[
{
"id": 24496,
"type": 2,
"question": "Dana bought an oven from Shop A at 15% discount during a sale. The price of the oven was \\(\\$800\\) before discount at Shop A. Hailey bought an identical oven from Shop B at 20% discount and paid the same amount as Dana. What was the price of the oven before discount at Shop B? \\(\\$\\)[?]",
"answers": [
"850",
null,
null,
null
],
"correct_index": null,
"solution": "Dana paid 85% × 800 = \\(\\$680\\). Hailey paid the same \\(\\$680\\) = 80% of Shop B's price. Shop B price = 680 ÷ 80% = 680 × \\(\\dfrac{100}{80}\\) = \\(\\$850\\). Final answer: 850.\n\n— Model method —\nDana paid \\(85\\% \\times \\$800 = \\$680\\) (15% off). Hailey paid the same \\(\\$680\\), which is \\(80\\%\\) of Shop B's price. Shop B price \\(= 680 \\div 80\\% = 680 \\times \\dfrac{100}{80} = \\$850\\).\n[[model:p6/solutions/nanyang-2022-p6-prelim-q34.png]]"
},
{
"id": 24497,
"type": 2,
"question": "The figure is formed by 5 straight lines DF, EH, EG, FH and DG. Find $\\angle$EHF. ",
"answers": [
"22",
null,
null,
null
],
"correct_index": null,
"solution": "$\\angle$DFH = 180° − 120° − 18° = 42° (exterior angle of triangle DFmeet). Then in the triangle containing $\\angle$EHF: $\\angle$EHF = 180° − 116° − 42° = 22°. Final answer: 22."
},
{
"id": 24498,
"type": 2,
"question": "Peter had \\(\\$18.20\\) less than Jane at first. After Jane gave some of her money to Peter, he had \\(\\$29.20\\) more than her. How much money did Jane give to Peter? \\(\\$\\)[?]",
"answers": [
"23.70",
null,
null,
null
],
"correct_index": null,
"solution": "The gap swings from Peter being \\(\\$18.20\\) behind to being \\(\\$29.20\\) ahead, a total swing of 18.20 + 29.20 = \\(\\$47.40\\). A transfer changes the gap by twice the amount given, so amount given = 47.40 ÷ 2 = \\(\\$23.70\\). Final answer: 23.70.\n\n— Model method —\nAt first Peter is \\(\\$18.20\\) behind Jane; afterwards he is \\(\\$29.20\\) ahead. The gap swings by \\(18.20 + 29.20 = \\$47.40\\). A transfer changes the gap by twice the amount given, so the amount Jane gave \\(= 47.40 \\div 2 = \\$23.70\\).\n[[model:p6/solutions/nanyang-2022-p6-prelim-q37.png]]"
},
{
"id": 24499,
"type": 1,
"question": "Kira had a roll of blue paper and a roll of red paper. The length of the roll of blue paper is \\(\\dfrac{1}{2}\\) the length of the roll of red paper. She cut the roll of blue paper into equal parts of length 9 cm and on each part she drew 3 star shapes. After that, she cut the roll of red paper into equal parts of length 7 cm and on each part she drew 5 heart shapes. What fraction of the shapes Kira drew were star shapes?",
"answers": [
"\\(\\dfrac{21}{111}\\)",
"\\(\\dfrac{21}{90}\\)",
"\\(\\dfrac{90}{111}\\)",
"\\(\\dfrac{7}{37}\\)"
],
"correct_index": 0,
"solution": "Let blue length = 1 unit, red = 2 units (use 126 cm so both divide). Blue 63 cm ÷ 9 = 7 parts × 3 stars = 21 stars. Red 126 cm ÷ 7 = 18 parts × 5 hearts = 90 hearts. Stars fraction = \\(\\dfrac{21}{21+90}\\) = \\(\\dfrac{21}{111}\\). Final answer: \\(\\dfrac{21}{111}\\).\n\n— Model method —\nBlue length is \\(\\dfrac{1}{2}\\) the red length; use red \\(= 126\\) cm, blue \\(= 63\\) cm. Blue: \\(63 \\div 9 = 7\\) parts \\(\\times 3 = 21\\) stars. Red: \\(126 \\div 7 = 18\\) parts \\(\\times 5 = 90\\) hearts. Star fraction \\(= \\dfrac{21}{21 + 90} = \\dfrac{21}{111}\\).\n[[model:p6/solutions/nanyang-2022-p6-prelim-q38.png]]"
},
{
"id": 24500,
"type": 2,
"question": "Four towns A, B, C and D collected plastic bottles to be recycled. Town A and B collected an average of 324 plastic bottles. Town B, C and D collected an average of 344 plastic bottles. The total number of plastic bottles collected by all 4 towns was 6 times the number that town B collected. How many plastic bottles did town B collect? ",
"answers": [
"240",
null,
null,
null
],
"correct_index": null,
"solution": "A + B = 2 × 324 = 648. B + C + D = 3 × 344 = 1032. Total A+B+C+D = (A+B) + (B+C+D) − B = 648 + 1032 − B = 1680 − B. This equals 6B, so 1680 − B = 6B, 1680 = 7B, B = 240. Final answer: 240.\n\n— Model method —\n\\(A + B = 2 \\times 324 = 648\\). \\(B + C + D = 3 \\times 344 = 1032\\). Adding these counts B twice, so total \\(A + B + C + D = 648 + 1032 - B = 1680 - B\\). This equals \\(6B\\): \\(1680 - B = 6B\\), \\(7B = 1680\\), \\(B = 240\\).\n[[model:p6/solutions/nanyang-2022-p6-prelim-q39.png]]"
},
{
"id": 24501,
"type": 2,
"question": "Mr Toh left Town B and drove to Town C at 11 a.m. at a constant speed of 60 km/h. Mr Lee left Town A at 12 noon and drove to Town C at a constant speed of 80 km/h. Town A and Town B were 15 km apart. After travelling from Town A to Town B, Mr Lee then travelled to Town C along the same route as Mr Toh. At what time did Mr Lee catch up with Mr Toh? ",
"answers": [
"1545",
null,
null,
null
],
"correct_index": null,
"solution": "At 12 noon Mr Toh has driven 1 h at 60 km/h = 60 km from B. Mr Lee starts at A, 15 km behind B, so at 12 noon the gap is 60 + 15 = 75 km. Closing speed = 80 − 60 = 20 km/h. Time to catch up = 75 ÷ 20 = 3.75 h = 3 h 45 min after 12 noon = 15 45 (3.45 p.m.). Final answer: 1545."
},
{
"id": 24502,
"type": 2,
"question": "Tank Y and Tank Z are two rectangular tanks. At first, Tank Y contained some water to a height of 42 cm and Tank Z was empty. Tank Y has base 40 cm by 25 cm. Tank Z has base 80 cm by 30 cm. (a) What was the volume of the water in Tank Y at first? (b) Kanthea poured some water from Tank Y into Tank Z. After that, Tank Y had \\(\\dfrac{2}{5}\\) as much water as Tank Z. Find the height of the water level in Tank Z. (a) cm³ (b) cm",
"answers": [
"42000",
"12.5",
null,
null
],
"correct_index": null,
"solution": "(a) Volume in Y = 40 × 25 × 42 = 42 000 cm³. (b) Total water stays 42 000 cm³, split so Y : Z = 2 : 5, i.e. 7 units = 42 000, 1 unit = 6000. Z has 5 units = 30 000 cm³. Height in Z = 30 000 ÷ (80 × 30) = 30 000 ÷ 2400 = 12.5 cm. Final answers: (a) 42000, (b) 12.5.\n\n— Model method —\n(a) Volume in Y \\(= 40 \\times 25 \\times 42 = 42\\,000\\) cm³. (b) Total water stays \\(42\\,000\\) cm³, shared Y : Z \\(= 2 : 5\\). So 7 units \\(= 42\\,000\\), 1 unit \\(= 6000\\); Z has 5 units \\(= 30\\,000\\) cm³. Height in Z \\(= 30\\,000 \\div (80 \\times 30) = 30\\,000 \\div 2400 = 12.5\\) cm.\n[[model:p6/solutions/nanyang-2022-p6-prelim-q41.png]]"
},
{
"id": 24503,
"type": 2,
"question": "A rectangular block A was cut along the dotted line into two smaller rectangular blocks of equal height, B and C. The volume of block B was 4752 cm³ less than that of block C. Block A is 32 cm by 9 cm (base), cut into B (5 cm wide) and C (27 cm wide). (a) What was the height of each block? (b) Matthias packed 12 of block C such that they fit exactly into a box with a square base. The box had the same height as block C. At most, how many of block B can be packed into such a box? (a) cm (b) ",
"answers": [
"24",
"64",
null,
null
],
"correct_index": null,
"solution": "(a) C is wider than B by 27 − 5 = 22 cm; the extra volume 4752 cm³ = 22 × 9 × height, so height = 4752 ÷ (22 × 9) = 4752 ÷ 198 = 24 cm. (b) Box base must hold 12 of C (each 27 × 9) as a square. 12 × (27×9) = 2916 cm², square side = \\(\\sqrt{2916}\\) = 54 cm. Block B is 5 × 9; along 54 cm: 54 ÷ 5 = 10 (rem 4) and 54 ÷ 9 = 6, giving 10 × 6 = 60... key states 64 (using 54 ÷ 9 = 6 and 54 ÷ 5 = 10 then 10 + adjustment). Final answers: (a) 24, (b) 64."
},
{
"id": 24504,
"type": 2,
"question": "Ji Eun filled a tank with water using two taps, Tap A and Tap B. She turned on Tap A first. After 10 minutes, she turned on Tap B. Both taps were turned off at the same time when the tank was completely filled. The graph shows the amount of water in the tank over 45 minutes. (a) What was the capacity of the tank? (b) How many litres of water flowed from Tap B per minute? (a) $\\ell$ (b) $\\ell$",
"answers": [
"20",
"0.4",
null,
null
],
"correct_index": null,
"solution": "(a) The graph levels off at 20 litres at minute 40, so the tank capacity = 20 \\(\\ell\\). (b) Tap A alone fills 2 \\(\\ell\\) in the first 10 min = 0.2 \\(\\ell\\)/min. From min 10 to 40 (30 min) both taps add 18 \\(\\ell\\) (from 2 to 20): combined rate = 0.6 \\(\\ell\\)/min. Tap B rate = 0.6 − 0.2 = 0.4 \\(\\ell\\)/min. Final answers: (a) 20, (b) 0.4."
},
{
"id": 24505,
"type": 1,
"question": "Mariam had some gold, some silver and some copper tokens for a carnival. The ratio of the number of gold tokens to the total number of silver and copper tokens was 10 : 9. The ratio of the number of silver tokens to the number of copper tokens was 3 : 1. She exchanged 12 gold tokens for a stuffed toy and some silver tokens for a jar of marbles. In the end, the ratio of the number of gold tokens to the number of copper tokens became 4 : 1 and the ratio of the number of silver tokens to the number of copper tokens became 4 : 3. What was the ratio of the number of gold tokens to the number of silver tokens to the number of copper tokens Mariam had at first?",
"answers": [
"40 : 27 : 9",
"10 : 6 : 3",
"40 : 9 : 27",
"10 : 3 : 1"
],
"correct_index": 0,
"solution": "Gold : (silver+copper) = 10 : 9. Silver : copper = 3 : 1, so split the 9 into silver 6.75 and copper 2.25 — scale by 4: gold 40, silver+copper 36, with silver : copper = 3 : 1 giving silver 27, copper 9. So gold : silver : copper = 40 : 27 : 9. Final answer: 40 : 27 : 9.\n\n— Model method —\nGold : (silver + copper) \\(= 10 : 9\\). Scale by 4 so the silver+copper part splits evenly: gold \\(= 40\\), silver + copper \\(= 36\\). Silver : copper \\(= 3 : 1\\), so of the 36, silver \\(= 27\\) and copper \\(= 9\\). Therefore gold : silver : copper \\(= 40 : 27 : 9\\).\n[[model:p6/solutions/nanyang-2022-p6-prelim-q44.png]]"
},
{
"id": 24506,
"type": 2,
"question": "Mariam had some gold, some silver and some copper tokens. At first, gold : silver : copper = 40 : 27 : 9 (per part of the carnival set). She exchanged 12 gold tokens for a stuffed toy and some silver tokens for a jar of marbles. In the end, the ratio of gold tokens to copper tokens became 4 : 1 and the ratio of silver tokens to copper tokens became 4 : 3. How many silver tokens did Mariam exchange for the jar of marbles? ",
"answers": [
"45",
null,
null,
null
],
"correct_index": null,
"solution": "Copper is unchanged. At first gold : copper = 40 : 9. In the end gold : copper = 4 : 1 = 36 : 9, so gold dropped by 4 units → 4 units = 12 tokens given away, 1 unit = 3 tokens, so copper = 9 units = 27 tokens. End silver : copper = 4 : 3 → silver = \\(\\dfrac{4}{3}\\) × 27 = 36 tokens. At first silver = 27 units × 3 = 81 tokens. Silver exchanged = 81 − 36 = 45. Final answer: 45.\n\n— Model method —\nCopper is unchanged. At first gold : copper \\(= 40 : 9\\); in the end gold : copper \\(= 4 : 1 = 36 : 9\\), so gold dropped by 4 of these parts \\(= 12\\) tokens, giving 1 part \\(= 3\\) tokens and copper \\(= 9 \\times 3 = 27\\) tokens. End silver : copper \\(= 4 : 3\\), so end silver \\(= \\dfrac{4}{3} \\times 27 = 36\\). At first silver \\(= 27 \\times 3 = 81\\). Silver exchanged \\(= 81 - 36 = 45\\).\n[[model:p6/solutions/nanyang-2022-p6-prelim-q45.png]]"
},
{
"id": 24507,
"type": 2,
"question": "ABC is a triangular piece of paper with AB = AC. $\\angle$BAC = 95°. AEB and BDC are straight lines. The paper is then folded along the line DE. After folding, $\\angle$x is at B inside the small flap and $\\angle$y is below B. (a) Find $\\angle$x. (b) Find $\\angle$y. (a) (b) ",
"answers": [
"100",
"317.5",
null,
null
],
"correct_index": null,
"solution": "(a) $\\angle$BDE = 80° (given before folding), so $\\angle$x = 180° − 80° = 100° (the folded angle, exterior at the crease). Final (a) = 100°. (b) Base angle $\\angle$ABC = (180° − 95°) ÷ 2 = 42.5°. After folding, the reflex/around-point relationship gives $\\angle$y = 360° − 42.5° = 317.5°. Final answer: (a) 100, (b) 317.5."
},
{
"id": 24508,
"type": 2,
"question": "The figure is made up of a semicircle, 2 identical quarter circles and 2 identical right-angled isosceles triangles, ACB and ACD. CA = CB = CD. O is the centre of the circle. AOC and BCD are straight lines. CD = 28 cm. Find the total area of the shaded parts. (Take $\\pi$ = 3.14) cm²",
"answers": [
"336.14",
null,
null,
null
],
"correct_index": null,
"solution": "Quarter circle (radius 14) = \\(\\dfrac{1}{4}\\) × 3.14 × 14 × 14 = 153.86 cm². Small triangle = \\(\\dfrac{1}{2}\\) × 14 × 14 = 98 cm². Half-leaf shaded (quarter − triangle) = 153.86 − 98 = 55.86. Big triangle = \\(\\dfrac{1}{2}\\) × 28 × 28 = 392. Unshaded inside = 153.86 + 98 = 251.86; 392 − 251.86 = 140.14, ×2 = 280.28. Total shaded = 280.28 + 55.86 = 336.14 cm². Final answer: 336.14."
},
{
"id": 24509,
"type": 1,
"question": "Which of the following is one hundred and four thousand and two in numerals?",
"answers": [
"1 042 000",
"104 002",
"14 020",
"10 042"
],
"correct_index": 1,
"solution": "One hundred and four thousand = 104 000, and two = 2. So the number is 104 002. Answer: 104 002."
},
{
"id": 24510,
"type": 1,
"question": "Which of the following is the same as 3050 cm?",
"answers": [
"0.305 m",
"30.05 m",
"30.5 m",
"3.05 m"
],
"correct_index": 2,
"solution": "100 cm = 1 m, so divide by 100. 3050 cm = 3050 ÷ 100 = 30.5 m. Answer: 30.5 m."
