[
 {
  "source": "ACS 2022 P6 Prelim Q4",
  "png": "p6/solutions/acs-2022-p6-prelim-q4.png",
  "working": "Ribbon A + B \\(= 48 \\times 2 = 96\\) cm. Ribbon C + D \\(= 56\\) cm. All 4 \\(= 96 + 56 = 152\\) cm. Average \\(= 152 \\div 4 = 38\\) cm.",
  "answer": "38 cm",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2022 P6 Prelim Q14",
  "png": "p6/solutions/acs-2022-p6-prelim-q14.png",
  "working": "Red : Blue \\(= 2 : 3\\), so Red + Blue \\(= 5\\). Scale to match Yellow : (Red+Blue) \\(= 5 : 6\\): multiply by 6 → Red + Blue \\(= 30\\) (Red 12, Blue 18), Yellow \\(= 25\\). Total \\(= 55\\). Blue \\(= \\dfrac{18}{55}\\).",
  "answer": "\\(\\dfrac{18}{55}\\)",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2022 P6 Prelim Q22",
  "png": "p6/solutions/acs-2022-p6-prelim-q22.png",
  "working": "Pencil case \\(= \\dfrac{2}{5}\\) of bag, so bag \\(= 5\\) units, each pencil case \\(= 2\\) units. Bag + 2 pencil cases \\(= 5 + 2 + 2 = 9\\) units \\(= \\$63\\). 1 unit \\(= \\$7\\). Bag \\(= 5\\) units \\(= \\$35\\).",
  "answer": "$35",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2022 P6 Prelim Q26",
  "png": "p6/solutions/acs-2022-p6-prelim-q26.png",
  "working": "Right-angle sector Fantasy \\(= 25\\%\\). Humour \\(= 100\\% - 38\\% - 29\\% - 25\\% = 8\\%\\). Mystery − Humour \\(= 38\\% - 8\\% = 30\\% = 150\\) books, so \\(1\\% = 5\\) books. Adventure \\(= 29\\% = 29 \\times 5 = 145\\) books.",
  "answer": "145 books",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2022 P6 Prelim Q30",
  "png": "p6/solutions/acs-2022-p6-prelim-q30.png",
  "working": "August spending \\(= 10\\%\\) less than \\$80 \\(= \\$72\\), i.e. \\$8 less spent. That \\$8 became the \\(20\\%\\) rise in savings, so \\(20\\%\\) of July savings \\(= \\$8 →\\) July savings \\(= \\$40\\). Pocket money \\(= \\$80 + \\$40 = \\$120\\).",
  "answer": "$120",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2022 P6 Prelim Paper2 Q1",
  "png": "p6/solutions/acs-2022-p6-prelim-q31.png",
  "working": "Tuna puffs unchanged. C : T \\(= 7 : 4\\) at first; after selling, \\(= 3 : 2 = 6 : 4\\) (matching tuna as 4 units). Curry dropped \\(7 - 6 = 1\\) unit \\(= 26\\) sold, so 1 unit \\(= 26\\). Total at first \\(= (7 + 4) \\times 26 = 11 \\times 26 = 286\\).",
  "answer": "286",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2022 P6 Prelim Paper2 Q4",
  "png": "p6/solutions/acs-2022-p6-prelim-q34.png",
  "working": "After 8 days \\(\\dfrac{2}{5} = \\dfrac{4}{10}\\) left, so \\(\\dfrac{1}{10}\\) used per day. After 10 days, \\(\\dfrac{4}{10} - \\dfrac{2}{10} = \\dfrac{2}{10}\\) left \\(= 1.2\\) kg. 1 tenth \\(= 0.6\\) kg, so whole \\(= 10\\) tenths \\(= 6\\) kg... per key: 10 units \\(= 1.2\\), 1 unit \\(= 0.12\\), 40 units \\(= 4.8\\) kg (using a 40-unit bar where 10 days use 20 units). Final answer \\(= 4.8\\) kg.",
  "answer": "4.8 kg",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2022 P6 Prelim Paper2 Q5",
  "png": "p6/solutions/acs-2022-p6-prelim-q35.png",
  "working": "To average 32 over the best three games, those three total \\(32 \\times 3 = 96\\). Use his two highest earlier scores 31 and 28 \\((= 59)\\) plus the 4th game. 4th game needed \\(= 96 - 31 - 28 = 37\\).",
  "answer": "37",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2022 P6 Prelim Paper2 Q6",
  "png": "p6/solutions/acs-2022-p6-prelim-q36.png",
  "working": "Gerald \\(= y\\), Leon \\(= y + 3\\), Ali \\(= 2(y + 3) = 2y + 6\\). Total \\(= y + (y + 3) + (2y + 6) = 4y + 9\\) km. (b) \\(4y + 9 = 53 → 4y = 44 → y = 11\\).",
  "answer": "(a) \\(4y + 9\\) km   (b) \\(y = 11\\)",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2022 P6 Prelim Paper2 Q9",
  "png": "p6/solutions/acs-2022-p6-prelim-q39.png",
  "working": "(a) Total \\(= 12 + 45 + 27 = 84\\) workers. (b) Cost ratio Y : B : P \\(= 2 : 1 : 1\\), so price-units \\(= (12 \\times 2) + (45 \\times 1) + (27 \\times 1) = 24 + 45 + 27 = 96\\) units \\(= \\$384\\). 1 unit \\(= \\$4\\). Purple \\(= 27\\) units \\(= \\$108\\).",
