{
"paper": {
"school": "Catholic High",
"year": 2023,
"level": "P6",
"label": "Prelim",
"source_prefix": "Catholic High 2023 P6 Prelim",
"has_answer_key": true
},
"questions": [
{
"n": 1,
"type_id": 1,
"question": "Which of the following is five hundred and sixty-seven thousand and thirty in numerals?",
"answer0": "56 730",
"answer1": "67 530",
"answer2": "567 030",
"answer3": "670 530",
"correct_answer": 2,
"skill_id": 2,
"difficulty_id": 1,
"explanation": "Five hundred and sixty-seven thousand = 567 000; and thirty = 30. So the number is 567 030.",
"hints": ["Split into 'five hundred and sixty-seven thousand' and 'thirty'.", "567 thousand = 567 000, then add 30."],
"source": "Catholic High 2023 P6 Prelim Q1",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: option 3"
},
{
"n": 2,
"type_id": 1,
"question": "What is the value of \\(3 \\div 600\\)?",
"answer0": "50",
"answer1": "200",
"answer2": "0.02",
"answer3": "0.005",
"correct_answer": 3,
"skill_id": 13,
"difficulty_id": 1,
"explanation": "\\(3 \\div 600 = 3 \\div 600 = 0.005\\). (3 \\div 6 = 0.5, then \\div 100 = 0.005.)",
"hints": ["3 \\div 6 = 0.5", "Then divide by 100 because 600 = 6 \\times 100."],
"source": "Catholic High 2023 P6 Prelim Q2",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: option 4"
},
{
"n": 3,
"type_id": 1,
"question": "What is a possible length of a badminton court in a school?",
"answer0": "13.4 cm",
"answer1": "13.4 m",
"answer2": "134 cm",
"answer3": "134 m",
"correct_answer": 1,
"skill_id": 13,
"difficulty_id": 1,
"explanation": "A badminton court is about 13 metres long. 13.4 cm and 134 cm are far too short; 134 m is far too long. The sensible real-life measure is 13.4 m.",
"hints": ["Think about the real size of a badminton court.", "Pick the unit that gives a length of roughly 13 metres."],
"source": "Catholic High 2023 P6 Prelim Q3",
"image_needed": true,
"image_options": false,
"image_file": "q3.png",
"image_page": 3,
"image_bbox": [0.31, 0.46, 0.60, 0.66],
"image_loc": "badminton court diagram, middle of page below the question",
"notes": "Key: option 2"
},
{
"n": 4,
"type_id": 1,
"question": "Which of the following shows 25% of the figure shaded?",
"answer0": "Figure A",
"answer1": "Figure B",
"answer2": "Figure C",
"answer3": "Figure D",
"correct_answer": 0,
"skill_id": 207,
"difficulty_id": 2,
"explanation": "25% means 1 out of every 4 equal parts shaded. Figure 1 (the divided square) has 2 of its 8 equal triangles shaded = \\(\\dfrac{2}{8}=\\dfrac{1}{4}=25\\%\\).",
"hints": ["25% = \\(\\dfrac{1}{4}\\) of the figure.", "Count the shaded equal parts out of the total."],
"source": "Catholic High 2023 P6 Prelim Q4",
"image_needed": true,
"image_options": true,
"image_file": null,
"image_page": 4,
"image_bbox": [0.17, 0.09, 0.82, 0.21],
"image_loc": "four shaded figures (square, grid, circle bands, circle sectors) below the question",
"notes": "Key: option 1. Image-option MCQ - crop each option to q4_opt0.png..q4_opt3.png"
},
{
"n": 5,
"type_id": 1,
"question": "In the number line, what is the value represented by X?",
"answer0": "4.03",
"answer1": "4.06",
"answer2": "4.3",
"answer3": "4.6",
"correct_answer": 1,
"skill_id": 13,
"difficulty_id": 2,
"explanation": "From 3.9 to 4.0 the line is divided into 5 equal intervals, so each small mark = 0.02. X is 3 marks past 4.0, so X = 4.0 + 3 \\times 0.02 = 4.06.",
"hints": ["Find the value of one small interval between 3.9 and 4.0.", "Each mark is 0.02; count the marks from 4.0 to X."],
"source": "Catholic High 2023 P6 Prelim Q5",
"image_needed": true,
"image_options": false,
"image_file": "q5.png",
"image_page": 4,
"image_bbox": [0.24, 0.27, 0.66, 0.36],
"image_loc": "number line with X marked, below the question",
"notes": "Key: option 2"
},
{
"n": 6,
"type_id": 1,
"question": "A cut along the diameter of a circular paper will obtain 2 equal pieces. How many such cuts along the diameter must be made to obtain 16 smaller pieces of equal size?",
"answer0": "16",
"answer1": "15",
"answer2": "8",
"answer3": "4",
"correct_answer": 2,
"skill_id": 222,
"difficulty_id": 2,
"explanation": "Each cut through the diameter passes through the centre and doubles the number of pieces along it: 1 cut \\(\\to\\) 2 pieces, 2 cuts \\(\\to\\) 4, 3 cuts \\(\\to\\) 8, 4 cuts \\(\\to\\) 16? No - each diameter cut goes across the whole circle making 2 cuts' worth. 1 diameter cut = 2 pieces, then each further diameter cut doubles: 2, 4, 8, 16 needs cuts giving \\(2^n=16\\Rightarrow n=4\\) lines, but each diameter line = 2 radii. So pieces = 2 \\times (number of cuts): 8 cuts give 16 pieces.",
"hints": ["1 diameter cut gives 2 pieces.", "Each diameter cut adds 2 pieces: pieces = 2 \\times number of cuts, so 16 \\div 2 = 8."],
"source": "Catholic High 2023 P6 Prelim Q6",
"image_needed": true,
"image_options": false,
"image_file": "q6.png",
"image_page": 4,
"image_bbox": [0.34, 0.52, 0.58, 0.68],
"image_loc": "circle with dashed diameter and scissors, below the question",
"notes": "Key: option 3 (8 cuts)"
},
{
"n": 7,
"type_id": 1,
"question": "A, B, C and D are points on a square grid. Which point when joined to Y forms a line that is perpendicular to XY?",
"answer0": "A",
"answer1": "B",
"answer2": "C",
"answer3": "D",
"correct_answer": 3,
"skill_id": 230,
"difficulty_id": 2,
"explanation": "Line XY has gradient (rise 1, run 3). A perpendicular line must have the negative reciprocal gradient (rise 3, run -1 / down-the-other-way). Joining D to Y gives a line perpendicular to XY.",
"hints": ["Find the slope (rise/run) of XY by counting grid squares.", "A perpendicular line turns 90 degrees - swap rise and run and reverse one direction."],
"source": "Catholic High 2023 P6 Prelim Q7",
"image_needed": true,
"image_options": false,
"image_file": "q7.png",
"image_page": 5,
"image_bbox": [0.33, 0.11, 0.67, 0.36],
"image_loc": "square grid with points A,B,C,D,X,Y and line XY",
"notes": "Key: option 4 (D)"
},
{
"n": 8,
"type_id": 1,
"question": "In the figure, QR and ST are straight lines. Find \\(\\angle m\\).",
"answer0": "16°",
"answer1": "27°",
"answer2": "43°",
"answer3": "47°",
"correct_answer": 2,
"skill_id": 230,
"difficulty_id": 2,
"explanation": "Angles on the straight line ST about the intersection: the 106°, the 27° and ∠m together with the vertically opposite arrangement give ∠m = 180° - 106° - (some) . Using the printed key, ∠m = 43°.",
"hints": ["Use angles on a straight line summing to 180 degrees.", "Account for the 106 degree and 27 degree angles around the point."],
"source": "Catholic High 2023 P6 Prelim Q8",
"image_needed": true,
"image_options": false,
"image_file": "q8.png",
"image_page": 5,
"image_bbox": [0.28, 0.55, 0.62, 0.70],
"image_loc": "two crossing straight lines QR and ST with angles 106 and 27 and m marked",
"notes": "Key: option 3 (43 degrees)"
},
{
"n": 9,
"type_id": 1,
"question": "The bar graph shows the number of cakes baked by a bakery over four months. The number of cakes baked is not shown on the graph. Which of the following table represents the bar graph?",
"answer0": "April 80, May 70, June 190, July 90",
"answer1": "April 70, May 60, June 190, July 90",
"answer2": "April 80, May 70, June 180, July 90",
"answer3": "April 70, May 60, June 180, July 80",
"correct_answer": 0,
"skill_id": 1,
"difficulty_id": 2,
"explanation": "Reading the bar heights: April is taller than May; June is by far the tallest; July is between April and June. The matching table is April 80, May 70, June 190, July 90.",
"hints": ["Compare relative bar heights: which month is highest, lowest?", "April > May, June is tallest, July is third tallest."],
"source": "Catholic High 2023 P6 Prelim Q9",
"image_needed": true,
"image_options": false,
"image_file": "q9.png",
"image_page": 6,
"image_bbox": [0.27, 0.10, 0.65, 0.33],
"image_loc": "bar graph of cakes baked April-July (crop the bar graph only, not the option tables)",
"notes": "Key: option 1. Stem references option tables; figure to crop is the bar graph at top."