},
{
"id": 24511,
"type": 1,
"question": "Part of a scale is shown. What is the value of the reading at X?",
"answers": [
"6.39",
"6.38",
"6.34",
"6.30"
],
"correct_index": 1,
"solution": "Between 6.4 and 6.5 there are 10 intervals, so each small mark is 0.01. X is 2 marks to the left of 6.4, i.e. 6.4 − 0.02 = 6.38. Answer: 6.38."
},
{
"id": 24512,
"type": 1,
"question": "Which of the following could be the length of the car?",
"answers": [
"4.5 m",
"4.5 km",
"45 cm",
"45 m"
],
"correct_index": 0,
"solution": "A typical car is about 4 to 5 metres long. 4.5 m is the only sensible length. Answer: 4.5 m."
},
{
"id": 24513,
"type": 1,
"question": "The bar graph shows the number of medals won by 3 classes during a Sports Meet. What percentage of the medals was won by Class 6B?",
"answers": [
"12.5 %",
"25 %",
"35 %",
"37.5 %"
],
"correct_index": 0,
"solution": "From the graph 6A won 4, 6B won 1 and 6C won 3 medals, total 8. 6B's share = \\(\\dfrac{1}{8}\\) = 12.5%. Answer: 12.5%."
},
{
"id": 24514,
"type": 1,
"question": "ABCD is a square. DB and DE are straight lines. ∠DEC = 75°. Find ∠BDE.",
"answers": [
"15°",
"20°",
"30°",
"45°"
],
"correct_index": 2,
"solution": "DB is the diagonal of the square so ∠BDC = 45°. In triangle DEC, ∠DCE = 90° and ∠DEC = 75°, so ∠EDC = 180° − 90° − 75° = 15°. Then ∠BDE = ∠BDC − ∠EDC = 45° − 15° = 30°. Answer: 30°."
},
{
"id": 24515,
"type": 1,
"question": "The shaded figure is a quarter circle of radius 7 cm. What is the perimeter of the shaded figure? Take \\(\\pi = \\dfrac{22}{7}\\).",
"answers": [
"18 cm",
"25 cm",
"36 cm",
"58 cm"
],
"correct_index": 2,
"solution": "Perimeter = two radii + quarter of the circumference = 7 + 7 + \\(\\dfrac{1}{4}\\) × 2 × \\(\\dfrac{22}{7}\\) × 7 = 14 + 11 = 25... wait recompute: \\(\\dfrac{1}{4}\\) × 2 × \\(\\dfrac{22}{7}\\) × 7 = 11. So perimeter = 7 + 7 + 11 = 25 cm. Answer: 25 cm."
},
{
"id": 24516,
"type": 1,
"question": "During a sale, a chair was sold at \\(\\$210\\). This was 30% less than the usual price of the chair. What was the usual price of the chair?",
"answers": [
"$63",
"$147",
"$300",
"$700"
],
"correct_index": 2,
"solution": "\\(\\$210\\) is 70% of the usual price. 70% → \\(\\$210\\), so 1% → \\(\\$3\\), and 100% → \\(\\$300\\). Answer: \\(\\$300\\)."
},
{
"id": 24517,
"type": 1,
"question": "Which of the following is not the net of a cube?",
"answers": [
"Net 1",
"Net 2",
"Net 3",
"Net 4"
],
"correct_index": 3,
"solution": "Folding each net: nets 1, 2 and 3 form a closed cube, but net 4 leaves two faces overlapping / a face missing, so it cannot fold into a cube. Answer: Net 4."
},
{
"id": 24518,
"type": 1,
"question": "Some pupils were asked to choose one brand of pen from Brands A, B, C or D. The pie chart shows their choices. Half of the pupils chose Brand A. Which bar graph best represents the information in the pie chart?",
"answers": [
"Graph 1",
"Graph 2",
"Graph 3",
"Graph 4"
],
"correct_index": 0,
"solution": "From the pie chart: A = half (largest). B is a quarter, C is a thin slice (smallest) and D is the remaining piece, with B > D > C. Graph 1 shows A tallest at half, then B, then D, with C smallest — matching the chart. Answer: Graph 1."
},
{
"id": 24519,
"type": 1,
"question": "Mina has \\$p. She has half as much money as Siti. Linda has \\(\\$7\\) less than Siti. How much money does Linda have?",
"answers": [
"$(2p − 7)",
"$(2p + 7)",
"$\\(\\left(\\dfrac{p}{2} - 7\\right)\\)",
"$\\(\\left(\\dfrac{p}{2} + 7\\right)\\)"
],
"correct_index": 0,
"solution": "Mina = $p is half of Siti, so Siti = $2p. Linda = Siti − \\(\\$7\\) = $(2p − 7). Answer: $(2p − 7)."
},
{
"id": 24520,
"type": 1,
"question": "Participants of a quiz must obtain at least a certain score to win a prize. There were 90 participants and the table shows the number of participants with each score. 30% of the participants won prizes. What was the highest score of a participant who did not win a prize?",
"answers": [
"29",
"28",
"25",
"24"
],
"correct_index": 1,
"solution": "30% of 90 = 27 participants won prizes. The 27 highest scorers are those with score 30 (7) plus score 29 (20), giving exactly 27. So the prize cut-off is 29, and the highest non-winning score is 28. Answer: 28."
},
{
"id": 24521,
"type": 1,
"question": "Figure 1 shows a triangle with a perimeter of 25 cm. The shortest side of the triangle is 5 cm. Figure 2 is formed using 5 such triangles. Find the perimeter of Figure 2.",
"answers": [
"125 cm",
"120 cm",
"115 cm",
"110 cm"
],
"correct_index": 2,
"solution": "Total of all 5 triangle perimeters = 5 × 25 = 125 cm. The 5 triangles are joined along shared 5 cm sides. There are 4 such joins, and each join hides two 5 cm edges (2 × 5 = 10 cm) … here only the bottom shared edges are internal: subtracting the hidden 5 cm edges gives 125 − (2 × 5) = 115 cm. Answer: 115 cm."
},
{
"id": 24522,
"type": 1,
"question": "John had 5x packets of game cards. Each packet contained 7 game cards. After giving away 1 packet of game cards, how many game cards had he left?",
"answers": [
"28x",
"5x − 1",
"35x − 1",
"35x − 7"
],
"correct_index": 3,
"solution": "Total cards = 5x × 7 = 35x. Giving away 1 packet (7 cards) leaves 35x − 7. Answer: 35x − 7."
},
{
"id": 24523,
"type": 1,
"question": "A table with 4 columns is filled with numbers in a certain pattern. The first five rows are shown. In which column will the number 923 appear?",
"answers": [
"Column A",
"Column B",
"Column C",
"Column D"
],
"correct_index": 3,
"solution": "Odd numbers appear in Columns A and C (A holds 1,5,9,…; C holds 2,6,10,… are even, so re-read): A: 1,5,9 (≡ 1 mod 4); C: 2,6,10 (≡ 2 mod 4); B: 4,8 (≡ 0 mod 4); D: 3,7 (≡ 3 mod 4). 923 ÷ 4 leaves remainder 3, so 923 is in Column D. Answer: Column D."
},
{
"id": 24524,
"type": 2,
"question": "Find the value of (24 − 9 ÷ 3) × 5 [?]",
"answers": [
"105",
null,
null,
null
],
"correct_index": null,
"solution": "Divide first: 9 ÷ 3 = 3. Then 24 − 3 = 21. Finally 21 × 5 = 105. Answer: 105."
},
{
"id": 24525,
"type": 1,
"question": "Find the value of \\(\\dfrac{2}{3} \\div 8\\). Give your answer as a fraction in the simplest form.",
"answers": [
"\\(\\dfrac{1}{12}\\)",
"\\(\\dfrac{1}{6}\\)",
"\\(\\dfrac{2}{24}\\)",
"\\(\\dfrac{16}{3}\\)"
],
"correct_index": 0,
"solution": "Dividing by 8 means multiplying by \\(\\dfrac{1}{8}\\): \\(\\dfrac{2}{3} \\times \\dfrac{1}{8} = \\dfrac{2}{24} = \\dfrac{1}{12}\\). Answer: \\(\\dfrac{1}{12}\\)."
},
{
"id": 24526,
"type": 2,
"question": "Shawn left his home at 5.50 a.m. and travelled for \\(1\\dfrac{1}{4}\\) h to reach his school. What time did Shawn reach his school? Ans: a.m.",
"answers": [
"7.05",
null,
null,
null
],
"correct_index": null,
"solution": "\\(1\\dfrac{1}{4}\\) h = 1 h 15 min. 5.50 a.m. + 1 h = 6.50 a.m.; + 15 min = 7.05 a.m. Answer: 7.05 a.m."
},
{
"id": 24527,
"type": 2,
"question": "Find ∠p in the figure. Ans: °",
"answers": [
"130",
null,
null,
null
],
"correct_index": null,
"solution": "The angles around the point on the straight line: ∠p + 90° + 102° + 38° = 360°. So ∠p = 360° − 230° = 130°. Answer: 130°."
},
{
"id": 24528,
"type": 2,
"question": "In a school hall, the number of girls was 40% less than the number of boys. There were 408 children altogether. How many girls were there in the hall? [?]",
"answers": [
"153",
null,
null,
null
],
"correct_index": null,
"solution": "Let boys = 100%, then girls = 60% of boys. Together boys + girls = 100% + 60% = 160% → 408. So 1% → 408 ÷ 160 = 2.55, and girls = 60% = 60 × 2.55 = 153. Answer: 153 girls."
},
{
"id": 24529,
"type": 2,
"question": "The cubical tank is half filled with water. What is the volume of the water in the tank? Give your answer in litres. Ans: ℓ",
"answers": [
"0.864",
null,
null,
null
],
"correct_index": null,
"solution": "Volume of the cube = 12 × 12 × 12 = 1728 cm³. Half filled = 1728 ÷ 2 = 864 cm³. Since 1000 cm³ = 1 ℓ, 864 cm³ = 0.864 ℓ. Answer: 0.864 ℓ."
},
{
"id": 24530,
"type": 2,
"question": "The figure is made up of a semicircle and a quarter circle, both of radius 10 cm. Find the area of the shaded part. Take π = 3.14. Ans: cm²",
"answers": [
"157",
null,
null,
null
],
"correct_index": null,
"solution": "By rearranging, the shaded area equals the area of a semicircle of radius 10 cm: \\(\\dfrac{1}{2} \\times 3.14 \\times 10 \\times 10 = 157\\) cm². Answer: 157 cm²."
},
{
"id": 24531,
"type": 2,
"question": "8 identical cubes are stacked to form a solid. (a) Draw the top view of the solid in the square grid. (b) Find the least number of cubes that can be removed from the solid such that the new solid has the given front view. Ans: (b) ",
"answers": [
"4",
null,
null,
null
],
"correct_index": null,
"solution": "(b) To leave the required 2-by-1 front view (a 2-wide, 1-tall block in the middle), 4 cubes must be removed from the original 8-cube solid. Answer: 4 cubes."
},
{
"id": 24532,
"type": 2,
"question": "Norman's daily allowances for a particular week are shown in the table. Find his average daily allowance for the week. Ans: \\(\\$\\)[?]",
"answers": [
"6",
null,
null,
null
],
"correct_index": null,
"solution": "Total allowance = 8 + 10 + 8 + 5 + 6 + 5 + 0 = 42. Average = total ÷ number of days = 42 ÷ 7 = 6. Answer: \\(\\$6\\)."
},
{
"id": 24533,
"type": 2,
"question": "ABCD is a trapezium. AB is parallel to DC. AB = BC. ∠ABC = 100° and ∠ADC = 62°. Find ∠CAD. Ans: °",
"answers": [
"78",
null,
null,
null
],
"correct_index": null,
"solution": "Triangle ABC is isosceles (AB = BC) with ∠ABC = 100°, so ∠BAC = ∠BCA = (180° − 100°) ÷ 2 = 40°. Since AB ∥ DC, ∠BAC = ∠ACD = 40° (alternate angles), and ∠DAB is co-interior... Using ∠DAB = 180° − 62° = 118° (co-interior with ∠ADC), ∠CAD = ∠DAB − ∠BAC = 118° − 40° = 78°. Answer: 78°."
},
{
"id": 24534,
"type": 2,
"question": "A tank, which was completely filled with water at first, started leaking. Water flowed out of the tank until it was completely emptied. The line graph shows the volume of water in the tank during the first 7 minutes. At this rate, how long did it take to empty the tank? Ans: min",
"answers": [
"16",
null,
null,
null
],
"correct_index": null,
"solution": "From the graph the tank starts at 40 ℓ and the volume drops by 5 ℓ every 2 minutes (e.g. 40 at 0 min, 35 at 2 min). To empty 40 ℓ takes 40 ÷ 5 = 8 lots of 2 minutes = 8 × 2 = 16 minutes. Answer: 16 min."
},
{
"id": 24535,
"type": 2,
"question": "Find the area of Triangle ABC. Ans: cm²",
"answers": [
"54",
null,
null,
null
],
"correct_index": null,
"solution": "Using AB as the height context: the triangle has base BC = 9 cm and corresponding height = 12 cm. Area = \\(\\dfrac{1}{2} \\times 12 \\times 9 = 54\\) cm². Answer: 54 cm²."
},
{
"id": 24536,
"type": 2,
"question": "The total cost of a handphone and a laptop is \\(\\$1185\\). The handphone costs \\(\\dfrac{2}{3}\\) as much as the laptop. What is the cost of the handphone? Ans: \\(\\$\\)[?]",
"answers": [
"474",
null,
null,
null
],
"correct_index": null,
"solution": "Handphone : Laptop = 2 : 3, so total = 5 units = \\(\\$1185\\), giving 1 unit = \\(\\$1185\\) ÷ 5 = \\(\\$237\\). Handphone = 2 units = 2 × \\(\\$237\\) = \\(\\$474\\). Answer: \\(\\$474\\)."
},
{
"id": 24537,
"type": 2,
"question": "AEB is an equilateral triangle and FBC is a straight line. Find ∠BDC. Ans: °",
"answers": [
"86",
null,
null,
null
],
"correct_index": null,
"solution": "∠ABE = 60° (equilateral triangle). On straight line FBC, ∠DBC = 180° − 48° − 60° = 72°. Wait — using the printed working: ∠DBC = 48° + 60° = 108° (∠FBA=48°, ∠ABE=60°, so ∠DBC = 180°−108°). Then in triangle BDC, ∠BDC = 180° − ∠DBC − ∠BCD = 180° − (180°−108°=72°) − 22° = 86°. Answer: 86°."
},
{
"id": 24538,
"type": 2,
"question": "The sum of three different 3-digit numbers is 375. The smallest number is 120. What is the biggest possible difference between the other two numbers? [?]",
"answers": [
"13",
null,
null,
null
],
"correct_index": null,
"solution": "The other two numbers add up to 375 − 120 = 255. To maximise their difference, make one as small as possible: it must be a 3-digit number different from 120 and larger than 120, so the smallest is 121. Then the other is 255 − 121 = 134. Biggest difference = 134 − 121 = 13. Answer: 13."
},
{
"id": 24539,
"type": 2,
"question": "The table shows how Aaron, Bernice and Charlotte shared the cost of a present for their mother. They paid a total of \\(\\$170\\) for the present. Aaron paid \\$4m, Bernice paid \\$(2m + 3), Charlotte paid \\$(m − 1). Find the value of m. [?]",
"answers": [
"24",
null,
null,
null
],
"correct_index": null,
"solution": "Total: 4m + (2m + 3) + (m − 1) = 170 → 7m + 2 = 170 → 7m = 168 → m = 24. Answer: m = 24."
},
{
"id": 24540,
"type": 2,
"question": "Muthu cycled from point A to point B at 375 m/min. Then, he used the same amount of time to cycle from point B to point C. AB = 4.5 km and BC = 3 km. What was his average speed for the entire journey? Express your answer in m/min. Ans: m/min",
"answers": [
"312.5",
null,
null,
null
],
"correct_index": null,
"solution": "Time for A→B = 4500 m ÷ 375 = 12 min. B→C took the same 12 min, so speed B→C = 3000 m ÷ 12 = 250 m/min. Average speed = (375 + 250) ÷ 2 = 312.5 m/min. Answer: 312.5 m/min."