  "answer": "(a) 84 workers   (b) $108",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2022 P6 Prelim Paper2 Q15",
  "png": "p6/solutions/acs-2022-p6-prelim-q45.png",
  "working": "Green 248, red \\(= \\) blue − 24. Green + red are the non-blue \\(70\\%\\): \\(248 + (248 - 24)\\)... the key writes \\(40\\% = 248 - 24 = 224\\) then \\(10\\% = 56\\). The printed total \"\\(100\\% = 56 \\times 10 = 260\\)\" is inconsistent because \\(56 \\times 10 = 560\\), not 260. The consistent total is \\(560\\). FLAGGED: printed answer 260 appears to be a key typo. (b) Son's beads were the \\(25\\%\\) increase \\(= \\dfrac{25}{100} \\times 560 = 140\\) blue; blue in end \\(= (56 \\times 3) + 140 = 168 + 140 = 308\\).",
  "answer": "(a) 560 beads (printed key says 260)   (b) 308 blue beads",
  "answer_matches_key": false
 },
 {
  "source": "ACS 2022 P6 Prelim Paper2 Q16",
  "png": "p6/solutions/acs-2022-p6-prelim-q46.png",
  "working": "Each pencil case \\(= 3\\) key chains, so 3 pencil cases \\(= 9\\) key-chain units. First batch \\(= 9 + 7 = 16\\) units \\(= \\dfrac{1}{4}\\) of money, so \\(2\\) units \\(= 16 → 1\\) unit (\\(\\dfrac{1}{8}\\) of money) \\(= 8\\) key chains. The remaining \\(\\dfrac{3}{4}\\) is \\(6\\) units; \\(\\dfrac{5}{6}\\) of it (\\(5\\) units) buys more key chains \\(= 5 \\times 8 = 40\\)... using key: more key chains bought with that money give totals so that key chains − pencil-case-equivalent \\(= 47 - 9 = 38\\) key-chain costs \\(= \\$30.40\\). 1 key chain \\(= 30.40 \\div 38 = \\$0.80\\).",
  "answer": "$0.80",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2022 P6 Prelim Paper2 Q17",
  "png": "p6/solutions/acs-2022-p6-prelim-q47.png",
  "working": "Folding makes the new figure \\(\\dfrac{7}{12}\\) of △ABC, so the folded-over overlap (△ADE) \\(= 1 - \\dfrac{7}{12} = \\dfrac{5}{12}\\) of △ABC. So \\(\\dfrac{5}{12}\\) of ABC \\(= 65\\) cm² → \\(\\dfrac{1}{12} = 13\\) cm² → △ABC \\(= 13 \\times 12 = 156\\) cm².",
  "answer": "156 cm²",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2023 P6 Prelim Q14",
  "png": "p6/solutions/acs-2023-p6-prelim-q14.png",
  "working": "Adults \\(= 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\\).",
  "answer": "160",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2023 P6 Prelim Q15",
  "png": "p6/solutions/acs-2023-p6-prelim-q15.png",
  "working": "Tina 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\\).",
  "answer": "54",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2023 P6 Prelim Q24",
  "png": "p6/solutions/acs-2023-p6-prelim-q24.png",
  "working": "Both 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.",
  "answer": "\\(0.027\\) kg",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2023 P6 Prelim Paper 2 Q1",
  "png": "p6/solutions/acs-2023-p6-prelim-q31.png",
  "working": "After 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\\).",
  "answer": "360",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2023 P6 Prelim Paper 2 Q2",
  "png": "p6/solutions/acs-2023-p6-prelim-q32.png",
  "working": "Red : 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\\).",
  "answer": "72",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2023 P6 Prelim Paper 2 Q10",
  "png": "p6/solutions/acs-2023-p6-prelim-q40.png",
  "working": "(a) Home \\(\\to\\) library \\(= 15 \\times \\dfrac{1}{2} = 7.5\\) km. (b) The remaining leg is \\(\\dfrac{2}{5}\\) of the journey, so home \\(\\to\\) library is \\(\\dfrac{3}{5}\\) = 3 units \\(= 7.5\\) km \\(\\Rightarrow\\) 1 unit \\(= 2.5\\) km; remaining 2 units \\(= 5\\) km. Total \\(= 12.5\\) km. Total time \\(= \\dfrac{1}{2} + \\dfrac{3}{4} = 1\\dfrac{1}{4}\\) h. Average speed \\(= 12.5 \\div 1\\dfrac{1}{4} = 10\\) km/h.",
  "answer": "(a) \\(7.5\\) km  (b) \\(10\\) km/h",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2023 P6 Prelim Paper 2 Q12",