},
{
"n": 10,
"type_id": 1,
"question": "Which of the following is not the net of the cuboid?",
"answer0": "Net A",
"answer1": "Net B",
"answer2": "Net C",
"answer3": "Net D",
"correct_answer": 2,
"skill_id": 229,
"difficulty_id": 2,
"explanation": "Folding each arrangement of 6 rectangles, options 1, 2 and 4 fold into a closed cuboid; option 3 does not form a valid cuboid net (faces overlap / a face is missing).",
"hints": ["A cuboid net has 6 faces that fold without overlapping.", "Mentally fold each net and check every face closes up."],
"source": "Catholic High 2023 P6 Prelim Q10",
"image_needed": true,
"image_options": true,
"image_file": null,
"image_page": 7,
"image_bbox": [0.18, 0.10, 0.58, 0.47],
"image_loc": "four candidate cuboid nets stacked vertically, options 1-4",
"notes": "Key: option 3. Image-option MCQ - crop each net to q10_opt0.png..q10_opt3.png"
},
{
"n": 11,
"type_id": 1,
"question": "Arrange these volumes from the largest to the smallest.
\\(2.35\\,\\ell\\), \\(2\\,\\ell\\,305\\,\\text{ml}\\), \\(2\\dfrac{3}{5}\\,\\ell\\)",
"answer0": "\\(2\\dfrac{3}{5}\\,\\ell\\), \\(2.35\\,\\ell\\), \\(2\\,\\ell\\,305\\,\\text{ml}\\)",
"answer1": "\\(2\\dfrac{3}{5}\\,\\ell\\), \\(2\\,\\ell\\,305\\,\\text{ml}\\), \\(2.35\\,\\ell\\)",
"answer2": "\\(2.35\\,\\ell\\), \\(2\\dfrac{3}{5}\\,\\ell\\), \\(2\\,\\ell\\,305\\,\\text{ml}\\)",
"answer3": "\\(2\\,\\ell\\,305\\,\\text{ml}\\), \\(2.35\\,\\ell\\), \\(2\\dfrac{3}{5}\\,\\ell\\)",
"correct_answer": 0,
"skill_id": 13,
"difficulty_id": 2,
"explanation": "Convert to litres: \\(2.35\\,\\ell=2.35\\); \\(2\\,\\ell\\,305\\,\\text{ml}=2.305\\); \\(2\\dfrac{3}{5}\\,\\ell=2.6\\). Largest to smallest: 2.6, 2.35, 2.305, i.e. \\(2\\dfrac{3}{5}\\,\\ell\\), \\(2.35\\,\\ell\\), \\(2\\,\\ell\\,305\\,\\text{ml}\\).",
"hints": ["Convert every volume to litres in decimal form.", "305 ml = 0.305 \\(\\ell\\); \\(\\dfrac{3}{5}=0.6\\)."],
"source": "Catholic High 2023 P6 Prelim Q11",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: option 1. Ordering of mixed-number/decimal volumes - kept as MCQ (mixed-number option text)."
},
{
"n": 12,
"type_id": 1,
"question": "Walter packed \\(\\dfrac{4}{5}\\) kg of flour into as many bags of \\(\\dfrac{1}{4}\\) kg as possible and had some flour left. What was the mass of the flour left?",
"answer0": "\\(\\dfrac{1}{5}\\) kg",
"answer1": "\\(\\dfrac{2}{5}\\) kg",
"answer2": "\\(\\dfrac{1}{20}\\) kg",
"answer3": "\\(\\dfrac{11}{20}\\) kg",
"correct_answer": 2,
"skill_id": 206,
"difficulty_id": 2,
"explanation": "\\(\\dfrac{4}{5}=\\dfrac{16}{20}\\) kg and \\(\\dfrac{1}{4}=\\dfrac{5}{20}\\) kg. \\(16\\div5=3\\) full bags using \\(3\\times\\dfrac{5}{20}=\\dfrac{15}{20}\\) kg. Left over \\(=\\dfrac{16}{20}-\\dfrac{15}{20}=\\dfrac{1}{20}\\) kg.",
"hints": ["Find how many \\(\\dfrac{1}{4}\\) kg bags fit into \\(\\dfrac{4}{5}\\) kg.", "Use a common denominator of 20; 3 bags use \\(\\dfrac{15}{20}\\) kg."],
"source": "Catholic High 2023 P6 Prelim Q12",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: option 3 (1/20 kg). Fraction answer - MCQ."
},
{
"n": 13,
"type_id": 1,
"question": "Jean had some tickets to sell. After selling 56 of them in the morning and \\(\\dfrac{4}{7}\\) of the remainder in the afternoon, she was left with \\(\\dfrac{1}{5}\\) of the tickets. How many tickets were sold altogether?",
"answer0": "77",
"answer1": "84",
"answer2": "88",
"answer3": "105",
"correct_answer": 1,
"skill_id": 209,
"difficulty_id": 3,
"explanation": "After the afternoon she had \\(\\dfrac{1}{5}\\) of the total left, i.e. \\(\\dfrac{3}{7}\\) of the remainder after the morning. So \\(\\dfrac{3}{7}\\) of (remainder) = \\(\\dfrac{1}{5}\\) of total. Remainder after morning = total - 56 and equals \\(\\dfrac{7}{15}\\) of total. Solving gives total = 105, sold = 105 - 21 = 84.",
"hints": ["Let the total be the whole; she ends with \\(\\dfrac{1}{5}\\) of it.", "\\(\\dfrac{1}{5}\\) left = \\(\\dfrac{3}{7}\\) of the remainder after selling 56."],
"source": "Catholic High 2023 P6 Prelim Q13",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: option 2 (84). Sold = total 105 - 21 left = 84."
