 × 5 = 200. From the graph Brand A = 100. Brand B = 560 − 200 − 120 − 100 = 140.",
      "hints": ["C : D = 3 : 5; treat 3 C-pens + 5 D-pens as one set with a fixed cost.", "Use 560 total and Brand A from the graph to find Brand B."],
      "source": "MGS 2025 P6 Prelim P2 Q10",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 26,
      "image_bbox": [0.18, 0.13, 0.80, 0.42],
      "image_loc": "bar graph of pens sold for Brands A-D (partly torn off)",
      "notes": "Key: (a) 120 Brand C, (b) 140 Brand B. Brand A = 100 read from graph. Two answer blanks."
    },
    {
      "n": 41,
      "type_id": 2,
      "question": "The figure is made up of 2 identical squares and 3 triangles. EH is parallel to FG. ∠EHG = 128° and ∠EFG = 78°.<br>(a) Find ∠CBH. [?]°<br>(b) Find ∠GEF. [?]°",
      "answer0": "64",
      "answer1": "76",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 70,
      "difficulty_id": 3,
      "explanation": "(a) The triangle at B is isosceles (sides are square sides). The angle at B inside it = 180° − 128° = 52° region gives base angles (180° − 52°) ÷ 2 = 64°, so ∠CBH = 64°. (b) EH // FG: ∠ at E and the 78° are related. 180° − 78° = 102° (co-interior), and subtracting the square's 26° component gives ∠GEF = 102° − 26° = 76°.",
      "hints": ["The triangles formed by the squares are isosceles.", "Use parallel-line angle properties for EH // FG."],
      "source": "MGS 2025 P6 Prelim P2 Q11",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 28,
      "image_bbox": [0.16, 0.13, 0.70, 0.44],
      "image_loc": "figure of 2 squares and 3 triangles with points A-H, angles 128° and 78°",
      "notes": "Key: (a) 64°, (b) 76°. Two answer blanks."
    },
    {
      "n": 42,
      "type_id": 2,
      "question": "Rae and Jane took part in a 14 km race. Both of them started running at the same time. Rae arrived at the finishing point 10 minutes before Jane who was 1.75 km behind her. Jane did not change her speed throughout the race and she completed the race at 1015.<br>(a) What time did both of them start running? [?]<br>(b) What was Rae's average speed for the race in m/min? [?]",
      "answer0": "0855",
      "answer1": "200",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 65,
      "difficulty_id": 3,
      "explanation": "(a) When Rae finished, Jane was 1.75 km = 1750 m behind, and Jane took 10 more minutes to cover that. Jane's speed = 1750 ÷ 10 = 175 m/min. Jane's total time = 14000 ÷ 175 = 80 min = 1 h 20 min. She finished at 1015, so they started at 1015 − 1 h 20 min = 0855. (b) Rae's time = 80 − 10 = 70 min. Rae's speed = 14000 ÷ 70 = 200 m/min.",
      "hints": ["Jane's speed comes from the 1750 m she covered in the extra 10 min.", "Rae finished 10 min before Jane."],
      "source": "MGS 2025 P6 Prelim P2 Q12",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: (a) 0855 (8.55 am), (b) 200 m/min. (a) answer entered as 24-h time 0855. Two answer blanks."
    },
    {
      "n": 43,
      "type_id": 2,
      "question": "The graphs show the cost of sending packages to Indonesia for the first 10 kg by 2 courier companies (Company A and Company B).<br>(a) How much does it cost to courier a package for the first 2 kg when using Company A's service? $[?]<br>(b) Adele used Company B's service and paid $9 for her package. What was the mass of her package? [?] kg<br>(c) Yanti and Dewi each sent a 8.7-kg package to Indonesia. Yanti used Company A's service while Dewi used Company B's service. What is the difference in the amounts they paid? $[?]",
      "answer0": "5",
      "answer1": "3",
      "answer2": "1.70",
      "answer3": null,
      "correct_answer": null,
      "skill_id": 78,
      "difficulty_id": 3,
      "explanation": "(a) From Company A's graph, the cost is flat at $5 up to 2 kg, so $5. (b) On Company B's graph, $9 corresponds to a mass of 3 kg. (c) For 8.7 kg: Company B charges its flat maximum $15 (reached at 5 kg). Company A: $12 at 4 kg then $1 per extra kg, so $12 + 4.7 × $1 = $16.70. Difference = $16.70 − $15 = $1.70.",
      "hints": ["Read the flat starting cost off Company A's graph for part (a).", "For (c) note Company B caps at $15 and Company A rises $1/kg after 4 kg."],
      "source": "MGS 2025 P6 Prelim P2 Q13",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 30,
      "image_bbox": [0.18, 0.13, 0.82, 0.82],
      "image_loc": "two cost-vs-mass line graphs: Company A (top) and Company B (bottom)",
      "notes": "Key: (a) $5, (b) 3 kg, (c) $1.70 (Company B cheaper). Three answer blanks (all numeric/decimal)."
    },
    {
      "n": 44,
      "type_id": 2,
      "question": "At a talent show, 23 participants invited either 1, 2 or 3 guests. The number of participants who invited 1 guest to those who invited 2 guests was 2 : 1. There were 44 guests altogether. Find the number of participants who invited 3 guests. [?]",
      "answer0": "8",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 60,
      "difficulty_id": 3,
      "explanation": "Let participants inviting 1 guest = 2u and 2 guests = u (ratio 2 : 1). Each (2u + u) set of 3 participants contributes 2×1 + 1×2 = 4 guests. Testing values until the totals match 23 participants and 44 guests gives 2u = 20, u = 10 is too many; the answer key's table yields 8 participants inviting 3 guests (with the 1- and 2-guest groups satisfying both 23 participants and 44 guests).",
      "hints": ["Let the 1-guest and 2-guest groups be 2u and u.", "Use both totals: 23 participants and 44 guests."],
      "source": "MGS 2025 P6 Prelim P2 Q14",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: 8 participants invited 3 guests (from the answer-key systematic table)."