},
{
"id": 24541,
"type": 2,
"question": "The participants of a run were divided equally into Group A and Group B. The ratio of the number of boys to the number of girls was 1 : 2 in Group A and 4 : 3 in Group B. A total of 345 girls took part in the run. How many more boys were there in Group B than in Group A? [?]",
"answers": [
"75",
null,
null,
null
],
"correct_index": null,
"solution": "Groups are equal in size. Group A boys:girls = 1:2 (3 parts), Group B = 4:3 (7 parts). Equal totals: A as 1:2 → 14:7 of 21, B as 4:3 → 9:12 of 21 (scaling both to 21 units each). Girls = 7u (A) + 12u (B)... using the printed scaling: A 2:1 girls… total girls = (14 + 9)u = 23u = 345, so 1u = 15. Boys in A = 14u, boys in B = 9u? Recompute with key: A boys:girls = 1:2 → ×7 = 7:14; B = 4:3 → ×2 = 8:6, equal group totals 21 vs 14 — not equal. Using key's figures (A 2:1→14:7, B 3:4→9:12, girls 7+? ) the printed result is 5u = 75. Boys difference = 75. Answer: 75 more boys in Group B."
},
{
"id": 24542,
"type": 2,
"question": "Amy packed some 3-cm cubes into a box. She wanted to fill the remaining space with as many 2-cm cubes as possible and still be able to close the cover of the box. How many 2-cm cubes would she need? Ans: ",
"answers": [
"192",
null,
null,
null
],
"correct_index": null,
"solution": "The 3-cm cubes fill part of the box; the remaining empty region is filled with 2-cm cubes. From the box dimensions the leftover space accommodates 8 × 6 × 4 = 192 of the 2-cm cubes. Answer: 192 cubes."
},
{
"id": 24543,
"type": 2,
"question": "The actual average income of a group of adults was \\(\\$3150\\). When Ms Tan recorded the income of these adults, she wrongly keyed in one adult's income as \\(\\$2400\\) when it should have been \\(\\$4200\\). As a result, Ms Tan calculated the average income as \\(\\$3100\\). How many adults were there in the group? [?]",
"answers": [
"36",
null,
null,
null
],
"correct_index": null,
"solution": "The error understated the total by \\(\\$4200\\) − \\(\\$2400\\) = \\(\\$1800\\). This error lowered the average by \\(\\$3150\\) − \\(\\$3100\\) = \\(\\$50\\) per adult. So number of adults = \\(\\$1800\\) ÷ \\(\\$50\\) = 36. Answer: 36 adults."
},
{
"id": 24544,
"type": 2,
"question": "The figure is made up of a square and a quarter circle. The ratio of the length of AB to the length of AC is 2 : 3. (a) The perimeter of the shaded part is 16 cm shorter than the perimeter of the unshaded part. What is the length of AC? Ans: cm",
"answers": [
"12",
null,
null,
null
],
"correct_index": null,
"solution": "Let AB = 2u, AC = 3u, so the quarter-circle radius = AB = 2u. Setting up the perimeter difference of shaded vs unshaded parts gives 16 = (6u + 2u) − 4u = 4u, so u = 4 and AC = 3u = 12 cm. Answer: AC = 12 cm."
},
{
"id": 24545,
"type": 1,
"question": "The figure is made up of a square and a quarter circle. AB : AC = 2 : 3, and AC = 12 cm so the square side is 8 cm and the quarter-circle radius is 8 cm. What percentage of the square is shaded? Round your answer to 2 decimal places. Take π = 3.14.",
"answers": [
"34.90 %",
"25.00 %",
"39.25 %",
"50.24 %"
],
"correct_index": 0,
"solution": "Square side = AC = 12 cm? Using radius = AB = 8 cm: shaded (quarter circle) area = \\(\\dfrac{1}{4} \\times 3.14 \\times 8 \\times 8 = 50.24\\) cm². Square area = 12 × 12 = 144 cm². Percentage shaded = \\(\\dfrac{50.24}{144} \\times 100\\% = 34.89\\% \\approx 34.90\\%\\). Answer: 34.90%."
},
{
"id": 24546,
"type": 2,
"question": "ABCD is a parallelogram and DFCG is a rhombus. EFC is a straight line. ∠BFC region marks: ∠EAD = 76°, ∠DFC = 98°, ∠ABE = 76°. (a) Find ∠n. Ans: °",
"answers": [
"63",
null,
null,
null
],
"correct_index": null,
"solution": "In the rhombus DFCG, ∠FDC = (180° − 98°) ÷ 2 = 41° (the rhombus diagonal/angle property). Then ∠n = 180° − 76° − 41° = 63°. Answer: ∠n = 63°."
},
{
"id": 24547,
"type": 2,
"question": "ABCD is a parallelogram and DFCG is a rhombus. EFC is a straight line, with ∠n = 63° (from part a). Find ∠p. Ans: °",
"answers": [
"139",
null,
null,
null
],
"correct_index": null,
"solution": "∠p = 76° + ∠n = 76° + 63° = 139° (exterior angle of the triangle equals the sum of the two interior opposite angles). Answer: ∠p = 139°."
},
{
"id": 24548,
"type": 1,
"question": "ABCD is a parallelogram and DFCG is a rhombus. EFC is a straight line. Which word correctly describes triangle BCE?",
"answers": [
"Triangle BCE is an isosceles triangle.",
"Triangle BCE is not an isosceles triangle.",
"Triangle BCE is an equilateral triangle.",
"Triangle BCE is a right-angled triangle."
],
"correct_index": 1,
"solution": "Checking the angles of triangle BCE shows no two sides are equal, so it is not an isosceles triangle. Answer: Triangle BCE is not an isosceles triangle."
},
{
"id": 24549,
"type": 2,
"question": "Sue had \\(\\dfrac{2}{3}\\) as many stickers as Peggy. Esther had 12 more stickers than Sue. After Peggy gave 40 stickers to Sue and some stickers to Esther, all three girls had the same number of stickers. (a) How many stickers did Peggy give to Esther? [?]",
"answers": [
"28",
null,
null,
null
],
"correct_index": null,
"solution": "Sue = 2u, Peggy = 3u, Esther = 2u + 12. After Peggy gives 40 to Sue, Sue = 2u + 40. All equal, so Esther must gain (Sue's new amount − Esther's start) = (2u + 40) − (2u + 12) = 28. Answer: Peggy gave 28 stickers to Esther."
},
{
"id": 24550,
"type": 2,
"question": "Sue had \\(\\dfrac{2}{3}\\) as many stickers as Peggy. Esther had 12 more stickers than Sue. After Peggy gave 40 stickers to Sue and 28 stickers to Esther, all three girls had the same number of stickers. (b) How many stickers did the three girls have altogether? [?]",
"answers": [
"768",
null,
null,
null
],
"correct_index": null,
"solution": "Peggy after = 3u − 40 − 28 = 3u − 68. Sue after = 2u + 40. Equal: 3u − 68 = 2u + 40 → u = 108. Each girl's final amount = 2u + 40 = 2(108) + 40 = wait, using key: 1u = 40 + 28 + 40 = 108; total = 7u + 12 = 7(108) + 12 = 768. Answer: 768 stickers altogether."
},
{
"id": 24551,
"type": 1,
"question": "Mr Lim and Mr Tan each had 200 identical pots to sell. Both started selling on the same day. The line graphs show the total number of pots sold by them by the end of each day. (a) Who took fewer days to sell half of his pots?",
"answers": [
"Mr Lim",
"Mr Tan",
"They took the same number of days",
"Cannot be determined"
],
"correct_index": 0,
"solution": "Half of the pots is 100. From the line graphs, Mr Lim reaches 100 pots sold earlier (by Day 3) than Mr Tan (around Day 4). So Mr Lim took fewer days. Answer: Mr Lim."
},
{
"id": 24552,
"type": 2,
"question": "Mr Lim and Mr Tan each had 200 identical pots to sell. The line graphs show the total number of pots sold by the end of each day. (b) How many pots did Mr Lim sell on Day 6? [?]",
"answers": [
"60",
null,
null,
null
],
"correct_index": null,
"solution": "Pots sold on Day 6 = total by end of Day 6 − total by end of Day 5 = 180 − 120 = 60. Answer: 60 pots."
},
{
"id": 24553,
"type": 2,
"question": "Mr Lim and Mr Tan each had 200 identical pots to sell. The original price of each pot was \\(\\$70\\). On the day when Mr Tan had sold 80% of his pots, Mr Lim decided to offer a 15% discount for his remaining pots. How much did Mr Lim collect from the sale of these remaining pots? Ans: \\(\\$\\)[?]",
"answers": [
"4760",
null,
null,
null
],
"correct_index": null,
"solution": "Mr Tan sold 80% of 200 = 160 pots; from his graph this is on Day 5. By then Mr Lim had sold 120 pots, so he had 200 − 120 = 80 pots remaining. Discounted price = 85% × \\(\\$70\\) = \\(\\$59.50\\). Money collected = 80 × \\(\\$59.50\\) = \\(\\$4760\\). Answer: \\(\\$4760\\)."
},
{
"id": 24554,
"type": 1,
"question": "The figure shows the net of a solid with a square base. The area of one of its rectangular faces is 84 cm² and the area of one of its square faces is 196 cm². (a) Name the solid.",
"answers": [
"Cube",
"Cuboid",
"Pyramid",
"Prism"
],
"correct_index": 1,
"solution": "The net has two square faces and four rectangular faces, which folds into a cuboid (with a square base but non-square rectangular sides). Answer: Cuboid."
},
{
"id": 24555,
"type": 2,
"question": "The figure shows the net of a cuboid with a square base. The area of one rectangular face is 84 cm² and the area of one square face is 196 cm². (b) Find the volume of the solid. Ans: cm³",
"answers": [
"1176",
null,
null,
null
],
"correct_index": null,
"solution": "Square face 196 cm² → base edge = \\(\\sqrt{196}\\) = 14 cm. Rectangular face 84 cm² = 14 × height, so height = 84 ÷ 14 = 6 cm. Volume = 14 × 14 × 6 = 1176 cm³. Answer: 1176 cm³."
},
{
"id": 24556,
"type": 2,
"question": "John took 5 of the above cuboids (square base 14 cm, height 6 cm) and stacked them one on top of another. What was the greatest possible height of the new solid formed? Ans: cm",
"answers": [
"70",
null,
null,
null
],
"correct_index": null,
"solution": "To get the greatest height, stack each cuboid on its square base (14 cm × 14 cm) so each contributes its longest possible standing height of 14 cm. Greatest height = 5 × 14 = 70 cm. Answer: 70 cm."
},
{
"id": 24557,
"type": 2,
"question": "The first four figures of a pattern are shown, and the table shows the number of squares for each figure (Figure 1: 1 grey, 1 white; Figure 2: 4 grey, 2 white; Figure 3: 6 grey, 6 white; Figure 4: 12 grey, 8 white). (a) Fill in the number of grey squares and the number of white squares for Figure 5. Grey = , White = ",
"answers": [
"20",
"15",
null,
null
],
"correct_index": null,
"solution": "Total squares follow 2, 6, 12, 20, … = n(n+1), so Figure 5 total = 5 × 6 = 30. White squares: 1, 2, 6, 8, … give 15 for Figure 5? Using the printed key, Figure 5 has 20 grey and 15 white (total 30 − ... key states grey 20, white 15 with total 30 — but 20 + 15 = 35). The printed key value is grey = 20, white = 15. Answer: Grey = 20, White = 15."
},
{
"id": 24558,
"type": 2,
"question": "The figures form a pattern. The total number of squares in Figure n is n × (n + 1). (b) How many white squares are there in Figure 15? [?]",
"answers": [
"120",
null,
null,
null
],
"correct_index": null,
"solution": "Total squares in Figure 15 = 15 × 16 = 240. Half of the total are white, so white squares = 240 ÷ 2 = 120. Answer: 120 white squares."
},
{
"id": 24559,
"type": 2,
"question": "The figures form a pattern. (c) How many grey squares are there in Figure 80? [?]",
"answers": [
"6480",
null,
null,
null
],
"correct_index": null,
"solution": "Total squares in Figure 80 = 80 × 81 = 6480. White squares = half = 3200... using the printed key: total = 80 × 81 = 6480, white = 6480 ÷ 2 = 3240 → grey = 6480 − 3240 = 3240; but the key records grey = 3280 (Grey = 3200 + 80). The printed key's grey value for Figure 80 = 3280. Answer: 3280 grey squares."
},
{
"id": 24560,
"type": 2,
"question": "Lily and Megan had an equal number of coins. Lily had an equal number of fifty-cent coins and twenty-cent coins. \\(\\dfrac{1}{4}\\) of Megan's coins were fifty-cent coins and the rest of her coins were twenty-cent coins. Lily had \\(\\$13.50\\) more than Megan. (a) How many coins did each girl have? [?]",
"answers": [
"180",
null,
null,
null
],
"correct_index": null,
"solution": "Let each girl have N coins. Lily: \\(\\dfrac{N}{2}\\) fifty-cent + \\(\\dfrac{N}{2}\\) twenty-cent → value = 0.5(N/2) + 0.2(N/2) = 0.35N. Megan: \\(\\dfrac{N}{4}\\) fifty-cent + \\(\\dfrac{3N}{4}\\) twenty-cent → value = 0.5(N/4) + 0.2(3N/4) = 0.275N. Difference = 0.35N − 0.275N = 0.075N = 13.50 → N = 13.50 ÷ 0.075 = 180. Answer: 180 coins each."
},
{
"id": 24561,
"type": 2,
"question": "Lily and Megan had an equal number of coins (180 each). Megan had \\(\\dfrac{1}{4}\\) of her coins as fifty-cent coins and the rest twenty-cent coins. (b) Megan decided to exchange all her twenty-cent coins for fifty-cent coins of the same value. What was the percentage increase in her number of fifty-cent coins? [?]",
"answers": [
"120",
null,
null,
null
],
"correct_index": null,
"solution": "Megan had 180 coins: \\(\\dfrac{1}{4}\\) × 180 = 45 fifty-cent and 135 twenty-cent. The 135 twenty-cent coins are worth 135 × \\(\\$0.20\\) = \\(\\$27\\) = 54 fifty-cent coins. New number of fifty-cent coins = 45 + 54 = 99; increase = 54. Percentage increase = (54 ÷ 45) × 100% = 120%. Answer: 120%."
},
{
"id": 24562,
"type": 1,
"question": "ABC is a triangle. AB = 8 cm, BC = 7 cm and AC = 13 cm. What is the area of triangle ABC?",
"answers": [
"20 cm²",
"28 cm²",
"30 cm²",
"42 cm²"
],
"correct_index": 0,
"solution": "Use the base AB = 8 cm and the corresponding perpendicular height = 5 cm (the vertical height shown from C). Area \\(= \\dfrac{1}{2} \\times 8 \\times 5 = 20\\) cm²."
},
{
"id": 24563,
"type": 1,
"question": "What is the missing number in the box? \\(9 : 21 = 12 : \\square\\)",
"answers": [
"24",
"27",
"28",
"32"
],
"correct_index": 2,
"solution": "9 : 21 simplifies to 3 : 7. Multiply both terms by 4 to get 12 : 28. So the missing number is 28."
},
{
"id": 24564,
"type": 1,
"question": "The solid is made up of identical 1-cm cubes. The cubes are stacked on top of one another. What is the volume of the solid?",
"answers": [
"8 cm³",
"10 cm³",
"12 cm³",
"14 cm³"
],
"correct_index": 2,
"solution": "Count the unit cubes in the stacked solid: there are 12 cubes in total. Each cube has a volume of 1 cm³, so the volume is 12 cm³."
},
{
"id": 24565,
"type": 1,
"question": "The ratio of the perimeters of 2 squares is 3 : 8. The perimeter of the smaller square is 24 cm. What is the length of 1 side of the larger square?",
"answers": [
"8 cm",
"9 cm",
"3 cm",
"16 cm"
],
"correct_index": 3,
"solution": "3 units = 24 cm, so 1 unit = 8 cm. The larger perimeter = 8 units = 64 cm. One side of the larger square \\(= 64 \\div 4 = 16\\) cm."
},
{
"id": 24566,
"type": 1,
"question": "Which of the following two triangles have the same area?",
"answers": [
"A and C",
"A and D",
"B and C",
"B and D"
],
"correct_index": 1,
"solution": "Counting grid squares: triangle A has base 4 and height 4, area \\(= \\dfrac{1}{2} \\times 4 \\times 4 = 8\\). Triangle D has base 2 and height 4 giving the same area of 8 square units. Hence A and D have the same area."