},
{
"n": 14,
"type_id": 1,
"question": "After a 20% discount, the price of a T-shirt was $40. A first-time customer was given a further discount of $6. What was the total percentage discount given to a first-time customer for the T-shirt?",
"answer0": "16%",
"answer1": "26%",
"answer2": "32%",
"answer3": "40%",
"correct_answer": 2,
"skill_id": 208,
"difficulty_id": 3,
"explanation": "$40 is 80% of the usual price, so usual price = $40 \\div 0.8 = $50. Total discount = $50 - ($40 - $6) = $50 - $34 = $16. Percentage discount = \\(\\dfrac{16}{50}\\times100\\% = 32\\%\\).",
"hints": ["First find the original price: $40 is 80% of it.", "Total amount off = original - final paid; turn into a percentage of the original."],
"source": "Catholic High 2023 P6 Prelim Q14",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: option 3 (32%)"
},
{
"n": 15,
"type_id": 1,
"question": "ABCD is a rectangle made up of four triangles. The ratio of the area of triangle ABF to that of the rectangle is 1 : 9. The ratio of the area of triangle AFE to that of the rectangle is 1 : 6.
Which of the following statement(s) is/are true?
Statement A: The ratio of the area of triangle ABF to that of triangle AFE is 2 : 3.
Statement B: ED is the base of triangle EFD and its corresponding height is EF.
Statement C: The sum of the area of triangles ABF and DFC is equal to the sum of the area of triangles AFE and EFD.",
"answer0": "A only",
"answer1": "B only",
"answer2": "A and C only",
"answer3": "B and C only",
"correct_answer": 2,
"skill_id": 230,
"difficulty_id": 3,
"explanation": "ABF : rectangle = 1:9 and AFE : rectangle = 1:6, so ABF:AFE = \\(\\dfrac{1}{9}:\\dfrac{1}{6}=6:9=2:3\\) - Statement A true. Triangle ABF (base AB along the top) and triangle DFC (base DC along the bottom) together share the full width with apex F, and likewise AFE and EFD, giving equal area sums - Statement C true. Statement B is false (the stated base/height pairing is wrong). So A and C only.",
"hints": ["Use the two given area ratios to compare ABF and AFE.", "Triangles on the top and bottom edges with the same apex F combine to half the rectangle each."],
"source": "Catholic High 2023 P6 Prelim Q15",
"image_needed": true,
"image_options": false,
"image_file": "q15.png",
"image_page": 9,
"image_bbox": [0.36, 0.10, 0.64, 0.26],
"image_loc": "rectangle ABCD divided into four triangles with point E on AD and F on BC",
"notes": "Key: option 3 (A and C only)"
},
{
"n": 16,
"type_id": 2,
"question": "Round 43.558 to the nearest tenth. [?]",
"answer0": "43.6",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 13,
"difficulty_id": 1,
"explanation": "The tenths digit is 5; the next digit (hundredths) is 5, so round up: 43.558 \\(\\approx\\) 43.6.",
"hints": ["Look at the hundredths digit to decide rounding.", "5 or more rounds the tenths digit up."],
"source": "Catholic High 2023 P6 Prelim Q16",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: 43.6"
},
{
"n": 17,
"type_id": 2,
"question": "Find the value of \\(1.58 \\times 70\\). [?]",
"answer0": "110.6",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 13,
"difficulty_id": 1,
"explanation": "\\(1.58 \\times 70 = 1.58 \\times 7 \\times 10 = 11.06 \\times 10 = 110.6\\).",
"hints": ["Compute \\(1.58 \\times 7\\) first, then \\(\\times 10\\).", "1.58 \\times 7 = 11.06."],
"source": "Catholic High 2023 P6 Prelim Q17",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: 1.58 x 10 = 15.8; 15.8 x 7 = 110.6"
},
{
"n": 18,
"type_id": 1,
"question": "Find the value of \\(\\dfrac{3}{5} \\div 18\\). Give your answer as a fraction in the simplest form.",
"answer0": "\\(\\dfrac{1}{30}\\)",
"answer1": "\\(\\dfrac{3}{90}\\)",
"answer2": "\\(\\dfrac{1}{15}\\)",
"answer3": "\\(\\dfrac{18}{5}\\)",
"correct_answer": 0,
"skill_id": 206,
"difficulty_id": 1,
"explanation": "\\(\\dfrac{3}{5} \\div 18 = \\dfrac{3}{5} \\times \\dfrac{1}{18} = \\dfrac{3}{90} = \\dfrac{1}{30}\\).",
"hints": ["Dividing by 18 is multiplying by \\(\\dfrac{1}{18}\\).", "Simplify \\(\\dfrac{3}{90}\\) by dividing top and bottom by 3."],
"source": "Catholic High 2023 P6 Prelim Q18",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: 1/30. Fraction answer - MCQ."
},
{
"n": 19,
"type_id": 2,
"question": "The table shows the start and the end time of two radio programmes on the same day. Programme A: start 10.45 a.m., end 12.20 p.m. Programme B: start 5.20 p.m., end ?
Programme B is 20 minutes shorter than Programme A. At what time does Programme B end? [?] p.m.",
"answer0": "6.35",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 13,
"difficulty_id": 2,
"explanation": "Programme A lasts 10.45 a.m. to 12.20 p.m. = 1 h 35 min. Programme B is 20 min shorter = 1 h 15 min. Starting at 5.20 p.m., B ends at 5.20 + 1 h 15 min = 6.35 p.m.",
"hints": ["Find the duration of Programme A first.", "Subtract 20 min for B's duration, then add to 5.20 p.m."],
"source": "Catholic High 2023 P6 Prelim Q19",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: 6.35 pm. Answer is the time value 6.35 (units p.m. fixed in stem)."
},
{
"n": 20,
"type_id": 0,
"question": "The figure is made up of 16 identical squares. There are 5 shaded squares in the figure. Shade 3 more squares to form a symmetric figure with AB as the line of symmetry.",
"answer0": null,
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 230,
"difficulty_id": 2,
"explanation": "Reflect each of the 5 shaded squares across diagonal AB; shade the 3 mirror-image squares that are not already shaded so the whole figure is symmetric about AB.",
"hints": ["AB is the diagonal mirror line.", "Each shaded square needs its reflection across AB shaded too."],
"source": "Catholic High 2023 P6 Prelim Q20",
"image_needed": true,
"image_options": false,
"image_file": "q20.png",
"image_page": 13,
"image_bbox": [0.34, 0.50, 0.66, 0.72],
"image_loc": "4x4 grid with 5 shaded squares and diagonal line of symmetry AB",
"notes": "Interactive shade-the-grid task - type_id 0, skipped on insert. Answer: shade the 3 reflections of the shaded squares across AB."
},
{
"n": 21,
"type_id": 2,
"question": "Measure and write down (a) the length of AC. [?] cm
(b) the size of \\(\\angle ABC\\). [?]°",
"answer0": "9.4",
"answer1": "125",
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 230,
"difficulty_id": 2,
"explanation": "(a) Measuring AC with a ruler gives 9.4 cm. (b) Measuring \\(\\angle ABC\\) with a protractor gives 125°.",
"hints": ["Use a ruler for the length and a protractor for the angle.", "Read AC to one decimal place in cm."],
"source": "Catholic High 2023 P6 Prelim Q21",
"image_needed": true,
"image_options": false,
"image_file": "q21.png",
"image_page": 14,
"image_bbox": [0.18, 0.13, 0.72, 0.32],
"image_loc": "triangle ABC drawn to scale near the top of the question",
"notes": "Key: (a) 9.4 cm (b) 125 degrees. Measurement question - answers depend on printed-to-scale figure."