    },
    {
      "n": 45,
      "type_id": 2,
      "question": "Bala had 120 more stamps than Charles at first. Each of them gave Ali some of their stamps. The number of stamps Charles gave Ali was 30% of the stamps Bala had at first. The number of stamps Bala gave to Ali was \\(\\dfrac{3}{5}\\) of the number of stamps Charles had at first. Both Bala and Charles had an equal number of stamps left. How many stamps did Bala have at first? [?]",
      "answer0": "640",
      "answer1": null,
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 60,
      "difficulty_id": 3,
      "explanation": "Let Charles at first = 10u, so Bala at first = 10u + 120. Bala gave \\(\\dfrac{3}{5}\\) of Charles' = \\(\\dfrac{3}{5}\\times10u = 6u\\), leaving Bala with (10u + 120) − 6u = 4u + 120. Charles gave 30% of Bala's first = ... the key models Charles giving 3u + 36, leaving 7u − 36. Equal left: 7u − 36 = 4u + 120 → 3u = 156 → u = 52. Bala at first = 10u + 120 = 520 + 120 = 640.",
      "hints": ["Let Charles' first amount be 10u so the fractions are whole.", "Set the two 'left' expressions equal."],
      "source": "MGS 2025 P6 Prelim P2 Q15",
      "image_needed": false,
      "image_options": false,
      "image_file": null,
      "image_page": null,
      "image_bbox": null,
      "image_loc": null,
      "notes": "Key: 640. (Key's table: Bala first 10u+120, gave 6u, left 4u+120; Charles first 10u, gave 3u+36, left 7u-36; solving gives 1u=52, Bala=640.)"
    },
    {
      "n": 46,
      "type_id": 2,
      "question": "The water tank is made up of 3 rectangular cuboids joined together. At first, the tank was filled with 1800 ml of water. Both taps were turned on for 2 minutes. Tap A flowed at 1.2 litres per minute while Tap B flowed at 2 litres per minute. The base cuboid is 50 cm by 10 cm by 13 cm; the left side cuboid is 12 cm wide and the right side cuboid is 5 cm wide, each 10 cm deep.<br>(a) What was the total volume of water in the tank at the end of the 2 minutes? [?] litres<br>(b) What was the height of the water in the tank in the end? [?] cm",
      "answer0": "8.2",
      "answer1": "23",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 76,
      "difficulty_id": 3,
      "explanation": "(a) Water from taps in 2 min = 2 × (1.2 + 2) = 6.4 litres. Total = 6.4 + 1.8 = 8.2 litres. (b) Base cuboid capacity = 50 × 10 × 13 = 6500 cm³. Total water = 8200 cm³, so 8200 − 6500 = 1700 cm³ rises in the two side cuboids. Combined base area of side cuboids = 12×10 + 5×10 = 170 cm². Height in sides = 1700 ÷ 170 = 10 cm. Total height = 13 + 10 = 23 cm.",
      "hints": ["Total inflow = 2 min × (Tap A + Tap B), add the starting 1800 ml.", "Fill the base cuboid first, then the two side columns."],
      "source": "MGS 2025 P6 Prelim P2 Q16",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 34,
      "image_bbox": [0.16, 0.20, 0.82, 0.52],
      "image_loc": "U-shaped water tank with Tap A (12 cm), Tap B (5 cm), base 50 cm x 10 cm x 13 cm",
      "notes": "Key: (a) 8.2 litres, (b) 23 cm. Two answer blanks."
    },
    {
      "n": 47,
      "type_id": 2,
      "question": "Ken built a racetrack for toy cars by using 3 different rectangular pieces and 10 identical curved pieces. A curved piece is a quarter-annulus of width 7 cm (Figure 1). The width of the racetrack is 7 cm.<br>(a) Find the length of x. [?] cm<br>(b) Find the area of the entire track. (Take \\(\\pi=\\dfrac{22}{7}\\)) [?] cm²",
      "answer0": "36",
      "answer1": "1659",
      "answer2": null,
      "answer3": null,
      "correct_answer": null,
      "skill_id": 74,
      "difficulty_id": 3,
      "explanation": "(a) x = 15 + 14 + 7 = 36 cm. (b) Area of 3 rectangles = (15 + 21 + 36) × 7 = 504 cm². Outer radius of curves = (14 + 7 + 7) ÷ 2 = 14 cm; area of 10 big quarter-circles = 10 × \\(\\dfrac{1}{4}\\times\\dfrac{22}{7}\\times14\\times14\\) = 1540 cm². Area of 10 small quarter-circles (radius 7) = 10 × \\(\\dfrac{1}{4}\\times\\dfrac{22}{7}\\times7\\times7\\) = 385 cm². Curved area = 1540 − 385 = 1155 cm². Total = 504 + 1155 = 1659 cm².",
      "hints": ["For x, add the vertical segments: 15 + 14 + 7.", "Curved area = big quarter-circles minus small quarter-circles."],
      "source": "MGS 2025 P6 Prelim P2 Q17",
      "image_needed": true,
      "image_options": false,
      "image_file": null,
      "image_page": 36,
      "image_bbox": [0.18, 0.27, 0.70, 0.62],
      "image_loc": "racetrack figure with curved pieces, 15 cm, 14 cm, 7 cm and x marked",
      "notes": "Key: (a) 36 cm, (b) 1659 cm². Two answer blanks. Figure 1 (quarter-annulus) is on the same page above the racetrack."
    }
  ]
}