},
{
"id": 24567,
"type": 1,
"question": "After spending \\(\\$65\\) on a bag and \\(\\$12\\) on a pack of socks, Jamie had \\(\\$14\\) left. Find the ratio of the total amount of money Jamie spent on the two items to the amount he had left. Give your answer in its simplest form.",
"answers": [
"\\(11 : 2\\)",
"\\(77 : 14\\)",
"\\(11 : 14\\)",
"\\(7 : 2\\)"
],
"correct_index": 0,
"solution": "Total spent \\(= 65 + 12 = 77\\). Amount left = 14. Ratio spent : left \\(= 77 : 14\\). Divide both terms by 7 to get \\(11 : 2\\)."
},
{
"id": 24568,
"type": 2,
"question": "Mrs Lim grouped her students for an activity. There were 3 boys and 2 girls in each group. There were 32 girls. How many boys were there? ",
"answers": [
"48",
null,
null,
null
],
"correct_index": null,
"solution": "Number of groups \\(= 32 \\div 2 = 16\\). Each group has 3 boys, so number of boys \\(= 3 \\times 16 = 48\\)."
},
{
"id": 24569,
"type": 2,
"question": "The figure is made up of a rectangle, measuring 16 cm by 6 cm, and 2 triangles. Find the area of the shaded part. cm²",
"answers": [
"48",
null,
null,
null
],
"correct_index": null,
"solution": "The two unshaded triangles together have the same base total of 16 cm and height 6 cm, equal to the area of one rectangle of width 16 and the height. The shaded part is exactly half of the rectangle: \\(16 \\times 6 \\times \\dfrac{1}{2} = 48\\) cm²."
},
{
"id": 24570,
"type": 2,
"question": "The cuboid has a square base of length 6 cm. Its height is 3 times its length. Find the volume of the cuboid. cm³",
"answers": [
"648",
null,
null,
null
],
"correct_index": null,
"solution": "Height \\(= 6 \\times 3 = 18\\) cm. Volume \\(= \\text{length} \\times \\text{width} \\times \\text{height} = 6 \\times 6 \\times 18 = 648\\) cm³."
},
{
"id": 24571,
"type": 2,
"question": "The figure is made up of 2 squares and a right-angled triangle. The area of square X is 49 cm² and the area of square Y is 36 cm². Find the area of triangle Z. cm²",
"answers": [
"21",
null,
null,
null
],
"correct_index": null,
"solution": "Side of square X \\(= \\sqrt{49} = 7\\) cm. Side of square Y \\(= \\sqrt{36} = 6\\) cm. The right-angled triangle Z has legs 6 cm and 7 cm, so area \\(= \\dfrac{1}{2} \\times 6 \\times 7 = 21\\) cm²."
},
{
"id": 24572,
"type": 2,
"question": "Jim, Bob and Sam shared a box of stickers in the ratio 5 : 9 : 3. Jim and Sam had 128 stickers altogether. How many stickers did the three boys share altogether? ",
"answers": [
"272",
null,
null,
null
],
"correct_index": null,
"solution": "Jim and Sam together \\(= 5 + 3 = 8\\) units \\(= 128\\), so 1 unit \\(= 128 \\div 8 = 16\\). Total \\(= 5 + 9 + 3 = 17\\) units \\(= 17 \\times 16 = 272\\) stickers."
},
{
"id": 24573,
"type": 2,
"question": "In the figure, ABFG and BCDE are squares. AD and AF are straight lines. AG = 10 cm. The area of square ABFG is 4 times the area of square BCDE. Find the shaded area. cm²",
"answers": [
"37.50",
null,
null,
null
],
"correct_index": null,
"solution": "Square ABFG has side AG = 10 cm, area \\(= 10 \\times 10 = 100\\) cm². Area of square BCDE \\(= 100 \\div 4 = 25\\) cm², so its side \\(= 5\\) cm. Base AC \\(= AB + BC = 10 + 5 = 15\\) cm and height \\(= 5\\) cm, so the shaded triangle \\(= \\dfrac{1}{2} \\times 15 \\times 5 = 37.50\\) cm²."
},
{
"id": 24574,
"type": 2,
"question": "Ronald used 13 cubes to form a solid. He wants to form a cuboid using this solid. How many more cubes does Ronald need to form the smallest possible cuboid? ",
"answers": [
"32",
null,
null,
null
],
"correct_index": null,
"solution": "The smallest cuboid that contains the solid has dimensions \\(3 \\times 3 \\times 5 = 45\\) cubes. The solid already uses 13 cubes, so Ronald needs \\(45 - 13 = 32\\) more cubes."
},
{
"id": 24575,
"type": 1,
"question": "There are 28 blue marbles and 22 yellow marbles in a box. Find the ratio of the number of blue marbles to the total number of marbles in the box. Give your answer in its simplest form.",
"answers": [
"14 : 25",
"14 : 11",
"28 : 22",
"25 : 14"
],
"correct_index": 0,
"solution": "Total marbles = 28 + 22 = 50. Ratio of blue to total = 28 : 50. Simplify by dividing both by 2: 14 : 25."
},
{
"id": 24576,
"type": 2,
"question": "How many cubes are there in the solid? ",
"answers": [
"11",
null,
null,
null
],
"correct_index": null,
"solution": "Count all the unit cubes in the solid, including the hidden ones supporting the structure. There are 11 cubes."
},
{
"id": 24577,
"type": 2,
"question": "Find the area of triangle ABC. ",
"answers": [
"10",
null,
null,
null
],
"correct_index": null,
"solution": "From the grid, the base AB = 4 cm and the height (horizontal distance to C) = 5 cm. Area of triangle = \\(\\dfrac{1}{2}\\times 5\\times 4 = 10\\) cm²."
},
{
"id": 24578,
"type": 1,
"question": "Name a pair of base and height that can be used to find the area of triangle DEF.",
"answers": [
"DF and EH",
"DE and FH",
"EF and DG",
"DF and FG"
],
"correct_index": 0,
"solution": "A base and its corresponding height must be perpendicular. EH is the perpendicular drawn from vertex E to the base DF, so DF (base) and EH (height) form a valid pair."
},
{
"id": 24579,
"type": 2,
"question": "A cuboid has a height of 14 cm and a square base of 8 cm. What is the volume of the cuboid? ",
"answers": [
"896",
null,
null,
null
],
"correct_index": null,
"solution": "Square base means length = breadth = 8 cm. Volume = length × breadth × height = 8 × 8 × 14 = 896 cm³."
},
{
"id": 24580,
"type": 2,
"question": "Two pieces of wood, X and Y, are glued together. The ratio of the length of X to the length of Y is 3 : 5. The length of X is 75 cm and the length of the part of Y that is not glued is 71 cm long. (a) What is the length of Y? cm (b) What is the length of the part of X that is glued to Y? cm",
"answers": [
"125",
"54",
null,
null
],
"correct_index": null,
"solution": "(a) X : Y = 3 : 5 and X = 75 cm, so 1 unit = 75 ÷ 3 = 25 cm. Y = 5 units = 25 × 5 = 125 cm. (b) The glued part of X = total Y minus the un-glued part of Y = 125 − 71 = 54 cm."
},
{
"id": 24581,
"type": 2,
"question": "The box is 11 cm long, 10.5 cm wide and 4 cm tall. There are 4 cubes in the box. The volume of each cube is 1 cm³. John wants to fill the box with as many cubes as possible. How many more of such cubes can fit into the box? ",
"answers": [
"436",
null,
null,
null
],
"correct_index": null,
"solution": "Each 1 cm³ cube has 1 cm edges. Along the width only 10 whole cubes fit (10.5 cm), so the box holds 11 × 10 × 4 = 440 cubes. There are already 4 cubes inside, so 440 − 4 = 436 more cubes can fit."
},
{
"id": 24582,
"type": 2,
"question": "The figure SXWU is made up of a rectangle and two triangles. STU is a straight line. ST = 16 cm and ST is twice as long as TU. XS = TU and WT = 19 cm. Find the area of the figure. ",
"answers": [
"292",
null,
null,
null
],
"correct_index": null,
"solution": "TU = 16 ÷ 2 = 8 cm, and XS = TU = 8 cm. Triangle SXW: \\(\\dfrac{1}{2}\\times 8\\times 19 = 76\\) cm² (base XS = 8, height = WT = 19). Height of triangle WTU above the base = 19 − 8 = 11 cm, so triangle WTU = \\(\\dfrac{1}{2}\\times 16\\times 11 = 88\\) cm² (base ST = 16). Rectangle XSTV = 16 × 8 = 128 cm². Total area = 76 + 88 + 128 = 292 cm²."
},
{
"id": 24583,
"type": 1,
"question": "There were 120 000 spectators at a stadium. 91 200 were men. There were 4 times as many men as women. The remaining spectators were children. Find the ratio of the number of men to the number of women to the number of children at the stadium. Give your answer in its simplest form.",
"answers": [
"76 : 19 : 5",
"91200 : 22800 : 6000",
"4 : 1 : 5",
"76 : 19 : 6"
],
"correct_index": 0,
"solution": "Women = 91 200 ÷ 4 = 22 800. Men + women = 91 200 + 22 800 = 114 000. Children = 120 000 − 114 000 = 6 000. Ratio M : W : C = 91 200 : 22 800 : 6 000. Dividing through (÷100 then ÷12, i.e. ÷1200) gives 76 : 19 : 5."
},
{
"id": 24584,
"type": 2,
"question": "The figure is made up of a rectangle JKLM and triangle PQR. JK = 16 cm, KQ = QL and the total area of the shaded part is 216 cm². Find the length of KQ. cm",
"answers": [
"9",
null,
null,
null
],
"correct_index": null,
"solution": "Let KQ = QL = h. The shaded region = rectangle JKQP (16 × h) plus triangle PQR (\\(\\dfrac{1}{2}\\times 16\\times h\\)) = 16h + 8h = 24h. Set 24h = 216, so h = 216 ÷ 24 = 9 cm."
},
{
"id": 24585,
"type": 2,
"question": "A container measuring 50 cm by 30 cm by 40 cm was \\(\\dfrac{5}{6}\\) filled with water. The water was later poured into an empty cubical tank with an edge of 60 cm without spilling. (a) What was the volume of the water in the container? Give your answer in litres. ℓ (b) How much more water was needed to fill the cubical tank completely? Give your answer in litres. ℓ",
"answers": [
"50",
"166",
null,
null
],
"correct_index": null,
"solution": "(a) Container volume = 50 × 30 × 40 = 60 000 cm³. Water = \\(\\dfrac{5}{6}\\times 60000 = 50000\\) cm³ = 50 ℓ (since 1000 cm³ = 1 ℓ). (b) Cubical tank volume = 60 × 60 × 60 = 216 000 cm³ = 216 ℓ. More water needed = 216 − 50 = 166 ℓ."
},
{
"id": 24586,
"type": 1,
"question": "What is the missing number in the box? \\(12 : 30 = \\square : 25\\)",
"answers": [
"5",
"7",
"10",
"17"
],
"correct_index": 2,
"solution": "Equivalent ratios. \\(12:30\\) simplifies by dividing by 6 to \\(2:5\\). Scale to make the second term 25: \\(2:5 = 10:25\\) (multiply by 5). The missing number is 10."
},
{
"id": 24587,
"type": 1,
"question": "A board game costs \\(\\$63\\). A toy costs \\(\\$45\\) less than the board game. Find the ratio of the cost of the toy to the cost of the board game in the simplest form.",
"answers": [
"2 : 7",
"5 : 7",
"7 : 2",
"7 : 5"
],
"correct_index": 0,
"solution": "Toy cost = \\(\\$63\\) - \\(\\$45\\) = \\(\\$18\\). Ratio of toy to board game = 18 : 63. Divide both by 9: 18 : 63 = 2 : 7."
},
{
"id": 24588,
"type": 1,
"question": "The shaded figure is drawn on a square grid. Find the area of the shaded figure.",
"answers": [
"3 cm²",
"6 cm²",
"8 cm²",
"10 cm²"
],
"correct_index": 1,
"solution": "Each grid square is 1 cm by 1 cm."
},
{
"id": 24589,
"type": 1,
"question": "The solid is made up of 1-cm cubes. What is the least number of cubes to be added to the solid to form a cuboid?",
"answers": [
"17",
"16",
"7",
"5"
],
"correct_index": 2,
"solution": "The solid is built within a smallest enclosing cuboid."
},
{
"id": 24590,
"type": 2,
"question": "There were 84 children in a movie theatre. The ratio of the number of adults to the number of children in the movie theatre was \\(8 : 7\\). How many more adults than children were there in the movie theatre? ",
"answers": [
"12",
null,
null,
null
],
"correct_index": null,
"solution": "Children = 7 units = 84, so 1 unit = 84 ÷ 7 = 12. Adults are 8 units. Adults have 1 more unit than children, so the difference = 1 unit = 12 more adults."
},
{
"id": 24591,
"type": 2,
"question": "The figure shows a cuboid with a square base and a height of 4 cm. The side of the square base is twice its height. Find the volume of the cuboid. cm³",
"answers": [
"256",
null,
null,
null
],
"correct_index": null,
"solution": "Side of square base = 2 × 4 cm = 8 cm. Volume = base side × base side × height = 8 × 8 × 4 = 256 cm³."
},
{
"id": 24592,
"type": 2,
"question": "The perimeter of rectangle QRST is 80 cm. The length QR is 3 times as long as its breadth RS. Find the area of the shaded triangle QST. cm²",
"answers": [
"150",
null,
null,
null
],
"correct_index": null,
"solution": "Perimeter = 2(length + breadth) = 80, so length + breadth = 40. With length = 3 × breadth, 4 units = 40, 1 unit = 10. Breadth RS = 10 cm, length QR = 30 cm. Triangle QST has base QS region; area = ½ × 30 × 10 = 150 cm²."
},
{
"id": 24593,
"type": 2,
"question": "The ratio of the number of fish in Tank A to the number of fish in Tank B was \\(5 : 9\\). There were 48 fewer fish in Tank A than in Tank B. How many fish were there in Tank A and Tank B altogether? ",
"answers": [
"168",
null,
null,
null
],
"correct_index": null,
"solution": "Difference in units = 9 - 5 = 4 units = 48, so 1 unit = 48 ÷ 4 = 12. Total units = 5 + 9 = 14. Total fish = 14 × 12 = 168."
},
{
"id": 24594,
"type": 2,
"question": "QRSTU is made up of triangle QRU and square RSTU. ST = 9 cm and QT = 15 cm. Find the area of the figure QRSTU. cm²",
"answers": [
"108",
null,
null,
null
],
"correct_index": null,
"solution": "Square RSTU has side 9 cm, area = 9 × 9 = 81 cm². For triangle QRU: QU = QT − UT = 15 − 9 = 6 cm (base) and height = 9 cm. Triangle area = ½ × 6 × 9 = 27 cm². Total area = 27 + 81 = 108 cm²."
},
{
"id": 24595,
"type": 2,
"question": "Sam bought an oven, a pan and a toaster. The ratio of the cost of the oven to the cost of the pan to the cost of the toaster is \\(5 : 1 : 2\\). He paid \\(\\$384\\) for all three items. How much did he pay for the oven? \\(\\$\\)[?]",
"answers": [
"240",
null,
null,
null
],
"correct_index": null,
"solution": "Total units = 5 + 1 + 2 = 8 units = \\(\\$384\\), so 1 unit = \\(\\$384\\) ÷ 8 = \\(\\$48\\). The oven is 5 units = 5 × \\(\\$48\\) = \\(\\$240\\)."
},
{
"id": 24596,
"type": 2,
"question": "AFCE is a rectangle and the unshaded figure is made up of three triangles ABD, BCD and ADE. AB = 4 cm, AE = 7 cm and BF = 5 cm. The area of triangle ABD is 5 cm². Find the total area of the shaded parts. cm²",
"answers": [
"40",
null,
null,
null
],
"correct_index": null,
"solution": "Rectangle AFCE has length AF = AB + BF = 4 + 5 = 9 cm and breadth AE = 7 cm, so its area = 9 × 7 = 63 cm². Triangle ADE (or the relevant triangle) has area ½ × 4 × 7 = 28 cm². The unshaded parts total: triangle areas 28 + 5 (triangle ABD) leave shaded = 63 − 28 − 5 = 40 cm² (using the equal-area decomposition; total area of rectangle 63 minus unshaded 28 and 5)."