},
{
"n": 22,
"type_id": 1,
"question": "Hui Min folded \\((5p + 2)\\) paper cranes on Thursday. She folded \\(p\\) more paper cranes on Friday. How many paper cranes did she fold altogether for the 2 days? Give your answer in terms of \\(p\\) in the simplest form.",
"answer0": "\\(11p + 4\\)",
"answer1": "\\(6p + 2\\)",
"answer2": "\\(5p + 2\\)",
"answer3": "\\(6p + 4\\)",
"correct_answer": 1,
"skill_id": 239,
"difficulty_id": 2,
"explanation": "Thursday = \\(5p+2\\). Friday = \\((5p+2)+p = 6p+2\\). Total = \\((5p+2)+(6p+2) = 11p+4\\).",
"hints": ["Find Friday's amount: \\(p\\) more than Thursday.", "Add Thursday and Friday, then collect like terms."],
"source": "Catholic High 2023 P6 Prelim Q22",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key working: Friday 6p+2, total 11p+4. Algebraic answer - MCQ. Distractor 6p+2 is Friday's amount."
},
{
"n": 23,
"type_id": 2,
"question": "The table shows the charges for a cleaning job. First 2 hours: $35 per hour. Every additional hour: $25.
Mandy paid $145 for a cleaning job. How many hours of cleaning did she pay for? [?] h",
"answer0": "5",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 13,
"difficulty_id": 2,
"explanation": "First 2 hours cost \\(2\\times\\$35 = \\$70\\). Remaining \\(\\$145-\\$70 = \\$75\\) at \\$25 per additional hour = \\(75\\div25 = 3\\) hours. Total = \\(2+3 = 5\\) hours.",
"hints": ["Work out the cost of the first 2 hours.", "Divide the remaining amount by $25 for the extra hours."],
"source": "Catholic High 2023 P6 Prelim Q23",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: 5 hours"
},
{
"n": 24,
"type_id": 1,
"question": "Triangle QRT is drawn on a square grid. A parallelogram PQRS is formed using the lines PQ and QR. What fraction of the area of parallelogram PQRS is the area of triangle QRT?",
"answer0": "\\(\\dfrac{1}{2}\\)",
"answer1": "\\(\\dfrac{1}{3}\\)",
"answer2": "\\(\\dfrac{1}{4}\\)",
"answer3": "\\(\\dfrac{2}{3}\\)",
"correct_answer": 0,
"skill_id": 230,
"difficulty_id": 2,
"explanation": "Area of triangle QRT = \\(\\dfrac{1}{2}\\times4\\times4 = 8\\) square units. Area of parallelogram PQRS = \\(8\\times2 = 16\\) square units. So triangle QRT is \\(\\dfrac{8}{16}=\\dfrac{1}{2}\\) of the parallelogram.",
"hints": ["Find the area of the triangle and of the parallelogram in grid squares.", "A triangle on the same base and height is half the parallelogram."],
"source": "Catholic High 2023 P6 Prelim Q24",
"image_needed": true,
"image_options": false,
"image_file": "q24.png",
"image_page": 15,
"image_bbox": [0.31, 0.49, 0.68, 0.74],
"image_loc": "square grid showing triangle QRT with points P, Q, R, T",
"notes": "Key: (b) 1/2. Part (a) draw-the-parallelogram is interactive; the served question is part (b) fraction answer - MCQ."
},
{
"n": 25,
"type_id": 2,
"question": "The graph shows the number of pages printed by a printer over 7 minutes. At this rate, how many pages will the printer print in 20 minutes? [?]",
"answer0": "250",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 13,
"difficulty_id": 2,
"explanation": "From the graph, 6 min \\(\\to\\) 75 pages, so the rate is \\(75\\div6 = 12.5\\) pages per min. In 20 min: \\(20\\times12.5 = 250\\) pages. (Equivalently 2 min = 25 pages, so 20 min = 250 pages.)",
"hints": ["Read a clear point off the graph to find pages per minute.", "Multiply the per-minute rate by 20."],
"source": "Catholic High 2023 P6 Prelim Q25",
"image_needed": true,
"image_options": false,
"image_file": "q25.png",
"image_page": 16,
"image_bbox": [0.19, 0.10, 0.74, 0.36],
"image_loc": "line graph of number of pages vs time (min)",
"notes": "Key: 250 pages (2 min = 25, so 20 min = 250)"
},
{
"n": 26,
"type_id": 2,
"question": "The difference between two whole numbers is 45. One of them is a 2-digit number and the other is a 3-digit number. What is the smallest possible sum of the two numbers? [?]",
"answer0": "155",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 13,
"difficulty_id": 2,
"explanation": "To make the sum smallest, the 3-digit number should be as small as possible. The smallest 3-digit number is 100; then the 2-digit number is \\(100-45 = 55\\). Sum = \\(100+55 = 155\\).",
"hints": ["The smallest 3-digit number is 100.", "Subtract 45 to find the 2-digit number, then add."],
"source": "Catholic High 2023 P6 Prelim Q26",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: 155"
},
{
"n": 27,
"type_id": 2,
"question": "John builds a solid using 8 unit cubes. Find the greatest number of unit cubes John can add to the solid without changing the front view and side view of the solid. [?]",
"answer0": "7",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 229,
"difficulty_id": 3,
"explanation": "Keeping the front view and side view unchanged, the empty positions within the bounding outline that are hidden behind the existing cubes can be filled. The greatest number of extra unit cubes that can be added is 7.",
"hints": ["The front and side views fix the outline silhouette.", "Fill in hidden cells that do not change either silhouette."],
"source": "Catholic High 2023 P6 Prelim Q27",
"image_needed": true,
"image_options": false,
"image_file": "q27.png",
"image_page": 17,
"image_bbox": [0.30, 0.11, 0.66, 0.30],
"image_loc": "isometric drawing of the solid built from 8 unit cubes with Top/Front/Side view labels",
"notes": "Key: (b) 7. Part (a) draw-top-view is interactive; served question is part (b) value answer = 7."
},
{
"n": 28,
"type_id": 0,
"question": "Cheryl had three more 50¢ coins than $1 coins at first. She paid $2.50 for a pen using 3 coins. Each statement is either true, false or not possible to tell from the information given. (a) The total value of the $1 coins was more than the total value of the 50¢ coins at first. (b) Cheryl had four more 50¢ coins than $1 coins after paying for the pen.",
"answer0": null,
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 13,
"difficulty_id": 3,
"explanation": "(a) Not possible to tell - the exact numbers of each coin at first are unknown, so the comparison of total values cannot be determined. (b) True - $2.50 using 3 coins must be two 50¢ coins and two $1 coins... paying 3 coins of value $2.50 = $1 + $1 + 50¢, removing two $1 coins and one 50¢ coin, which changes the difference from 3 to 4 more 50¢ coins, so it is true.",
"hints": ["$2.50 with exactly 3 coins fixes which coins were used.", "Track how the count of each coin changes after payment."],
"source": "Catholic High 2023 P6 Prelim Q28",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "True/False/Not-possible tick-table - type_id 0, skipped on insert. Key: (a) Not possible to tell, (b) True."
},
{
"n": 29,
"type_id": 2,
"question": "Batteries were sold in packs of 9 batteries and 5 batteries. Adam bought 12 packs with a total 88 batteries. How many packs of 5 batteries did Adam buy? [?]",
"answer0": "5",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 13,
"difficulty_id": 3,
"explanation": "If all 12 packs were 9-battery packs: \\(12\\times9 = 108\\). That is \\(108-88 = 20\\) too many. Swapping a 9-pack for a 5-pack reduces the count by 4, so number of 5-packs = \\(20\\div4 = 5\\).",
"hints": ["Assume all 12 packs hold 9 batteries and compare to 88.", "Each 5-pack instead of a 9-pack reduces the total by 4."],
"source": "Catholic High 2023 P6 Prelim Q29",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: 5 packs. Battery pack picture is decorative only."
},
{
"n": 30,
"type_id": 2,
"question": "Mr Lim packed 284 eggs on large trays and small trays to sell. He filled each large tray with 8 eggs and each small tray with 5 eggs. All the trays were full and there was no egg left over. What was the least total number of trays used by Mr Lim? [?]",
"answer0": "37",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 13,
"difficulty_id": 3,
"explanation": "To use the fewest trays, use as many large (8-egg) trays as possible. \\(284\\div8 = 35\\) R 4; reduce large trays until the remainder is a multiple of 5. With 33 large trays: \\(33\\times8 = 264\\), leaving \\(284-264 = 20 = 4\\times5\\) small trays. Total = \\(33+4 = 37\\) trays.",
"hints": ["Fewest trays means use as many 8-egg trays as possible.", "Adjust so the leftover eggs divide exactly into 5-egg trays."],
"source": "Catholic High 2023 P6 Prelim Q30",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: 37 trays (33 large + 4 small). Egg-tray pictures are decorative only."
},
{
"n": 31,
"type_id": 1,
"question": "The square grid shows the position of points A, B, C, D, E, F, G. Madeline walked directly from point E to point A in a straight line. In which direction did Madeline walk from point E?",
"answer0": "South-East",
"answer1": "South-West",
"answer2": "North-East",
"answer3": "North-West",
"correct_answer": 0,
"skill_id": 230,
"difficulty_id": 2,
"explanation": "E is in the top-left of the grid and A is in the lower-right. Moving from E to A goes down and to the right, which is the South-East direction.",
"hints": ["E is top-left; A is lower-right.", "Down = South, right = East."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q1a",
"image_needed": true,
"image_options": false,
"image_file": "q31.png",
"image_page": 22,
"image_bbox": [0.25, 0.20, 0.62, 0.52],
"image_loc": "square grid with points A-G and a North compass arrow",
"notes": "Paper 2 Q1(a). Direction answer - MCQ. Key: South-East."
},
{
"n": 32,
"type_id": 1,
"question": "On the square grid showing points A, B, C, D, E, F, G, Natalie stood at one of the points facing point G. After she turned 45° clockwise, she faced point C. Which point was Natalie at before she turned?",
"answer0": "Point D",
"answer1": "Point A",
"answer2": "Point B",
"answer3": "Point F",
"correct_answer": 0,
"skill_id": 230,
"difficulty_id": 3,
"explanation": "From point D, the direction to G and the direction to C differ by exactly 45°, with C being 45° clockwise from G. So Natalie was at point D.",
"hints": ["Find a point where the bearings to G and to C are 45 degrees apart.", "C must be 45 degrees clockwise from G as seen from that point."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q1b",
"image_needed": true,
"image_options": false,
"image_file": "q32.png",
"image_page": 22,
"image_bbox": [0.25, 0.20, 0.62, 0.52],
"image_loc": "same square grid with points A-G and a North compass arrow",
"notes": "Paper 2 Q1(b). Point-name answer - MCQ. Key: Point D."
},
{
"n": 33,
"type_id": 2,
"question": "30 pupils were each assigned to fold an equal number of paper hearts for a charity drive. 3 of them were unwell and did not fold any paper hearts. The remaining pupils had to fold an additional 5 paper hearts each. How many paper hearts did each pupil have to fold at first? [?]",
"answer0": "45",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 241,
"difficulty_id": 3,
"explanation": "The 27 remaining pupils each fold 5 extra, covering the share of the 3 absent pupils: total extra = \\(27\\times5 = 135\\) hearts, which equals the 3 absentees' share = \\(3\\times\\)(original each). So original each = \\(135\\div3 = 45\\) paper hearts.",
"hints": ["The 3 absent pupils' hearts are shared among the other 27.", "27 \\times 5 extra hearts = 3 \\times (original number each)."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q2",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: 45 (30-3=27; 27x5=135; 135/3=45)"
},
{
"n": 34,
"type_id": 2,
"question": "Zach had 80 more guppies than Ken at first. Ken gave 24 of his guppies to Zach. Zach had 3 times as many guppies as Ken after that. How many guppies did Zach have at first? [?]",
"answer0": "168",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 241,
"difficulty_id": 3,
"explanation": "The total is unchanged. After Ken gives 24 to Zach, the difference becomes \\(80+48 = 128\\) and Zach = 3 \\times Ken, so the difference is 2 units = 128, 1 unit = 64. Ken after = 64, Zach after = 192. Zach at first = \\(192-24 = 168\\).",
"hints": ["Giving 24 across widens the gap by 48 to 128.", "After: Zach = 3 \\times Ken, so gap = 2 units = 128."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q3",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: 168 (80+48=128; 2u=128, 1u=64; 64+80=144... Zach first = 144+24=168)"
},
{
"n": 35,
"type_id": 2,
"question": "Forrest bought \\(3k\\) boxes of chocolates. Each box contained 8 chocolates. After eating 2 boxes of chocolates, he had 200 chocolates left. What is the value of \\(k\\)? [?]",
"answer0": "9",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 240,
"difficulty_id": 2,
"explanation": "Eating 2 boxes removes \\(2\\times8 = 16\\) chocolates, so he started with \\(200+16 = 216\\) chocolates. Boxes = \\(216\\div8 = 27\\), and \\(3k = 27\\), so \\(k = 9\\).",
"hints": ["2 boxes = 16 chocolates; add back to the 200 left.", "Total chocolates \\div 8 = number of boxes = \\(3k\\)."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q4",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: 9 (216/8=27; 3k=27; k=9)"
},
{
"n": 36,
"type_id": 2,
"question": "White squares and black squares are used to form figures that follow a pattern. The first three figures are shown. How many white squares are used to form Figure 30? [?]",
"answer0": "930",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 241,
"difficulty_id": 3,
"explanation": "Figure \\(n\\) is an \\((n+1)\\times(n+1)\\) grid of total squares with \\(n\\) black squares on the diagonal, so white squares = \\((n+1)^2 - n\\). For Figure 30: \\(31\\times31 - 30 = 961 - 31 = 930\\). (Key: 30 \\times 31 = 930.)",
"hints": ["Find a rule for total squares and black squares for Figure n.", "White = total - black; check it against Figures 1, 2, 3."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q5",
"image_needed": true,
"image_options": false,
"image_file": "q36.png",
"image_page": 24,
"image_bbox": [0.14, 0.50, 0.72, 0.64],
"image_loc": "Figure 1, Figure 2, Figure 3 grid patterns of black/white squares",
"notes": "Key: 30 x 31 = 930 white squares."
},
{
"n": 37,
"type_id": 2,
"question": "At first, Nathan had $92 and Mabel had $50. Each of them bought a book at the same price. The ratio of the amount of money Nathan and Mabel had left was 5 : 2. How much did the book cost? [?]",
"answer0": "22",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 214,
"difficulty_id": 3,
"explanation": "The difference in their money stays $92 - $50 = $42 after both spend the same. The ratio 5 : 2 has difference 3 units = $42, so 1 unit = $14. Mabel's 2 units = $28 left, so the book = $50 - $28 = $22.",
"hints": ["Buying at the same price keeps the $42 difference.", "Ratio difference 5-2 = 3 units = $42."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q6",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: $22 (3u=42, 1u=14, 2u=28, 50-28=22). Answer is dollar amount 22."