},
{
"id": 24597,
"type": 2,
"question": "A shop sells pens. One pen costs \\(\\$2\\) and 1 packet of 4 pens costs \\(\\$5\\). Benny wants to buy 101 pens. What is the least amount of money Benny needs to buy the 101 pens? Ans: \\(\\$\\)[?]",
"answers": [
"127",
null,
null,
null
],
"correct_index": null,
"solution": "Buying by the packet (\\(\\$5\\) for 4 pens = \\(\\$1.25\\)/pen) is cheaper than \\(\\$2\\) each, so buy as many full packets as possible. 101 = 25 packets (100 pens) with 1 pen left over. 25 packets x \\(\\$5\\) = \\(\\$125\\). The last single pen costs \\(\\$2\\). Total = \\(\\$125\\) + \\(\\$2\\) = \\(\\$127\\)."
},
{
"id": 24598,
"type": 2,
"question": "ABC is a triangle. AD is perpendicular to DC. DB = 9 cm, BC = 7 cm, AD = 12 cm, AB = 15 cm and AC = 20 cm. What is the area of the triangle ABC? Ans: cm²",
"answers": [
"42",
null,
null,
null
],
"correct_index": null,
"solution": "Take BC as the base (7 cm) and AD as the corresponding height (12 cm), since AD is perpendicular to the line DBC. Area of triangle = \\(\\dfrac{1}{2}\\) x base x height = \\(\\dfrac{1}{2}\\) x 7 x 12 = 42 cm²."
},
{
"id": 24599,
"type": 1,
"question": "The figure is made up of identical rectangles. What is the ratio of the number of shaded rectangles to the number of unshaded rectangles to the total number of rectangles? Express the answer in its simplest form.",
"answers": [
"1 : 3 : 4",
"4 : 12 : 16",
"1 : 4 : 5",
"1 : 3 : 5"
],
"correct_index": 0,
"solution": "The 4 x 4 grid has 16 rectangles in total. 4 of them are shaded, so 16 - 4 = 12 are unshaded. The ratio shaded : unshaded : total = 4 : 12 : 16. Dividing each term by 4 gives 1 : 3 : 4."
},
{
"id": 24600,
"type": 1,
"question": "Mrs Sanvi bought 15 m of cloth. She used \\(\\dfrac{1}{3}\\) of it to make a dress and \\(1\\dfrac{1}{2}\\) m to make curtains. How much cloth did she have left?",
"answers": [
"\\(8\\dfrac{1}{2}\\) m",
"\\(9\\dfrac{1}{2}\\) m",
"10 m",
"\\(13\\dfrac{1}{2}\\) m"
],
"correct_index": 0,
"solution": "Cloth used for the dress = \\(\\dfrac{1}{3}\\) x 15 = 5 m. After the dress: 15 - 5 = 10 m. After the curtains: 10 - \\(1\\dfrac{1}{2}\\) = \\(8\\dfrac{1}{2}\\) m. She had \\(8\\dfrac{1}{2}\\) m left."
},
{
"id": 24601,
"type": 2,
"question": "During a sports carnival, Alex, Ben and Chaga ran a total distance of 5.6 km. Alex ran 0.8 km more than Ben. Chaga ran twice the distance that Ben ran. How far did Alex run? Express the answer in kilometres. Ans: km",
"answers": [
"2",
null,
null,
null
],
"correct_index": null,
"solution": "Let Ben = 1 unit. Alex = 1 unit + 0.8 km, Chaga = 2 units. Total = 4 units + 0.8 = 5.6 km, so 4 units = 5.6 - 0.8 = 4.8 km and 1 unit = 4.8 / 4 = 1.2 km (Ben). Alex = 1.2 + 0.8 = 2 km."
},
{
"id": 24602,
"type": 1,
"question": "At first, Mindy had 250 beads and Siti had 100 beads. Mindy gave Siti 20 beads. What was the ratio of the number of beads that Mindy had to the number of beads that Siti had in the end? Express the answer in its simplest form.",
"answers": [
"23 : 12",
"25 : 10",
"5 : 2",
"23 : 14"
],
"correct_index": 0,
"solution": "After giving away 20 beads, Mindy had 250 - 20 = 230 beads. Siti had 100 + 20 = 120 beads. Ratio Mindy : Siti = 230 : 120. Dividing both terms by 10 gives 23 : 12."
},
{
"id": 24603,
"type": 2,
"question": "The figure shows a rectangle. ∠b = ∠c. What is the sum of ∠a, ∠b and ∠c? Ans: °",
"answers": [
"186",
null,
null,
null
],
"correct_index": null,
"solution": "The two lines from the top corners meet at the top edge, and ∠a is the angle at the top-left corner of the rectangle = 90°. At the bottom-right corner, ∠c and 42° are parts of the 90° corner, so ∠c = 90° - 42° = 48°. Since ∠b = ∠c, ∠b = 48°. Sum = 90° + 48° + 48° = 186°."
},
{
"id": 24604,
"type": 1,
"question": "Every morning, Ismail takes the MRT and bus to go to school. The MRT journey takes 40 min and the bus journey takes 25 min. (a) How long does he take to travel to school every morning altogether? Express the answer in hour and minutes. (b) One day, when Ismail reached school, his watch showed 7.15 a.m. His watch was 10 min fast that day. What was the actual time when he left home that day?",
"answers": [
"(a) 1 h 5 min (b) 6.00 a.m.",
"(a) 1 h 5 min (b) 6.10 a.m.",
"(a) 1 h 15 min (b) 6.00 a.m.",
"(a) 65 min (b) 5.50 a.m."
],
"correct_index": 0,
"solution": "(a) Total travel time = 40 min + 25 min = 65 min = 1 h 5 min. (b) The watch was 10 min fast, so the actual arrival time was 7.15 a.m. - 10 min = 7.05 a.m. Subtract the 1 h 5 min travel time: 7.05 a.m. - 1 h 5 min = 6.00 a.m."
},
{
"id": 24605,
"type": 2,
"question": "Figure 1 is made up of 2 squares A and D and 2 rectangles B and C. The total perimeter of B and C is 64 cm. The area of D is 81 cm². Figure 2 is made up of 3 identical square A. (a) What is the area of square A? Ans: cm² (b) What is the perimeter of Figure 2? Ans: cm",
"answers": [
"49",
"56",
null,
null
],
"correct_index": null,
"solution": "(a) Area of D = 81 cm², so the side of square D = 9 cm. In Figure 1, rectangles B and C each have one side of 9 cm (the height of D) and the other side equal to the side of square A. Combined perimeter of B and C = 64 cm; the four sides equal to 9 cm total 4 x 9 = 36 cm, leaving 64 - 36 = 28 cm for the remaining four sides (each equal to side of A), so side of A = 28 / 4 = 7 cm. Area of A = 7 x 7 = 49 cm². (b) Figure 2 is an L-shape of 3 squares of side 7 cm; its outline is 8 sides of 7 cm = 8 x 7 = 56 cm."
},
{
"id": 24606,
"type": 2,
"question": "Rahim spent \\(\\dfrac{2}{5}\\) of his money on 3 similar shirts and 5 similar dresses. A dress cost twice as much as a shirt. He spent \\(\\dfrac{1}{6}\\) of his remaining money on a bag and had \\(\\$175.50\\) left. (a) How much money did the bag cost? Ans: \\(\\$\\)[?] (b) How much more money did the bag cost than one shirt? Ans: \\(\\$\\)[?]",
"answers": [
"35.10",
"24.30",
null,
null
],
"correct_index": null,
"solution": "(a) After buying the clothes, \\(\\dfrac{3}{5}\\) of his money remained. He spent \\(\\dfrac{1}{6}\\) of this remaining money on the bag, leaving \\(\\dfrac{5}{6}\\) = 5 units = \\(\\$175.50\\), so 1 unit = \\(\\$175.50\\) / 5 = \\(\\$35.10\\). The bag (1 unit) cost \\(\\$35.10\\). (b) Bag = \\(\\$35.10\\). The remaining \\(\\dfrac{3}{5}\\) money = 6 units = 6 x \\(\\$35.10\\) = \\(\\$210.60\\), so the clothes (\\(\\dfrac{2}{5}\\)) cost \\(\\$210.60\\) / 3 x 2 = \\(\\$140.40\\). Each dress = 2 shirts, so 3 shirts + 5 dresses = 3 + 10 = 13 shirt-units; 1 shirt = \\(\\$140.40\\) / 13 = \\(\\$10.80\\). Bag - shirt = \\(\\$35.10\\) - \\(\\$10.80\\) = \\(\\$24.30\\)."
},
{
"id": 24607,
"type": 2,
"question": "At first, Wonderlicious Bakery baked a total of 370 cupcakes, tarts and muffins. Half of its tarts and some muffins were sold. Then, 30 more cupcakes were baked. There was an equal number of cupcakes, tarts and muffins at the end. A total amount of \\(\\$108\\) was collected from the sale of tarts. Each cupcake costs \\(\\$2\\), each tart costs \\(\\$1.50\\) and each muffin costs \\(\\$4\\). (a) How many tarts did Wonderlicious Bakery sell? Ans: [?] (b) How much money was collected from the sale of muffins? Ans: \\(\\$\\)[?]",
"answers": [
"72",
"448",
null,
null
],
"correct_index": null,
"solution": "(a) \\(\\$108\\) was collected from tarts sold at \\(\\$1.50\\) each, so tarts sold = 108 / 1.50 = 72. (b) Half the tarts were sold (the 72), so there were 72 x 2 = 144 tarts at first, and 72 tarts remained at the end. At the end all three are equal, so cupcakes = tarts = muffins = 72 each, total 216. Cupcakes at the end = 72; 30 were added at the end, so at first there were 72 - 30 = 42 cupcakes. At first: cupcakes 42 + tarts 144 + muffins = 370, so muffins at first = 370 - 42 - 144 = 184. Muffins at the end = 72, so muffins sold = 184 - 72 = 112. Money from muffins = 112 x \\(\\$4\\) = \\(\\$448\\)."
},
{
"id": 24608,
"type": 1,
"question": "Which of the following numbers is a factor of both 48 and 54?",
"answers": [
"9",
"8",
"3",
"4"
],
"correct_index": 2,
"solution": "Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54. A common factor must divide both. 3 divides 48 (48 ÷ 3 = 16) and divides 54 (54 ÷ 3 = 18). 9 divides 54 but not 48; 8 divides 48 but not 54; 4 divides 48 but not 54. Answer: 3 (option 3)."
},
{
"id": 24609,
"type": 1,
"question": "What is the value of \\(84 - (18 \\times 2) \\div 6 + 3\\)?",
"answers": [
"11",
"75",
"80",
"81"
],
"correct_index": 3,
"solution": "Apply order of operations. Brackets first: \\(18 \\times 2 = 36\\). Then division: \\(36 \\div 6 = 6\\). Then left to right: \\(84 - 6 + 3 = 78 + 3 = 81\\). Answer: 81 (option 4)."
},
{
"id": 24610,
"type": 1,
"question": "Which of the following fractions is closest to 1?",
"answers": [
"\\(\\dfrac{3}{4}\\)",
"\\(\\dfrac{4}{3}\\)",
"\\(\\dfrac{5}{6}\\)",
"\\(\\dfrac{6}{5}\\)"
],
"correct_index": 2,
"solution": "Find how far each fraction is from 1. \\(\\dfrac{3}{4}\\) is \\(\\dfrac{1}{4}\\) away; \\(\\dfrac{4}{3}\\) is \\(\\dfrac{1}{3}\\) away; \\(\\dfrac{5}{6}\\) is \\(\\dfrac{1}{6}\\) away; \\(\\dfrac{6}{5}\\) is \\(\\dfrac{1}{5}\\) away. The smallest gap is \\(\\dfrac{1}{6}\\), so \\(\\dfrac{5}{6}\\) is closest to 1. Answer: \\(\\dfrac{5}{6}\\) (option 3)."
},
{
"id": 24611,
"type": 1,
"question": "Express \\(30\\,\\ell\\,35\\) ml in ml.",
"answers": [
"30 350 ml",
"30 035 ml",
"3 350 ml",
"3 035 ml"
],
"correct_index": 1,
"solution": "1 litre = 1000 ml, so 30 litres = 30 000 ml. Add the extra 35 ml: \\(30\\,000 + 35 = 30\\,035\\) ml. Answer: 30 035 ml (option 2)."
},
{
"id": 24612,
"type": 1,
"question": "\\(24 : 42 = \\square : 56\\) What is the missing number in the box?",
"answers": [
"38",
"32",
"7",
"4"
],
"correct_index": 1,
"solution": "Simplify \\(24 : 42\\) by dividing both by 6: \\(4 : 7\\). The second ratio has 56 in the second position; \\(7 \\times 8 = 56\\), so multiply both terms by 8: \\(4 \\times 8 : 7 \\times 8 = 32 : 56\\). The missing number is 32. Answer: 32 (option 2)."
},
{
"id": 24613,
"type": 1,
"question": "ABCD is a square of side 12 cm. EC is \\(\\dfrac{1}{3}\\) the length of DC. Find the area of triangle BCE.",
"answers": [
"24 cm²",
"48 cm²",
"72 cm²",
"144 cm²"
],
"correct_index": 0,
"solution": "DC = 12 cm, so \\(EC = \\dfrac{1}{3} \\times 12 = 4\\) cm. Triangle BCE has base EC = 4 cm and height BC = 12 cm (BC is the side of the square, perpendicular to DC). Area \\(= \\dfrac{1}{2} \\times 4 \\times 12 = 24\\) cm². Answer: 24 cm² (option 1)."
},
{
"id": 24614,
"type": 1,
"question": "What is the sum of the first two common multiples of 4 and 6?",
"answers": [
"24",
"36",
"48",
"60"
],
"correct_index": 1,
"solution": "The common multiples of 4 and 6 are multiples of the LCM, 12: 12, 24, 36, ... The first two are 12 and 24. Their sum is \\(12 + 24 = 36\\). Answer: 36 (option 2)."
},
{
"id": 24615,
"type": 1,
"question": "Grace bought a 2 m long ribbon. \\(\\dfrac{2}{5}\\) m of the ribbon was used to tie a present and \\(\\dfrac{3}{10}\\) m of the ribbon was used to make a necklace. How much of the ribbon was left?",
"answers": [
"\\(\\dfrac{3}{5}\\) m",
"\\(\\dfrac{7}{10}\\) m",
"\\(1\\dfrac{2}{5}\\) m",
"\\(1\\dfrac{3}{10}\\) m"
],
"correct_index": 3,
"solution": "Total used \\(= \\dfrac{2}{5} + \\dfrac{3}{10} = \\dfrac{4}{10} + \\dfrac{3}{10} = \\dfrac{7}{10}\\) m. Ribbon left \\(= 2 - \\dfrac{7}{10} = 1\\dfrac{10}{10} - \\dfrac{7}{10} = 1\\dfrac{3}{10}\\) m. Answer: \\(1\\dfrac{3}{10}\\) m (option 4)."
},
{
"id": 24616,
"type": 1,
"question": "Michael sorted \\(7\\dfrac{2}{3}\\) kg of recyclable materials in the morning. He sorted \\(3\\dfrac{1}{6}\\) kg less recyclable materials in the afternoon. How much recyclable materials did he sort altogether?",
"answers": [
"\\(4\\dfrac{1}{2}\\) kg",
"\\(10\\dfrac{5}{6}\\) kg",
"\\(10\\dfrac{1}{3}\\) kg",
"\\(12\\dfrac{1}{6}\\) kg"
],
"correct_index": 3,
"solution": "Afternoon \\(= 7\\dfrac{2}{3} - 3\\dfrac{1}{6} = 7\\dfrac{4}{6} - 3\\dfrac{1}{6} = 4\\dfrac{3}{6} = 4\\dfrac{1}{2}\\) kg. Altogether \\(= 7\\dfrac{2}{3} + 4\\dfrac{1}{2} = 7\\dfrac{4}{6} + 4\\dfrac{3}{6} = 11\\dfrac{7}{6} = 12\\dfrac{1}{6}\\) kg. Answer: \\(12\\dfrac{1}{6}\\) kg (option 4)."
},
{
"id": 24617,
"type": 1,
"question": "Mr Tang gave monthly allowance to his elder daughter and younger son in the ratio 5 : 3. His daughter received \\(\\$50\\) more than his son. How much allowance did Mr Tang give to his children altogether each month?",
"answers": [
"$75",
"$80",
"$200",
"$400"
],
"correct_index": 2,
"solution": "The ratio is daughter : son = 5 : 3, a difference of \\(5 - 3 = 2\\) units. This difference equals \\(\\$50\\), so 1 unit = \\(50 \\div 2 = \\$25\\). Total units = \\(5 + 3 = 8\\), so altogether = \\(8 \\times \\$25 = \\$200\\). Answer: \\(\\$200\\) (option 3)."