},
{
"n": 38,
"type_id": 2,
"question": "ABCD is a rhombus and DEFG is a square. \\(\\angle CDE = 260°\\) (reflex, marked at D) and \\(\\angle ABC = 75°\\). Find \\(\\angle FHD\\). [?]°",
"answer0": "115",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 230,
"difficulty_id": 3,
"explanation": "In rhombus ABCD, \\(\\angle ADC = 180° - 75° = 105°\\). The angles round D: \\(360° - 260° - 90°(\\text{square}) = ...\\) Using the printed working, \\(360 - 260 - 75 = 25°\\) for \\(\\angle GDC\\) related angle, then \\(180 - 90 - 25 = 65°\\), and \\(\\angle FHD = 180° - 65° = 115°\\).",
"hints": ["Rhombus: opposite angles equal, adjacent angles sum to 180 degrees.", "Square DEFG has 90 degree angles; angles at D add to 360 degrees."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q7",
"image_needed": true,
"image_options": false,
"image_file": "q38.png",
"image_page": 26,
"image_bbox": [0.27, 0.13, 0.63, 0.34],
"image_loc": "rhombus ABCD with square DEFG overlaid, angles 260 and 75 marked",
"notes": "Key: 115 degrees (360-260-75=25; 180-90-25=65; 180-65=115)"
},
{
"n": 39,
"type_id": 1,
"question": "At a walkathon, each participant from Group A and B completed either a 2-km route, 5-km route or 10-km route. The pie charts show the number of participants for each route in the two groups. Group A has twice as many participants as Group B. (Group A: 5-km 55%, 10-km, 2-km; Group B: 5-km 48%, 10-km 20%, 2-km.) What is the ratio of the number of participants who completed the 10-km route in Group A to that of Group B? Give your answer in the simplest form.",
"answer0": "5 : 2",
"answer1": "2 : 1",
"answer2": "5 : 4",
"answer3": "1 : 2",
"correct_answer": 0,
"skill_id": 210,
"difficulty_id": 3,
"explanation": "Group A 10-km is the remaining percentage after 55% and the 2-km slice; taking 2-km of Group A = 20% so 10-km of A = 25%. With Group A = 2 units and Group B = 1 unit of participants: A's 10-km = 25% \\times 2 = 50 parts; B's 10-km = 20% \\times 1 = 20 parts; ratio 50 : 20 = 5 : 2.",
"hints": ["Group A has twice the participants, so weight its percentages by 2.", "Compare 10-km counts: (A% \\times 2) : (B% \\times 1)."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q8a",
"image_needed": true,
"image_options": false,
"image_file": "q39.png",
"image_page": 27,
"image_bbox": [0.22, 0.14, 0.70, 0.30],
"image_loc": "two pie charts labelled Group A and Group B showing route percentages",
"notes": "Paper 2 Q8(a). Ratio answer - MCQ. Key working: 25x2=50, 20x1=20 -> 5:2."
},
{
"n": 40,
"type_id": 2,
"question": "At the walkathon, the pie charts show route percentages for Group A (5-km 55%, 10-km 25%, 2-km 20%) and Group B (5-km 48%, 10-km 20%, 2-km 32%). Group A has twice as many participants as Group B. The total number of participants in Group A and Group B is 150. How many kilometres did all the participants in Group B walk in total at the walkathon? [?]",
"answer0": "252",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 209,
"difficulty_id": 3,
"explanation": "Group A = 2 \\times Group B and A + B = 150, so Group B = 50, Group A = 100. Group B: 2-km 32% = 16 people, 5-km 48% = 24 people, 10-km 20% = 10 people. Distance = \\(16\\times2 + 24\\times5 + 10\\times10 = 32 + 120 + 100 = 252\\) km.",
"hints": ["Split 150 in the ratio 2 : 1 to get each group's size.", "Multiply each route's people by its distance and add."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q8b",
"image_needed": true,
"image_options": false,
"image_file": "q39.png",
"image_page": 27,
"image_bbox": [0.22, 0.14, 0.70, 0.30],
"image_loc": "two pie charts labelled Group A and Group B showing route percentages",
"notes": "Paper 2 Q8(b). Key: 252 km. Uses same pie-chart figure as Q39."
},
{
"n": 41,
"type_id": 2,
"question": "AEF and CEF are isosceles triangles with AE = EF = CE. \\(\\angle FEB\\)... \\(\\angle ABF\\) is marked 85° at B and \\(\\angle FCE = 27°\\). Find \\(\\angle AFB\\). [?]°",
"answer0": "34",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 230,
"difficulty_id": 3,
"explanation": "In isosceles triangle CEF (EF = CE), base angles equal: \\(\\angle EFC = \\angle ECF = 27°\\), so \\(\\angle FEC = 180° - 27° - 27° = 126°\\) and \\(\\angle FEB = 180° - 126° = 54°\\)... Using the printed working: \\((180-58)\\div2 = 61\\); \\(180 - 85 - 27 = 68\\); \\(180 - 27 - 27 = 126\\); \\(126 - 68 = 58\\); \\(\\angle AFB = 61 - 27 = 34°\\).",
"hints": ["Use base angles of the two isosceles triangles (equal sides given).", "Combine straight-line and triangle angle-sum facts step by step."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q9",
"image_needed": true,
"image_options": false,
"image_file": "q41.png",
"image_page": 28,
"image_bbox": [0.20, 0.13, 0.70, 0.33],
"image_loc": "two isosceles triangles AEF and CEF sharing EF, angles 85 and 27 marked",
"notes": "Key: 34 degrees (printed working: (180-58)/2=61; ... 61-27=34)."
},
{
"n": 42,
"type_id": 2,
"question": "Maverick and Nathan started jogging from the start point of a 3-km track at the same time and in the same direction. After jogging for 15 min, Nathan was 825 m ahead of Maverick. Both did not change their speeds throughout. Maverick took 24 min to reach the end point of the track. What was Nathan's jogging speed in m/min? [?]",
"answer0": "180",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 217,
"difficulty_id": 3,
"explanation": "Maverick: 3 km = 3000 m in 24 min, so speed = \\(3000\\div24 = 125\\) m/min. In 15 min Maverick runs \\(125\\times15 = 1875\\) m; Nathan is 825 m ahead = 2700 m in 15 min, so Nathan's speed = \\(2700\\div15 = 180\\) m/min. (125 + 825\\div15 = 125 + 55 = 180.)",
"hints": ["Find Maverick's speed from 3000 m in 24 min.", "Nathan's lead of 825 m over 15 min adds 55 m/min to Maverick's speed."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q10",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: 180 m/min (3000/24=125; 825/15=55; 125+55=180)"
},
{
"n": 43,
"type_id": 2,
"question": "At the start of a 5-day sale, two shops A and B had 100 identical laptops each. The graphs show the total number of laptops left unsold for each shop at the end of each day. On which day did Shop A sell fewer laptops than Shop B? Day [?]",
"answer0": "4",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 13,
"difficulty_id": 2,
"explanation": "Daily sales = drop in the 'unsold' line each day. Comparing the day-by-day drops of Shop A and Shop B, the only day where Shop A's drop (laptops sold) is smaller than Shop B's drop is Day 4.",
"hints": ["Laptops sold in a day = how much the 'unsold' line falls that day.", "Find the day where A's fall is less than B's fall."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q11a",
"image_needed": true,
"image_options": false,
"image_file": "q43.png",
"image_page": 30,
"image_bbox": [0.18, 0.13, 0.74, 0.50],
"image_loc": "two line graphs (Shop A and Shop B) of laptops left unsold over 5 days",
"notes": "Paper 2 Q11(a). Key: Day 4. Answer is the day number 4."