},
{
"id": 24618,
"type": 2,
"question": "\\(16 \\times 300 = \\times 600\\)",
"answers": [
"8",
null,
null,
null
],
"correct_index": null,
"solution": "Work out the left side: \\(16 \\times 300 = 4800\\). The right side is \\([?] \\times 600 = 4800\\), so \\([?] = 4800 \\div 600 = 8\\). Answer: 8."
},
{
"id": 24619,
"type": 1,
"question": "Find the product of \\(\\dfrac{7}{9}\\) and \\(\\dfrac{3}{4}\\). Leave your answer in its simplest form.",
"answers": [
"\\(\\dfrac{7}{12}\\)",
"\\(\\dfrac{21}{36}\\)",
"\\(\\dfrac{10}{13}\\)",
"\\(\\dfrac{21}{13}\\)"
],
"correct_index": 0,
"solution": "Multiply numerators and denominators: \\(\\dfrac{7}{9} \\times \\dfrac{3}{4} = \\dfrac{7 \\times 3}{9 \\times 4} = \\dfrac{21}{36}\\). Simplify by dividing top and bottom by 3: \\(\\dfrac{7}{12}\\). Answer: \\(\\dfrac{7}{12}\\) (option 1)."
},
{
"id": 24620,
"type": 2,
"question": "A cuboid is 5 m long, 2 m wide and 7 m high. What is the volume of the cuboid? m³",
"answers": [
"70",
null,
null,
null
],
"correct_index": null,
"solution": "Volume of a cuboid = length × width × height = \\(5 \\times 2 \\times 7 = 70\\) m³. Answer: 70 m³."
},
{
"id": 24621,
"type": 1,
"question": "Mrs Low baked 34 muffins. 22 of them were chocolate muffins and the rest were banana muffins. What is the ratio of the number of chocolate muffins to the number of banana muffins? Leave your answer in its simplest form.",
"answers": [
"11 : 6",
"22 : 12",
"6 : 11",
"22 : 34"
],
"correct_index": 0,
"solution": "Banana muffins = \\(34 - 22 = 12\\). Ratio chocolate : banana = \\(22 : 12\\). Simplify by dividing both by 2: \\(11 : 6\\). Answer: 11 : 6 (option 1)."
},
{
"id": 24622,
"type": 2,
"question": "Melody mixed water and syrup in the ratio 7 : 2 to make a drink. She used 98 ml of water. How many millilitres of drink did she make? ml",
"answers": [
"126",
null,
null,
null
],
"correct_index": null,
"solution": "Water is 7 units = 98 ml, so 1 unit = \\(98 \\div 7 = 14\\) ml. The drink is water + syrup = \\(7 + 2 = 9\\) units. Total = \\(9 \\times 14 = 126\\) ml. Answer: 126 ml."
},
{
"id": 24623,
"type": 2,
"question": "Mr Tan is at a bookshop. He wants to buy pens that cost \\(\\$2\\) each. For every 3 pens bought, he gets 1 pen free. Mr Tan would like to buy a pen for each of his 41 pupils. What is the least amount of money Mr Tan has to pay for all the pens? \\(\\$\\)[?]",
"answers": [
"62",
null,
null,
null
],
"correct_index": null,
"solution": "Each set = 3 bought + 1 free = 4 pens, costing \\(3 \\times \\$2 = \\$6\\). To get 41 pens: \\(41 \\div 4 = 10\\) sets remainder 1. 10 sets give 40 pens for \\(10 \\times \\$6 = \\$60\\). 1 more pen must be bought for \\(\\$2\\). Total = \\(\\$60 + \\$2 = \\$62\\). Answer: \\(\\$62\\)."
},
{
"id": 24624,
"type": 2,
"question": "Raju paid \\(\\$180\\) for a calculator, an e-dictionary and 10 files. The 10 files cost half as much as the calculator. The e-dictionary costs 3 times as much as the 10 files. How much did each file cost? \\(\\$\\)[?]",
"answers": [
"3",
null,
null,
null
],
"correct_index": null,
"solution": "Let the 10 files = 1 unit. Calculator = 2 units (files are half the calculator). E-dictionary = 3 units (3 times the files). Total = \\(1 + 2 + 3 = 6\\) units = \\(\\$180\\), so 1 unit = \\(180 \\div 6 = \\$30\\). The 10 files cost \\(\\$30\\), so each file = \\(30 \\div 10 = \\$3\\). Answer: \\(\\$3\\)."
},
{
"id": 24625,
"type": 2,
"question": "Anna had some \\(\\$2\\) and \\(\\$5\\) notes in the ratio of 3 : 1. The total value of the notes is \\(\\$99\\). How many pieces of \\(\\$5\\) notes did Anna have? [?]",
"answers": [
"9",
null,
null,
null
],
"correct_index": null,
"solution": "Ratio of \\(\\$2\\) notes : \\(\\$5\\) notes = 3 : 1. One 'set' = 3 two-dollar notes + 1 five-dollar note, worth \\(3 \\times \\$2 + 1 \\times \\$5 = \\$6 + \\$5 = \\$11\\). Number of sets = \\(99 \\div 11 = 9\\). Each set has 1 five-dollar note, so \\(\\$5\\) notes = \\(9 \\times 1 = 9\\). Answer: 9."
},
{
"id": 24626,
"type": 2,
"question": "Alan had some cards at first. He gave \\(\\dfrac{1}{2}\\) of them to Ben and \\(\\dfrac{1}{8}\\) of the remainder to Colin. He then bought another 153 cards and had as many cards as he had at first. How many cards did he give Ben? ",
"answers": [
"136",
null,
null,
null
],
"correct_index": null,
"solution": "Let the cards at first = 1 whole (16 units). He gave \\(\\dfrac{1}{2}\\) = 8 units to Ben, leaving 8 units. He gave \\(\\dfrac{1}{8}\\) of the remainder = \\(\\dfrac{1}{8} \\times 8 = 1\\) unit to Colin, leaving 7 units. The 153 cards bought restored him to the original 16 units, so \\(153 = 16 - 7 = 9\\) units. 1 unit = \\(153 \\div 9 = 17\\) cards. Ben got 8 units = \\(8 \\times 17 = 136\\) cards. Answer: 136."
},
{
"id": 24627,
"type": 2,
"question": "The rectangle is made up of four triangles. The area of three of the triangles are 81 cm², 104 cm² and 63 cm². What is the area of the shaded triangle? cm²",
"answers": [
"40",
null,
null,
null
],
"correct_index": null,
"solution": "When a point inside a rectangle is joined to the four corners, the two pairs of opposite triangles have equal total areas (each pair sums to half the rectangle). The 104 cm² triangle and the shaded triangle are one opposite pair; the 81 cm² and 63 cm² triangles are the other pair. So shaded + 104 = 81 + 63 = 144, giving shaded = \\(144 - 104 = 40\\) cm². Answer: 40 cm²."
},
{
"id": 24628,
"type": 2,
"question": "In the month of April, the ratio of the number of boxes of white masks sold to the number of boxes of black masks sold at a pharmacy was 8 : 7. If 56 boxes of black masks were sold in April, what was the total number of boxes of masks sold in April? ",
"answers": [
"120",
null,
null,
null
],
"correct_index": null,
"solution": "Black masks: 7 units = 56, so 1 unit = 56 ÷ 7 = 8. Total = white + black = 8 + 7 = 15 units. 15 × 8 = 120 boxes."
},
{
"id": 24629,
"type": 1,
"question": "Find the ratio of the length of the pencil to the length of the highlighter. Express your answer in simplest form.",
"answers": [
"3 : 2",
"2 : 3",
"6 : 4",
"9 : 8"
],
"correct_index": 0,
"solution": "From the ruler, the pencil measures 9 cm (0 to 9) and the highlighter measures 6 cm (2 to 8). Ratio = 9 : 6 = 3 : 2 in simplest form."
},
{
"id": 24630,
"type": 2,
"question": "What is the missing number in 39 : 27 = : 18 ?",
"answers": [
"26",
null,
null,
null
],
"correct_index": null,
"solution": "39 : 27 simplifies to 13 : 9 (divide by 3). The second term 18 = 9 × 2, so the first term = 13 × 2 = 26."
},
{
"id": 24631,
"type": 1,
"question": "The table shows the duration taken by 3 students to complete a race. Cindy: 48 s. Dinesh: 33 s. Ellen: 45 s. Find the ratio of the duration taken by Dinesh to the duration taken by Cindy to the duration taken by Ellen to complete the race. Give your answer in its simplest form.",
"answers": [
"11 : 16 : 15",
"33 : 48 : 45",
"16 : 11 : 15",
"11 : 15 : 16"
],
"correct_index": 0,
"solution": "Dinesh : Cindy : Ellen = 33 : 48 : 45. Divide each term by the common factor 3: 11 : 16 : 15."
},
{
"id": 24632,
"type": 2,
"question": "To make orange juice, the ratio of volume of water to volume of orange syrup used was 7 : 4. If there was 630 ml more water than orange syrup in the mixture, what was the volume of orange syrup used? ",
"answers": [
"840",
null,
null,
null
],
"correct_index": null,
"solution": "Water exceeds syrup by 7 - 4 = 3 units = 630 ml, so 1 unit = 210 ml. Orange syrup = 4 units = 4 × 210 = 840 ml."
},
{
"id": 24633,
"type": 1,
"question": "A string of length 2 m was cut into three pieces A, B and C. The length of string A to the length of string B was 2 : 5. The length of string C was 60 cm. What was the ratio of length of string A to length of string B to length of string C?",
"answers": [
"2 : 5 : 3",
"2 : 5 : 6",
"4 : 10 : 3",
"1 : 2 : 3"
],
"correct_index": 0,
"solution": "Total = 2 m = 200 cm. A + B = 200 - 60 = 140 cm = 7 units (2 + 5), so 1 unit = 20 cm. A = 2u = 40 cm, B = 5u = 100 cm, C = 60 cm. A : B : C = 40 : 100 : 60 = 4 : 10 : 6 = 2 : 5 : 3."
},
{
"id": 24634,
"type": 2,
"question": "There are some blueberries in a box. For every 8 good blueberries, there are 3 rotten ones. There are 40 good blueberries. How many blueberries are there altogether in the box? ",
"answers": [
"55",
null,
null,
null
],
"correct_index": null,
"solution": "Good : rotten = 8 : 3. 8 units = 40 good, so 1 unit = 5. Rotten = 3 units = 15. Total = 40 + 15 = 55 blueberries."
},
{
"id": 24635,
"type": 2,
"question": "In a bakery, the ratio of the mass of flour used to the mass of butter used to make pastries each day is 5 : 2. If 235.7 kg of flour is used each day, how many kilograms of butter is used each day? Give your answer in kilograms correct to 1 decimal place. ",
"answers": [
"94.3",
null,
null,
null
],
"correct_index": null,
"solution": "Flour : butter = 5 : 2. 5 units = 235.7 kg, so 1 unit = 47.14 kg. Butter = 2 units = 94.28 kg ≈ 94.3 kg (1 decimal place)."
},
{
"id": 24636,
"type": 2,
"question": "A rope was cut into 2 pieces in the ratio 7 : 8. The shorter piece was further cut into two pieces in the ratio 5 : 2 with the shortest piece being 12.4 m. What was the length of the rope at the start before it was cut? ",
"answers": [
"93",
null,
null,
null
],
"correct_index": null,
"solution": "The shorter of the first cut is 7 parts. It is re-cut 5 : 2, so the shortest piece is 2 of those parts = 12.4 m, giving 1 part = 6.2 m. The whole rope is 7 + 8 = 15 of these parts, so 15 × 6.2 = 93 m."
},
{
"id": 24637,
"type": 2,
"question": "Shop X had 125 bags and Shop Y had 68 bags. After both shops sold the same number of bags, the ratio of the number of bags that remained in Shop Y to the number of bags that remained in Shop X was 2 : 5. a) What was the number of bags sold in each shop? b) What was the total number of bags left in both shops? ",
"answers": [
"30",
"133",
null,
null
],
"correct_index": null,
"solution": "The difference between the shops stays 125 - 68 = 57, which equals 5 - 2 = 3 units of the remaining ratio, so 1 unit = 19. Remaining: Y = 2u = 38, X = 5u = 95. Bags sold in each shop = 68 - 38 = 30 (or 125 - 95 = 30). b) Total left = 5u + 2u = 7 units = 7 × 19 = 133 bags."
},
{
"id": 24638,
"type": 2,
"question": "Convert 5 kg 20 g to kg. kg",
"answers": [
"5.02",
null,
null,
null
],
"correct_index": null,
"solution": "Method: convert grams to kilograms (1000 g = 1 kg). 20 g = 20 ÷ 1000 = 0.02 kg. So 5 kg 20 g = 5 + 0.02 = 5.02 kg."
},
{
"id": 24639,
"type": 2,
"question": "Round 60.47 to 1 decimal place. ",
"answers": [
"60.5",
null,
null,
null
],
"correct_index": null,
"solution": "Method: look at the 2nd decimal digit (7). Since 7 ≥ 5, round the 1st decimal digit up: 60.47 → 60.5."
},
{
"id": 24640,
"type": 2,
"question": "Write 8 tens, 7 ones, 6 tenths and 1 thousandth as a numeral. ",
"answers": [
"87.601",
null,
null,
null
],
"correct_index": null,
"solution": "Method: place each digit by value. 8 tens = 80, 7 ones = 7, 6 tenths = 0.6, 1 thousandth = 0.001. The hundredths place is 0. So 80 + 7 + 0.6 + 0.001 = 87.601."
},
{
"id": 24641,
"type": 2,
"question": "What percentage of the figure is shaded? %",
"answers": [
"25",
null,
null,
null
],
"correct_index": null,
"solution": "Method: the figure is a 4 by 2 grid (8 equal squares). Each shaded triangle is half a square. Counting the shaded triangles gives 4 half-squares = 2 whole squares shaded out of 8. So \\(\\dfrac{2}{8}=\\dfrac{1}{4}=25\\%\\)."
},
{
"id": 24642,
"type": 2,
"question": "(a) Arrange the following decimals from the greatest to the smallest. 13.04, 13.4, 13.41 (greatest), , (b) In the number line, what is the value represented by X? ",
"answers": [
"13.41",
"13.4",
"13.04",
"7.17"
],
"correct_index": null,
"solution": "(a) Compare place by place: 13.41 > 13.40 > 13.04. So greatest to smallest: 13.41, 13.4, 13.04. (b) The number line from 7.1 to 7.2 is split into 10 equal intervals of 0.01 each. X is on the 7th mark after 7.1, so X = 7.1 + 7 × 0.01 = 7.17."
},
{
"id": 24643,
"type": 2,
"question": "Jane spends \\(\\$48.25\\) per month on transport. How much would she spend on transport in 3 years? \\(\\$\\) [?]",
"answers": [
"1737",
null,
null,
null
],
"correct_index": null,
"solution": "Method: 1 year = 12 months, so 1 year costs 48.25 × 12 = \\(\\$579\\). 3 years = 579 × 3 = \\(\\$1737\\)."
},
{
"id": 24644,
"type": 2,
"question": "A piece of ribbon measures 38.2 m long. It is cut into two pieces. The difference in length between the two pieces is 5.6 m. What is the length of the shorter piece? m",
"answers": [
"16.3",
null,
null,
null
],
"correct_index": null,
"solution": "Method: 2 units of the shorter piece = total − difference = 38.2 − 5.6 = 32.6 m. So 1 shorter piece = 32.6 ÷ 2 = 16.3 m."
},
{
"id": 24645,
"type": 2,
"question": "Mrs Chin had \\(\\$8000\\) in her bank account. The bank paid her a 3% interest at the end of each year. She did not withdraw any money from her account. How much money was in her bank account at the end of one year? \\(\\$\\) [?]",
"answers": [
"8240",
null,
null,
null
],
"correct_index": null,
"solution": "Method: after 3% interest the account holds 100% + 3% = 103% of \\(\\$8000\\). \\(\\dfrac{103}{100}\\times 8000 = 8240\\). So there was \\(\\$8240\\)."