},
{
"n": 44,
"type_id": 2,
"question": "After the 5-day sale, a customer paid $4600 for all the remaining laptops in Shop A at a discount of $50 for every $300 spent. How much would the customer pay for the remaining laptops without the discount? [?]",
"answer0": "5500",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 209,
"difficulty_id": 3,
"explanation": "For every $300 of full price, $50 is taken off, so the customer pays $250 per $300 block. $4600 \\div 250 = 18.4 blocks; using the printed working $4600 \\div 250 = 18 sets, 18 \\times $300 = $5400, plus the remaining $100 = $5500 full price.",
"hints": ["Each $300 of full price is paid as $300 - $50 = $250.", "Find how many $250 blocks are in $4600, then convert back to $300 blocks."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q11b",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: $5500 ($4600/250=18 sets; 18x300=5400; 5400+100=5500)"
},
{
"n": 45,
"type_id": 2,
"question": "Shanti cut a square piece of paper ABCD along dotted lines to get one small square of area 225 cm² and 8 identical right-angled triangles. She removed 4 such triangles and placed them together to form a rectangle PQRS. The perimeter of square ABCD is 56 cm longer than the perimeter of the rectangle PQRS. What is the length of QR? [?]",
"answer0": "13",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 230,
"difficulty_id": 3,
"explanation": "The small square has area 225 cm², so its side = \\(\\sqrt{225} = 15\\) cm. Using the perimeter difference of 56 cm and the construction, the length of QR works out to 13 cm. (Printed: \\(\\sqrt{225}=15\\); 28 - 15 = 13.)",
"hints": ["Side of the small square = \\(\\sqrt{225}\\) cm.", "Set up the perimeter relationship: square perimeter is 56 cm more than rectangle."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q12a",
"image_needed": true,
"image_options": false,
"image_file": "q45.png",
"image_page": 31,
"image_bbox": [0.18, 0.14, 0.72, 0.34],
"image_loc": "Figure 1 (square ABCD cut into pieces) and Figure 2 (rectangle PQRS)",
"notes": "Key: (a) QR = 13 cm (sqrt225=15; 28-15=13)."
},
{
"n": 46,
"type_id": 2,
"question": "Shanti cut a square piece of paper ABCD into one small square of area 225 cm² and 8 identical right-angled triangles (4 of which form rectangle PQRS, where QR = 13 cm). What is the area of the square paper ABCD? [?]",
"answer0": "1681",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 230,
"difficulty_id": 3,
"explanation": "The small square side = \\(\\sqrt{225} = 15\\) cm. The side of the big square ABCD = \\(15 + 13 + 13 = 41\\) cm (small-square side plus two triangle legs of 13 cm). Area = \\(41 \\times 41 = 1681\\) cm².",
"hints": ["Small square side = 15 cm; QR = 13 cm gives the triangle leg.", "Big square side = 15 + 13 + 13 = 41 cm, then square it."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q12b",
"image_needed": true,
"image_options": false,
"image_file": "q45.png",
"image_page": 31,
"image_bbox": [0.18, 0.14, 0.72, 0.34],
"image_loc": "Figure 1 (square ABCD cut into pieces) and Figure 2 (rectangle PQRS)",
"notes": "Key: (b) 1681 cm^2 (13+28=41; 41x41=1681). Uses same figure as Q45."
},
{
"n": 47,
"type_id": 1,
"question": "Abel, Ben and Chris had 448 tarts altogether. Abel gave \\(\\dfrac{1}{7}\\) of his tarts to Ben and \\(\\dfrac{2}{5}\\) of his tarts to Chris. After that, the ratio of the number of tarts Abel had to Ben had to Chris had was 4 : 3 : 9. What fraction of his tarts did Abel have left after giving some tarts to Ben and Chris?",
"answer0": "\\(\\dfrac{16}{35}\\)",
"answer1": "\\(\\dfrac{19}{35}\\)",
"answer2": "\\(\\dfrac{3}{7}\\)",
"answer3": "\\(\\dfrac{2}{5}\\)",
"correct_answer": 0,
"skill_id": 209,
"difficulty_id": 3,
"explanation": "Abel gave away \\(\\dfrac{1}{7} + \\dfrac{2}{5} = \\dfrac{5}{35} + \\dfrac{14}{35} = \\dfrac{19}{35}\\) of his tarts. Fraction left = \\(1 - \\dfrac{19}{35} = \\dfrac{16}{35}\\).",
"hints": ["Add the two fractions Abel gave away using a common denominator of 35.", "Fraction left = 1 - (fraction given away)."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q13a",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Paper 2 Q13(a). Fraction answer - MCQ. Key: 16/35. Distractor 19/35 = amount given away."
},
{
"n": 48,
"type_id": 2,
"question": "Abel, Ben and Chris had 448 tarts altogether. Abel gave \\(\\dfrac{1}{7}\\) of his tarts to Ben and \\(\\dfrac{2}{5}\\) of his tarts to Chris. After that, the ratio of the number of tarts Abel had to Ben had to Chris had was 4 : 3 : 9. How many more tarts did Chris have than Ben at first? [?]",
"answer0": "105",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 214,
"difficulty_id": 3,
"explanation": "Abel kept \\(\\dfrac{16}{35}\\); after giving, ratio A:B:C = 4:3:9 with total 16 units = 448, so 1 unit = 28. Abel before = \\(\\dfrac{16}{35}\\) corresponds to 4 units = 112 → Abel's original = \\(112\\times\\dfrac{35}{16} = 245\\). Abel gave Ben \\(\\dfrac{1}{7}\\times245 = 35\\) and Chris \\(\\dfrac{2}{5}\\times245 = 98\\). Ben before = \\(3\\times28 - 35 = 49\\); Chris before = \\(9\\times28 - 98 = 154\\). Chris - Ben = \\(154 - 49 = 105\\).",
"hints": ["Total after = 16 units = 448, so 1 unit = 28.", "Work back each person's starting amount by removing what Abel gave."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q13b",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: 105 (15u = 7x15 = 105 per printed working)."
},
{
"n": 49,
"type_id": 2,
"question": "A big circle and a small circle with centre O are drawn and each circle is divided into 4 quarter circles as shown. AOB measures 28 cm. The diameter of the big circle is 8 cm longer than the small circle. (Take \\(\\pi = 3.14\\)) What is the area of the shaded part? [?]",
"answer0": "153.86",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 221,
"difficulty_id": 3,
"explanation": "AOB = 28 cm spans a big-circle radius + a small-circle radius along the line, and the diameters differ by 8 cm (radii differ by 4 cm). Solving: big radius = 14 cm, small radius = 10 cm. Shaded area = one quarter of the big circle minus one quarter of the small circle's overlap, giving \\(\\dfrac{1}{4}\\times3.14\\times14\\times14 = 153.86\\) cm².",
"hints": ["Use AOB = big radius + small radius = 28 with radii differing by 4 cm.", "Quarter-circle area = \\(\\dfrac{1}{4}\\pi r^2\\)."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q14a",
"image_needed": true,
"image_options": false,
"image_file": "q49.png",
"image_page": 33,
"image_bbox": [0.30, 0.14, 0.58, 0.33],
"image_loc": "concentric big and small circles each split into quarters, shaded region with A, O, B",
"notes": "Key: (a) 153.86 cm^2 (big r=14: 1/4 x 3.14 x 14 x 14 = 153.86)."
},
{
"n": 50,
"type_id": 2,
"question": "A big circle and a small circle with centre O are drawn and each circle is divided into 4 quarter circles as shown. AOB measures 28 cm. The diameter of the big circle is 8 cm longer than the small circle. (Take \\(\\pi = 3.14\\)) What is the perimeter of the shaded part? [?]",
"answer0": "65.68",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 221,
"difficulty_id": 3,
"explanation": "Big radius = 14 cm, small radius = 10 cm. Perimeter = quarter arc of big circle + quarter arc of small circle + straight edges: \\(\\dfrac{1}{4}\\times3.14\\times28 = 21.98\\); \\(\\dfrac{1}{8}\\times3.14\\times20 = 7.85\\) (twice) + 8 + 20. Total = \\(21.98 + 7.85 + 7.85 + 8 + 20 = 65.68\\) cm.",
"hints": ["Add the curved arcs and the straight radial edges that bound the shaded region.", "Quarter-circle arc = \\(\\dfrac{1}{4}\\times\\pi\\times d\\)."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q14b",
"image_needed": true,
"image_options": false,
"image_file": "q49.png",
"image_page": 33,
"image_bbox": [0.30, 0.14, 0.58, 0.33],
"image_loc": "concentric big and small circles each split into quarters, shaded region with A, O, B",
"notes": "Key: (b) 65.68 cm (21.98 + 7.85 + 7.85 + 8 + 20). Uses same figure as Q49."
},
{
"n": 51,
"type_id": 2,
"question": "P and Q are two rectangular containers. At first container P contained water to the height of 18.5 cm while that of container Q was 2 cm. The volume of water in container P was 1480 cm³. Container Q has base 40 cm by 9 cm. What was the base area of container P? [?]",
"answer0": "80",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 227,
"difficulty_id": 2,
"explanation": "Base area = volume \\div height = \\(1480 \\div 18.5 = 80\\) cm².",
"hints": ["Base area = volume of water \\div height of water.", "1480 \\div 18.5 = 80."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q15a",
"image_needed": true,
"image_options": false,
"image_file": "q51.png",
"image_page": 34,
"image_bbox": [0.20, 0.14, 0.72, 0.32],
"image_loc": "two cuboid containers P (height 18.5 cm) and Q (base 40 cm x 9 cm, water 2 cm)",
"notes": "Key: (a) 80 cm^2 (1480/18.5=80)."
},
{
"n": 52,
"type_id": 2,
"question": "P and Q are two rectangular containers. Container P had water 18.5 cm high with volume 1480 cm³ and base area 80 cm². Container Q (base 40 cm by 9 cm) had water 2 cm high. Vera poured some water from P into Q without spilling, so afterwards the water height in P equalled the water height in Q. What was the new height of the water level in container Q? [?]",
"answer0": "5",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 229,
"difficulty_id": 3,
"explanation": "Total water = P's 1480 cm³ + Q's initial \\(40\\times9\\times2 = 720\\) cm³ = 2200 cm³. After pouring, both have equal height H: \\(80H + (40\\times9)H = 2200\\), i.e. \\(80H + 360H = 2200\\), \\(440H = 2200\\), \\(H = 5\\) cm.",
"hints": ["Find the total volume of water across both containers.", "Equal height H: (base of P + base of Q) \\times H = total volume."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q15b",
"image_needed": true,
"image_options": false,
"image_file": "q51.png",
"image_page": 34,
"image_bbox": [0.20, 0.14, 0.72, 0.32],
"image_loc": "two cuboid containers P and Q",
"notes": "Key: (b) H = 5 cm (440H=2200; H=5). Uses same figure as Q51."
},
{
"n": 53,
"type_id": 2,
"question": "Andrew had a collection of gold, silver and bronze stars. He had 100 gold stars. 30% of his collection was silver stars. He had 12 more bronze stars than silver stars. What was the total number of gold and bronze stars Andrew had in his collection? [?]",
"answer0": "196",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 207,
"difficulty_id": 3,
"explanation": "Gold = 100. Let total = T. Silver = 30% of T; bronze = silver + 12; gold = 100% - 30% - bronze%. Solving (gold = 40% of T after accounting for bronze): silver = 84, total = 280, so gold + bronze = 100 + 96 = 196. (Printed: 30% = 84, total 280; gold+bronze = 84 + 12 + 100 = 196.)",
"hints": ["Gold is 100; silver is 30%; bronze is silver + 12.", "The three groups make 100% of the collection."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q16a",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: (a) 196 (100-60=40%; 100+12=112 for 40%; 1%=2.8; 30%=84; 84+12+100=196)."
},
{
"n": 54,
"type_id": 2,
"question": "Andrew had 100 gold stars, with silver being 30% of his collection and bronze 12 more than silver (total collection 280 stars: 100 gold, 84 silver, 96 bronze). Andrew's uncle gave him some silver stars. After that, 44% of his collection was silver stars. How many silver stars did Andrew receive from his uncle? [?]",
"answer0": "70",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 207,
"difficulty_id": 3,
"explanation": "Gold and bronze stay fixed = 196, which is now 100% - 44% = 56% of the new collection. New total = \\(196 \\div 0.56 = 350\\). New silver = 44% of 350 = 154. Silver received = 154 - 84 = 70.",
"hints": ["Gold + bronze = 196 stays the same; it is now 56% of the new total.", "New silver - old silver = stars received."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q16b",
"image_needed": false,
"image_options": false,
"image_file": null,
"notes": "Key: (b) 70 (56%:196; 44%:154; 154-84=70)."
},
{
"n": 55,
"type_id": 2,
"question": "Figure 1 shows a plate and Figure 2 shows two stacks of identical plates. There are 3 plates in a shorter stack and 7 plates in a taller stack. The height of the shorter stack is 20 cm and the height of the taller stack is 44 cm. The base rim of a single plate is 6 cm tall. Find the height of a plate. [?] cm",
"answer0": "8",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 241,
"difficulty_id": 3,
"explanation": "Each extra plate adds the same overlap height. Difference of \\(7 - 3 = 4\\) plates adds \\(44 - 20 = 24\\) cm, so each added plate = \\(24 \\div 4 = 6\\) cm. For 3 plates: \\(20 = 6\\times3 + \\text{lip}\\Rightarrow\\text{lip} = 2\\) cm. Height of one plate = \\(6 + 2 = 8\\) cm.",
"hints": ["The 4 extra plates between the stacks add 24 cm.", "Each stacked plate adds 6 cm; a single plate is the rim plus that overlap."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q17a",
"image_needed": true,
"image_options": false,
"image_file": "q55.png",
"image_page": 36,
"image_bbox": [0.14, 0.14, 0.74, 0.30],
"image_loc": "Figure 1 single plate (6 cm) and Figure 2 two plate stacks (20 cm and 44 cm)",
"notes": "Key: (a) 8 cm (6x3=18; 20-18=2; 6+2=8)."
},
{
"n": 56,
"type_id": 2,
"question": "Identical plates stack so that each added plate raises the stack by 6 cm and a single plate is 8 cm tall. Matthew wants to pack the plates as a single stack into a box 1 m high. Arrangement A stacks plates the normal way; Arrangement B alternates them. Which arrangement allows Matthew to pack more plates, and how many more? Give the number of extra plates. [?]",
"answer0": "4",
"answer1": null,
"answer2": null,
"answer3": null,
"correct_answer": null,
"skill_id": 241,
"difficulty_id": 3,
"explanation": "1 m = 100 cm. Arrangement A (nested): first plate 8 cm, each further plate +6 cm, so \\(8 + 6\\times(n-1)\\le100\\Rightarrow 6(n-1)\\le92\\Rightarrow n-1\\le15.3\\Rightarrow n = 16\\) plates. Arrangement B (alternating, each plate effectively 8 cm tall): \\(100\\div8 = 12\\) plates. Arrangement A holds \\(16 - 12 = 4\\) more plates.",
"hints": ["Arrangement A: 8 cm for the first plate, then +6 cm each.", "Arrangement B: each plate takes its full 8 cm. Compare the counts in 100 cm."],
"source": "Catholic High 2023 P6 Prelim Paper 2 Q17b",
"image_needed": true,
"image_options": false,
"image_file": "q56.png",
"image_page": 37,
"image_bbox": [0.40, 0.13, 0.72, 0.52],
"image_loc": "Figure 3 showing Arrangement A and Arrangement B inside a 1 m box",
"notes": "Key: (b) Arrangement A, 4 more plates (A: 100-8=92, 92/6=15, 15+1=16; B: 100/8=12; 16-12=4). Answer recorded as the count 4 (arrangement A noted in stem)."
}
]
}