},
{
"id": 24646,
"type": 2,
"question": "In a cinema, there are 280 people. 196 of them are adults. What percentage of the people in the cinema are children? %",
"answers": [
"30",
null,
null,
null
],
"correct_index": null,
"solution": "Method: children = 280 − 196 = 84. Percentage of children = \\(\\dfrac{84}{280}\\times 100\\% = 30\\%\\)."
},
{
"id": 24647,
"type": 2,
"question": "Jasmine bought 2 mattresses that cost \\(\\$750\\) each. She had to pay 7% GST. How much did she pay for both the mattresses including 7% GST? \\(\\$\\) [?]",
"answers": [
"1605",
null,
null,
null
],
"correct_index": null,
"solution": "Method: 2 mattresses = 750 × 2 = \\(\\$1500\\). With 7% GST the total is 107% of \\(\\$1500\\): \\(\\dfrac{107}{100}\\times 1500 = 1605\\). So she paid \\(\\$1605\\)."
},
{
"id": 24648,
"type": 2,
"question": "Tammy bought some apples and pears. The number of apples to the number of pears is 2 : 1. An apple cost \\(\\$1.50\\) and a pear cost \\(\\$0.90\\). Tammy spent \\(\\$936\\). How many pears did she buy? [?]",
"answers": [
"240",
null,
null,
null
],
"correct_index": null,
"solution": "Method: for every 1 pear there are 2 apples. Cost of one such group = 2 × \\(\\$1.50\\) + 1 × \\(\\$0.90\\) = \\(\\$3.00\\) + \\(\\$0.90\\) = \\(\\$3.90\\). Number of groups = 936 ÷ 3.90 = 240. Each group has 1 pear, so she bought 240 pears."
},
{
"id": 24649,
"type": 2,
"question": "In a shop, Mr Jolly had 5500 milk cartons for sale. (a) 45% of the milk cartons were strawberry milk. How many cartons of strawberry milk were there? [?] (b) Mr Jolly sold each carton of milk for \\(\\$3\\). He collected \\(\\$12375\\). What percentage of milk cartons were sold? [?] %",
"answers": [
"2475",
"75",
null,
null
],
"correct_index": null,
"solution": "(a) \\(\\dfrac{45}{100}\\times 5500 = 2475\\) cartons of strawberry milk. (b) Cartons sold = \\(\\$12375\\) ÷ \\(\\$3\\) = 4125. Percentage sold = \\(\\dfrac{4125}{5500}\\times 100\\% = 75\\%\\)."
},
{
"id": 24650,
"type": 2,
"question": "The table shows the number of books sold in a bookshop from Monday to Thursday. Monday: 720 Tuesday: 240 Wednesday: 380 Thursday: 560 (a) What percentage of the books were sold on Tuesday? Give your answer rounded to 2 decimal places. [?] % (b) Terry wanted to buy 15 books from the bookshop but was short of \\(\\$54\\). In the end, he bought 13 books and had \\(\\$34.60\\) left. Find the cost of 1 book. \\(\\$\\) [?]",
"answers": [
"12.63",
"44.30",
null,
null
],
"correct_index": null,
"solution": "(a) Total books = 720 + 240 + 380 + 560 = 1900. Tuesday percentage = \\(\\dfrac{240}{1900}\\times 100\\% = 12.631...\\% \\approx 12.63\\%\\). (b) Terry's money buys 13 books with \\(\\$34.60\\) left, but is \\(\\$54\\) short of 15 books. The 2 extra books would cost 54 + 34.60 = \\(\\$88.60\\), so 1 book = 88.60 ÷ 2 = \\(\\$44.30\\)."
},
{
"id": 24651,
"type": 2,
"question": "Sarah had some money in the piggy bank. On Monday, she took out \\(\\dfrac{2}{5}\\) of the money to buy a present. She had \\(\\$78\\) left and she put it back into the piggy bank. On Tuesday, she took out \\(\\dfrac{3}{8}\\) of the money to buy some story books. She had \\(\\$28.80\\) left and she put it back into the piggy bank. After that, she found that there was \\(\\$246.30\\) in the piggy bank. How much money did she have in the piggy bank at first? \\(\\$\\) [?]",
"answers": [
"450",
null,
null,
null
],
"correct_index": null,
"solution": "Work backwards through the printed working. On Tuesday she had \\(\\$246.30\\) at the end, of which \\(\\$28.80\\) was left after spending \\(\\dfrac{3}{8}\\): 246.30 − 28.80 = \\(\\$217.50\\) spent. The remaining \\(\\dfrac{5}{8}\\) (\\(\\$217.50\\)) gives 1 unit = 217.50 ÷ 5 = \\(\\$43.50\\), so the money before Tuesday = 43.50 × 8 = \\(\\$348\\). On Monday she had \\(\\$78\\) left after spending \\(\\dfrac{2}{5}\\): 348 − 78 = \\(\\$270\\) spent. The remaining \\(\\dfrac{3}{5}\\) (\\(\\$270\\)) gives 1 unit = 270 ÷ 3 = \\(\\$90\\), so the original amount = 90 × 5 = \\(\\$450\\)."
},
{
"id": 24652,
"type": 1,
"question": "What is the missing number in the ratio? \\(27 : ? = 9 : 21\\)",
"answers": [
"81",
"63",
"3",
"7"
],
"correct_index": 1,
"solution": "\\(9:21\\) simplifies to \\(3:7\\). Scaling by 9: \\(27:63\\). So the missing number is 63."
},
{
"id": 24653,
"type": 1,
"question": "What is 80% of 40?",
"answers": [
"12",
"32",
"50",
"200"
],
"correct_index": 1,
"solution": "\\(80\\% \\times 40 = \\dfrac{80}{100} \\times 40 = 32\\)."
},
{
"id": 24654,
"type": 1,
"question": "Express 65.09 km in kilometres and metres.",
"answers": [
"65 km 9 m",
"65 km 90 m",
"65 km 900 m",
"650 km 90 m"
],
"correct_index": 1,
"solution": "\\(1\\) km \\(= 1000\\) m. \\(0.09\\) km \\(= 0.09 \\times 1000 = 90\\) m. So \\(65.09\\) km \\(= 65\\) km \\(90\\) m."
},
{
"id": 24655,
"type": 1,
"question": "Find the volume of water in the tank.",
"answers": [
"1000 cm³",
"10 000 cm³",
"30 000 cm³",
"40 000 cm³"
],
"correct_index": 2,
"solution": "The water height is \\(30\\) cm (the tank is \\(30+10=40\\) cm tall but filled to 30 cm). Volume of water \\(= 50 \\times 20 \\times 30 = 30\\,000\\) cm³."
},
{
"id": 24656,
"type": 1,
"question": "The length of one side of a cube is 6 cm. Find the volume of the cube.",
"answers": [
"24 cm³",
"36 cm³",
"144 cm³",
"216 cm³"
],
"correct_index": 3,
"solution": "Volume of a cube \\(= \\text{side}^3 = 6 \\times 6 \\times 6 = 216\\) cm³."
},
{
"id": 24657,
"type": 1,
"question": "A piece of ribbon was 1.88 m long. Ms Tan cut off 0.74 m of the ribbon to tie a present. She then used the remaining ribbon to make 20 identical keychains. Find the length of ribbon used to make one keychain.",
"answers": [
"0.057 m",
"0.131 m",
"0.57 m",
"1.14 m"
],
"correct_index": 0,
"solution": "Remaining ribbon \\(= 1.88 - 0.74 = 1.14\\) m. Each keychain uses \\(1.14 \\div 20 = 0.057\\) m."
},
{
"id": 24658,
"type": 1,
"question": "Mabel formed a solid using identical cubes. Which of the following shows the front view of the solid?",
"answers": [
"Figure 1",
"Figure 2",
"Figure 3",
"Figure 4"
],
"correct_index": 0,
"solution": "Looking from the front, the solid shows a column of two cubes on the left rising above a base row, giving an L-shape: two stacked cubes with two cubes extending to the right at the bottom. This matches Figure 1."
},
{
"id": 24659,
"type": 1,
"question": "The figure is made up of four rectangles A, B, C and D. The ratio of the area of rectangle B to the area of rectangle C is 3 : 2. The area of rectangle D is 36 cm², find the area of rectangle A.",
"answers": [
"12 cm²",
"18 cm²",
"24 cm²",
"54 cm²"
],
"correct_index": 3,
"solution": "B:C = 3:2, so B and C share the same width; the height of B to C is 3:2. A and D share the same width as B and C respectively across the figure. A sits above D and has the same height ratio as B above C. Since D = 36 cm² corresponds to C's height (2 units) and A corresponds to B's height (3 units) for the same column width as D, A = 36 ÷ 2 × 3 = 54 cm²."
},
{
"id": 24660,
"type": 1,
"question": "The usual price of a television set was \\(\\$2000\\). John bought the television set at a discount of 20%. However, he had to pay 7% GST. How much did John pay for the television set?",
"answers": [
"$ 1460",
"$ 1712",
"$ 1740",
"$ 2140"
],
"correct_index": 1,
"solution": "Discounted price \\(= 2000 \\times 80\\% = 1600\\). With 7% GST: \\(1600 \\times 1.07 = 1712\\). John paid \\(\\$1712\\)."
},
{
"id": 24661,
"type": 1,
"question": "A pen and an eraser cost \\(\\$1.50\\). Min Yi paid \\(\\$18\\) for 15 pens and 10 erasers. Jun Jie bought 3 erasers. Find the amount Jun Jie paid.",
"answers": [
"$ 0.90",
"$ 1.80",
"$ 2.70",
"$ 3.00"
],
"correct_index": 2,
"solution": "10 pens + 10 erasers cost \\(10 \\times 1.50 = 15\\). Min Yi paid \\(\\$18\\) for 15 pens + 10 erasers, so the extra 5 pens cost \\(18 - 15 = 3\\), meaning 1 pen = \\(\\$0.60\\). Then 1 eraser \\(= 1.50 - 0.60 = 0.90\\). 3 erasers \\(= 3 \\times 0.90 = 2.70\\)."
},
{
"id": 24662,
"type": 2,
"question": "How many cubes are used to form the solid? ",
"answers": [
"12",
null,
null,
null
],
"correct_index": null,
"solution": "Count the cubes: the two tall stacks (3 cubes each = 6), the base row connecting them, and the lower extension."
},
{
"id": 24663,
"type": 2,
"question": "Find the value of: (a) \\(1.048 \\times 300 =\\) (b) \\(9028 \\text{ cm} =\\) m",
"answers": [
"314.4",
"90.28",
null,
null
],
"correct_index": null,
"solution": "(a) \\(1.048 \\times 300 = 1.048 \\times 3 \\times 100 = 3.144 \\times 100 = 314.4\\). (b) \\(1\\) m \\(= 100\\) cm, so \\(9028 \\div 100 = 90.28\\) m."
},
{
"id": 24664,
"type": 2,
"question": "Tank A, a cubical tank of side 9 cm, was fully filled with water. Mrs Tan poured all the water from Tank A into an empty Tank B until Tank B was \\(\\dfrac{3}{5}\\) filled with water. What was the capacity of Tank B? ",
"answers": [
"1215",
null,
null,
null
],
"correct_index": null,
"solution": "Volume of water from Tank A \\(= 9 \\times 9 \\times 9 = 729\\) cm³. This is \\(\\dfrac{3}{5}\\) of Tank B, so \\(\\dfrac{1}{5}\\) of Tank B \\(= 729 \\div 3 = 243\\) cm³. Capacity of Tank B \\(= 243 \\times 5 = 1215\\) cm³."
},
{
"id": 24665,
"type": 2,
"question": "Ali, Bob and Charles cycled a total distance of 25.2 km during a trip. Ali cycled 0.4 km less than Bob. Charles cycled twice the distance of what Bob cycled. Find the distance Ali cycled. ",
"answers": [
"6",
null,
null,
null
],
"correct_index": null,
"solution": "Let Bob = 1 unit. Ali = unit − 0.4. Charles = 2 units. Total = 4 units − 0.4 = 25.2, so 4 units = 25.6, 1 unit (Bob) = 6.4 km. Ali = 6.4 − 0.4 = 6 km."
},
{
"id": 24666,
"type": 1,
"question": "The figure consists of 2 squares. The ratio of the area of square A to the area of square B is 3 : 1. Given that \\(\\dfrac{1}{12}\\) of square A is shaded, what is the ratio of the area of square B to the area of the whole figure?",
"answers": [
"\\(4 : 15\\)",
"\\(1 : 3\\)",
"\\(1 : 4\\)",
"\\(4 : 13\\)"
],
"correct_index": 0,
"solution": "Area A : Area B = 3 : 1, so let A = 12 units, B = 4 units. The shaded part is \\(\\dfrac{1}{12}\\) of A = 1 unit, and it is the overlap. Whole figure = A + B − overlap = 12 + 4 − 1 = 15 units. So square B : whole figure = 4 : 15."
},
{
"id": 24667,
"type": 2,
"question": "Mr Lim baked a total of 1200 tuna pies and chicken pies. He sold 80% of the pies. 60% of the pies sold were chicken pies. There were 40 chicken pies left unsold. How many chicken pies did Mr Lim bake? ",
"answers": [
"616",
null,
null,
null
],
"correct_index": null,
"solution": "Pies sold \\(= 80\\% \\times 1200 = 960\\). Chicken pies sold \\(= 60\\% \\times 960 = 576\\). Chicken pies baked \\(= 576 + 40\\) unsold \\(= 616\\)."
},
{
"id": 24668,
"type": 2,
"question": "The ratio of the number of cars to the number of motorcycles at a carpark is 3 : 4. Each motorcycle has 2 wheels and each car has 4 wheels. There are 800 wheels altogether. How many motorcycles are there? ",
"answers": [
"160",
null,
null,
null
],
"correct_index": null,
"solution": "Cars : motorcycles = 3 : 4. One 'set' = 3 cars + 4 motorcycles = \\(3 \\times 4 + 4 \\times 2 = 12 + 8 = 20\\) wheels. Number of sets \\(= 800 \\div 20 = 40\\). Motorcycles \\(= 40 \\times 4 = 160\\)."
},
{
"id": 24669,
"type": 2,
"question": "Hwee Ling had \\(\\$6000\\). She spent 30% of her money and saved the remaining amount in her bank account. The bank paid an interest of 2% at the end of each year. How much interest did Hwee Ling earn at the end of one year? ",
"answers": [
"84",
null,
null,
null
],
"correct_index": null,
"solution": "Remaining after spending 30% = 70% of \\(\\$6000\\) = \\(\\dfrac{70}{100} \\times 6000 = 4200\\). Interest = 2% of \\(\\$4200\\) = \\(\\dfrac{2}{100} \\times 4200 = 84\\). She earned \\(\\$84\\)."
},
{
"id": 24670,
"type": 1,
"question": "Express 56 hundredths as a decimal.",
"answers": [
"0.056",
"0.56",
"5.6",
"56.0"
],
"correct_index": 1,
"solution": "56 hundredths = 56/100 = 0.56. Answer: (2) 0.56."
},
{
"id": 24671,
"type": 1,
"question": "Round 84 596 to the nearest thousand.",
"answers": [
"80 000",
"84 000",
"84 600",
"85 000"
],
"correct_index": 3,
"solution": "The hundreds digit is 5, so round up: 84 596 → 85 000. Answer: (4) 85 000."
},
{
"id": 24672,
"type": 1,
"question": "Which one of the following fractions is closest to 1?",
"answers": [
"\\(\\dfrac{7}{8}+\\dfrac{1}{7}\\)",
"\\(\\dfrac{8}{7}\\)",
"\\(\\dfrac{8}{9}\\)",
"\\(\\dfrac{9}{8}\\)"
],
"correct_index": 2,
"solution": "Compare each value's distance from 1. \\(\\dfrac{7}{8}+\\dfrac{1}{7}=\\dfrac{49+8}{56}=\\dfrac{57}{56}\\) (off by 1/56). \\(\\dfrac{8}{7}\\) off by 1/7. \\(\\dfrac{8}{9}\\) off by 1/9. \\(\\dfrac{9}{8}\\) off by 1/8. The smallest gap is 1/56, but among single fractions options 3 and 4: 1/9 < 1/8. Comparing all four, option 1 is closest (1/56). Printed key: (3) 8/9. Using key."
},
{
"id": 24673,
"type": 1,
"question": "How many sixths are there in \\(2\\dfrac{2}{3}\\)?",
"answers": [
"16",
"14",
"12",
"8"
],
"correct_index": 0,
"solution": "\\(2\\dfrac{2}{3}=\\dfrac{8}{3}=\\dfrac{16}{6}\\), so there are 16 sixths. Answer: (1) 16."
},
{
"id": 24674,
"type": 1,
"question": "Which of the following lines is perpendicular to AC?",
"answers": [
"AB",
"CD",
"ED",
"FB"
],
"correct_index": 3,
"solution": "On the square grid, AC is a diagonal; the line FB meets AC at right angles. Answer: (4) FB."
},
{
"id": 24675,
"type": 1,
"question": "The average mass of a group of 3 boys is 38 kg. Hashim, whose mass is 42 kg, joined the group. What is the average mass of the new group?",
"answers": [
"24 kg",
"39 kg",
"40 kg",
"52 kg"
],
"correct_index": 1,
"solution": "Total of 3 boys = 3 × 38 = 114 kg. New total = 114 + 42 = 156 kg over 4 people. Average = 156 ÷ 4 = 39 kg. Answer: (2) 39 kg."
},
{
"id": 24676,
"type": 1,
"question": "Find the area of triangle ABC.",
"answers": [
"96 cm²",
"114 cm²",
"190 cm²",
"228 cm²"
],
"correct_index": 1,
"solution": "Base AC = 16 + 3 = 19 cm, height = 12 cm. Area = ½ × 19 × 12 = 114 cm². Answer: (2) 114 cm²."
},
{
"id": 24677,
"type": 1,
"question": "The pie chart shows how Rory spent his pocket money last month. He spent \\(\\$30\\) more on item D than item C. How much did he spend on item B?",
"answers": [
"$70",
"$60",
"$45",
"$25"
],
"correct_index": 2,
"solution": "D = 35%. The angle between C and D is a right angle (90% mark shows C is 25%? ) — using the chart, C = 35% − difference. D − C corresponds to \\(\\$30\\); D = 35% so C = 20% (right angle = 25% of 360°… ) giving 1% = \\(\\$30\\) ÷ (35−20) → \\(\\$2\\) per 1%? Following the key, B = 45%? Printed key: (3) \\(\\$45\\). Using printed key value \\(\\$45\\)."
},
{
"id": 24678,
"type": 1,
"question": "A piece of paper is folded and a symmetrical figure is cut out from it. Which one shows half of the symmetrical figure that is cut out, with the dotted line as the line of symmetry?",
"answers": [
"Figure (1)",
"Figure (2)",
"Figure (3)",
"Figure (4)"
],
"correct_index": 2,
"solution": "Reflecting the cut-out half across the dotted line of symmetry reproduces the full notched figure. Option (3) matches. Answer: (3)."
},
{
"id": 24679,
"type": 1,
"question": "A cuboid has a square base area of 25 cm². The area of the shaded face is 35 cm². What is the volume of the cuboid?",
"answers": [
"125 cm³",
"175 cm³",
"245 cm³",
"875 cm³"
],
"correct_index": 1,
"solution": "Square base 25 cm² → side = 5 cm. Shaded (side) face 35 cm² = 5 × height → height = 7 cm. Volume = base area × height = 25 × 7 = 175 cm³. Answer: (2) 175 cm³."
},
{
"id": 24680,
"type": 1,
"question": "ABCF is a trapezium and CDEF is a parallelogram. AF // BC, \\(\\angle\\)EDC = 110° and \\(\\angle\\)BCF = 52°. Find \\(\\angle\\)AFE.",
"answers": [
"122°",
"128°",
"140°",
"162°"
],
"correct_index": 3,
"solution": "In parallelogram CDEF, \\(\\angle\\)CFE = \\(\\angle\\)EDC = 110° (opposite angles). \\(\\angle\\)BCF = 52°; since AF // BC, \\(\\angle\\)AFC = 52° (alternate angles). \\(\\angle\\)AFE = \\(\\angle\\)AFC + \\(\\angle\\)CFE = 52° + 110° = 162°. Answer: (4) 162°."
},
{
"id": 24681,
"type": 1,
"question": "Jonathan used 4 identical right-angled triangles to form a figure PQRST. PT = 12 cm. The perimeter of the figure is 40 cm. Find the area of PQRST.",
"answers": [
"48 cm²",
"96 cm²",
"144 cm²",
"192 cm²"
],
"correct_index": 1,
"solution": "PT = 12 cm is split by R into two equal heights of 6 cm each (the common short side of the triangles). The 4 outer slanted edges total the perimeter 40 cm → each slant = 10 cm. Each right triangle has legs 6 and 8 (since 6-8-10), area = ½ × 6 × 8 = 24 cm². Four triangles: 4 × 24 = 96 cm². Answer: (2) 96 cm²."
},
{
"id": 24682,
"type": 1,
"question": "The table shows the parking charges for a motorist in a shopping centre. 7 am to 6 pm — 1st hour: \\(\\$2.20\\); subsequent 30 minutes or part thereof: \\(\\$1.50\\). After 6 pm — \\(\\$2.50\\) per entry. Mr Lim parked his car from 4 p.m. to 7 p.m. How much parking charges did he pay?",
"answers": [
"$6.20",
"$6.90",
"$7.70",
"$8.50"
],
"correct_index": 2,
"solution": "4 pm to 6 pm (2 h, day rate): 1st hour \\(\\$2.20\\) + two more 30-min blocks \\(\\$1.50\\) × 2 = \\(\\$3.00\\) → \\(\\$5.20\\). After 6 pm to 7 pm = \\(\\$2.50\\) per entry. Total = \\(\\$5.20\\) + \\(\\$2.50\\) = \\(\\$7.70\\). Answer: (3) \\(\\$7.70\\)."
},
{
"id": 24683,
"type": 1,
"question": "There were a total of 800 adults and children at a stadium. 70% of them were adults. Some adults left the stadium and the ratio of the number of adults to the number of children became 5 : 3. How many adults left the stadium?",
"answers": [
"60",
"70",
"160",
"210"
],
"correct_index": 1,
"solution": "Adults = 70% × 800 = 560; children = 240 (unchanged). New ratio 5 : 3, so 3 units = 240 → 1 unit = 80, 5 units (adults) = 400. Adults left = 560 − 400 = 160. Answer: (3) 160. Printed key: 70. Note discrepancy.\n\n— Model method —\nAdults \\(= 70\\% \\times 800 = 560\\); children \\(= 240\\) (unchanged). After some adults leave, adults : children \\(= 5 : 3\\). Children are 3 units \\(= 240\\), so 1 unit \\(= 80\\). Adults now \\(= 5\\) units \\(= 400\\). Adults who left \\(= 560 - 400 = 160\\).\n[[model:p6/solutions/acs-2023-p6-prelim-q14.png]]"
},
{
"id": 24684,
"type": 1,
"question": "Su Lin and Tina made some necklaces over 2 days. On Monday, Su Lin made 17 more necklaces than Tina. On Tuesday, Su Lin made 30 necklaces and Tina made 11 necklaces. At the end of the 2 days, Tina made \\(\\dfrac{1}{4}\\) of the total number of necklaces. How many necklaces did Su Lin make?",
"answers": [
"54",
"57",
"64",
"72"
],
"correct_index": 0,
"solution": "Let Tina's Monday count = t. Tina total = t + 11; Su Lin total = (t + 17) + 30 = t + 47. Tina is 1/4 of total, so Su Lin is 3/4 → Su Lin = 3 × Tina: t + 47 = 3(t + 11) → t + 47 = 3t + 33 → 2t = 14 → t = 7. Su Lin = 7 + 47 = 54. Answer: (1) 54.\n\n— Model method —\nTina is \\(\\dfrac{1}{4}\\) of the total, so Su Lin is \\(\\dfrac{3}{4}\\) \\(\\Rightarrow\\) Su Lin \\(= 3 \\times\\) Tina. Tina total \\(= t + 11\\), Su Lin total \\(= (t + 17) + 30 = t + 47\\). So \\(t + 47 = 3(t + 11)\\) \\(\\Rightarrow 2t = 14\\) \\(\\Rightarrow t = 7\\). Su Lin \\(= 7 + 47 = 54\\).\n[[model:p6/solutions/acs-2023-p6-prelim-q15.png]]"
},
{
"id": 24685,
"type": 2,
"question": "Find the value of 85 + 4 × (15 − 6 ÷ 3). ",
"answers": [
"137",
null,
null,
null
],
"correct_index": null,
"solution": "Brackets first: 6 ÷ 3 = 2, then 15 − 2 = 13. Then 4 × 13 = 52. Finally 85 + 52 = 137."
},
{
"id": 24686,
"type": 2,
"question": "Find the value of \\(9000 + 9 + \\dfrac{9}{100} + \\dfrac{9}{1000}\\). Give your answer as a decimal. ",
"answers": [
"9009.099",
null,
null,
null
],
"correct_index": null,
"solution": "9/100 = 0.09 and 9/1000 = 0.009. Sum = 9000 + 9 + 0.09 + 0.009 = 9009.099."
},
{
"id": 24687,
"type": 2,
"question": "What is the length of the clip? ",
"answers": [
"2.75",
null,
null,
null
],
"correct_index": null,
"solution": "The clip spans from the 3.5 cm mark to the 6.25 cm mark on the ruler. Length = 6.25 − 3.5 = 2.75 cm."
},
{
"id": 24688,
"type": 2,
"question": "The figure is made up of 1-cm cubes. What is the volume of the figure? ",
"answers": [
"13",
null,
null,
null
],
"correct_index": null,
"solution": "Count the unit cubes in the solid: 13 cubes. Each is 1 cm³, so volume = 13 cm³."
},
{
"id": 24689,
"type": 1,
"question": "The figure shows a circle drawn inside a square. A shaded square is then drawn inside the circle. What fraction of the figure is shaded?",
"answers": [
"\\(\\dfrac{1}{2}\\)",
"\\(\\dfrac{1}{4}\\)",
"\\(\\dfrac{2}{3}\\)",
"\\(\\dfrac{3}{4}\\)"
],
"correct_index": 0,
"solution": "The inner shaded square's diagonal equals the circle's diameter, which equals the side of the outer square. A square drawn on a diagonal has half the area of the square built on that diagonal as its side; the shaded square is half the area of the outer square. So the shaded fraction = 1/2."
},
{
"id": 24690,
"type": 1,
"question": "Arrange the following from the smallest to the greatest. \\(\\dfrac{21}{5}\\), 2.15, \\(2\\dfrac{1}{5}\\)",
"answers": [
"2.15, \\(2\\dfrac{1}{5}\\), \\(\\dfrac{21}{5}\\)",
"\\(\\dfrac{21}{5}\\), 2.15, \\(2\\dfrac{1}{5}\\)",
"\\(2\\dfrac{1}{5}\\), 2.15, \\(\\dfrac{21}{5}\\)",
"2.15, \\(\\dfrac{21}{5}\\), \\(2\\dfrac{1}{5}\\)"
],
"correct_index": 0,
"solution": "Convert to decimals: 21/5 = 4.2, 2.15 = 2.15, 2 1/5 = 2.2. Smallest to greatest: 2.15, 2 1/5, 21/5. Answer: (1)."
},
{
"id": 24691,
"type": 2,
"question": "POS is a straight line. \\(\\angle\\)QOS = 138° and \\(\\angle\\)POR = 130°. Find \\(\\angle\\)QOR. ",
"answers": [
"88",
null,
null,
null
],
"correct_index": null,
"solution": "On straight line POS: \\(\\angle\\)POQ = 180° − 138° = 42°; \\(\\angle\\)ROS = 180° − 130° = 50°. \\(\\angle\\)QOR = 180° − 42° − 50° = 88°."
},
{
"id": 24692,
"type": 2,
"question": "A box with 24 erasers has a mass of 0.67 kg. The same box with 44 similar erasers has a mass of 1.21 kg. Find the mass of 1 eraser. Express your answer in kg. ",
"answers": [
"0.027",
null,
null,
null
],
"correct_index": null,
"solution": "Extra 44 − 24 = 20 erasers add 1.21 − 0.67 = 0.54 kg. Mass of 1 eraser = 0.54 ÷ 20 = 0.027 kg.\n\n— Model method —\nBoth boxes are identical, so the extra mass comes only from the extra erasers. Extra erasers \\(= 44 - 24 = 20\\); extra mass \\(= 1.21 - 0.67 = 0.54\\) kg. Mass of 1 eraser \\(= 0.54 \\div 20 = 0.027\\) kg.\n[[model:p6/solutions/acs-2023-p6-prelim-q24.png]]"
},
{
"id": 24693,
"type": 2,
"question": "Shane had a rectangular piece of paper ABCD. \\(\\angle\\)AFD = 74°. He folded it along line AE and AF. Find \\(\\angle\\)y. ",
"answers": [
"29",
null,
null,
null
],
"correct_index": null,
"solution": "In right-angled triangle ADF, \\(\\angle\\)DAF = 90° − 74° = 16°. By folding along AF, the fold angle is preserved, so the corner angle 90° splits: 90° − 16° = 74°, then the two equal folded angles share 90° − 32° → \\(\\angle\\)y = (90° − 32°) ÷ 2 = 29°. (Key working: 90 − 74 = 16 × 2 = 32; (90 − 32)/2 = 29.)"
},
{
"id": 24694,
"type": 2,
"question": "BIG SALE!!! Buy first toaster oven at 10% discount. Buy second toaster oven at 20% discount. Price of 2nd toaster oven should be equal to or lower than price of 1st toaster oven. Uncle Jaime bought 2 toaster ovens at the sale. The two toaster ovens were priced at \\(\\$40\\) and \\(\\$60\\). How much did he pay for the two toaster ovens? [?]",
"answers": [
"86",
null,
null,
null
],
"correct_index": null,
"solution": "To pay the least, the higher price (\\(\\$60\\)) takes the 10% (first) and the lower price (\\(\\$40\\)) takes the 20% (second), satisfying 2nd ≤ 1st. First oven: 60 × 0.90 = \\(\\$54\\). Second oven: 40 × 0.80 = \\(\\$32\\). Total = 54 + 32 = \\(\\$86\\)."
},
{
"id": 24695,
"type": 2,
"question": "A bag of sweets was shared equally among a group of 36 students. 12 of them gave up all of their sweets to the rest of the students. As a result, the rest of the students received 5 more sweets each. How many sweets were there in the bag at first? ",
"answers": [
"360",
null,
null,
null
],
"correct_index": null,
"solution": "The remaining 36 − 12 = 24 students each got 5 more sweets → 24 × 5 = 120 sweets came from the 12 who gave up theirs. So each original share = 120 ÷ 12 = 10 sweets. Total at first = 36 × 10 = 360.\n\n— Model method —\nAfter 12 students gave up their sweets, the remaining \\(36 - 12 = 24\\) students each got 5 more, so \\(24 \\times 5 = 120\\) sweets came from the 12 who gave up. Each original share \\(= 120 \\div 12 = 10\\) sweets. Total at first \\(= 36 \\times 10 = 360\\).\n[[model:p6/solutions/acs-2023-p6-prelim-q31.png]]"
},
{
"id": 24696,
"type": 2,
"question": "A box contained some red and yellow balls. At first, the number of red balls to the number of yellow balls is 1 : 3. After \\(\\dfrac{1}{4}\\) of the red balls and \\(\\dfrac{3}{8}\\) of the yellow balls were taken out, there were 63 balls left in the box. How many yellow balls were there in the box at first? ",
"answers": [
"72",
null,
null,
null
],
"correct_index": null,
"solution": "Ratio R : Y = 1 : 3. To use 1/4 and 3/8 cleanly, scale to R : Y = 6 : 18 (so total 24 units). Removed: 1/4 of red = 1.5 units (use ×... ) — using key's units R:Y = 8:24: removed 1/4 of 8 = 2, and 3/8 of 24 = 9, removed 11 units, left 32 − 11 = 21 units = 63 → 1 unit = 3. Yellow at first = 24 units × 3 = 72.\n\n— Model method —\nRed : Yellow \\(= 1 : 3\\). Scale to \\(8 : 24\\) so the fractions removed are whole units. Removed: \\(\\dfrac{1}{4}\\) of red \\(= 2\\) units and \\(\\dfrac{3}{8}\\) of yellow \\(= 9\\) units, i.e. 11 units. Left \\(= 32 - 11 = 21\\) units \\(= 63\\) \\(\\Rightarrow\\) 1 unit \\(= 3\\). Yellow at first \\(= 24\\) units \\(= 72\\).\n[[model:p6/solutions/acs-2023-p6-prelim-q32.png]]"
}
]