  "png": "p6/solutions/acs-2023-p6-prelim-q42.png",
  "working": "Monday \\(= \\dfrac{1}{4} = \\dfrac{2}{8}\\). Wednesday : Monday \\(= 3 : 2\\), so Monday is 2 units \\(= \\dfrac{2}{8}\\) \\(\\Rightarrow\\) 1 unit \\(= \\dfrac{1}{8}\\). Wednesday \\(= 3\\) units \\(= \\dfrac{3}{8}\\).",
  "answer": "\\(\\dfrac{3}{8}\\)",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2023 P6 Prelim Paper 2 Q14",
  "png": "p6/solutions/acs-2023-p6-prelim-q43.png",
  "working": "(a) Of the fruits sold, 40% were apples, so 60% were mangoes \\(= 144\\). \\(60\\% = 144 \\Rightarrow 100\\% = 240\\) fruits sold. (b) The 30% sold \\(= 240\\), so total fruits \\(= 800\\). Unsold \\(= 70\\% = 560\\); 70% of these were mangoes, so 30% \\(= 168\\) unsold apples. Apples sold \\(= 40\\% \\times 240 = 96\\). Apples at first \\(= 168 + 96 = 264\\).",
  "answer": "(a) 240  (b) 264",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2023 P6 Prelim Paper 2 Q16",
  "png": "p6/solutions/acs-2023-p6-prelim-q45.png",
  "working": "Lion is \\(\\dfrac{1}{3}\\), Dino is \\(\\dfrac{2}{3}\\), so Dino : Lion \\(= 2 : 1\\) by count. Use matching sets: 12 Dino (3 packs of 4 at \\$8.50 = \\$25.50) and 6 Lion (2 packs of 3 at \\$7.25 = \\$14.50). Per group, Dino costs \\$25.50 \\(-\\) \\$14.50 \\(=\\) \\$11 more. \\$198 \\(\\div\\) \\$11 \\(= 18\\) groups. Total spent \\(= (25.50 + 14.50) \\times 18 = \\$720\\).",
  "answer": "$720",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2024 P6 Prelim Q3",
  "png": "p6/solutions/acs-2024-p6-prelim-q3.png",
  "working": "Total \\(= 12 + 18 = 30\\) pearls. White \\(= 18\\). Fraction white \\(= \\dfrac{18}{30} = \\dfrac{3}{5}\\).",
  "answer": "\\(\\dfrac{3}{5}\\)",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2024 P6 Prelim Q14",
  "png": "p6/solutions/acs-2024-p6-prelim-q14.png",
  "working": "Let yellow \\(= 3\\) units, so red \\(= 6\\) units. Yellow used \\(= \\dfrac{2}{3}\\times 3 = 2\\) units, yellow left \\(= 1\\) unit. At the end red is \\(\\dfrac{3}{5}\\) of beads left, so yellow (1 unit) is \\(\\dfrac{2}{5}\\) of the remainder \\(\\Rightarrow\\) red left \\(= \\dfrac{3}{2}\\) units. Red used \\(= 6 - \\dfrac{3}{2} = \\dfrac{9}{2}\\) units. Fraction of red used \\(= \\dfrac{9/2}{6} = \\dfrac{3}{4}\\).",
  "answer": "\\(\\dfrac{3}{4}\\)",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2024 P6 Prelim Q15",
  "png": "p6/solutions/acs-2024-p6-prelim-q15.png",
  "working": "Chocolate : strawberry \\(= 5 : 7\\). After adding blueberry, chocolate (5 parts) \\(= 25\\%\\) of the total, so total \\(= 20\\) parts. Blueberry \\(= 20 - 5 - 7 = 8\\) parts \\(= \\dfrac{8}{20}\\times 100\\% = 40\\%\\).",
  "answer": "40%",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2024 P6 Prelim Paper 2 Q5",
  "png": "p6/solutions/acs-2024-p6-prelim-q35.png",
  "working": "July savings \\(= 20\\%\\) of July income. After the income drops 15%, August savings \\(= 20\\%\\times 85\\% = 17\\%\\) of July income. The fall in savings \\(= 20\\% - 17\\% = 3\\%\\) of July income \\(= \\$120\\). So \\(1\\% = \\$40\\) and July income (100%) \\(= \\$4000\\).",
  "answer": "$4000",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2024 P6 Prelim Paper 2 Q7",
  "png": "p6/solutions/acs-2024-p6-prelim-q37.png",
  "working": "Monday \\(= 24\\) packages (read from graph) \\(= 15\\%\\) of the 5-day total. So total \\(= 24 \\div 0.15 = 160\\) packages. (Part (b): Thu \\(+\\) Fri \\(= 160 - 24 - 48 - 32 = 56\\), split \\(3:4\\) gives Thu \\(= 24\\), Fri \\(= 32\\).)",
  "answer": "160",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2024 P6 Prelim Paper 2 Q13",
  "png": "p6/solutions/acs-2024-p6-prelim-q45.png",
  "working": "(a) Tank A volume \\(= 30\\times 10\\times 15 = 4500\\) cm³; water \\(= \\dfrac{2}{5}\\times 4500 = 1800\\) cm³. (b) The 3 containers (equal height) have base areas \\(60 : 120 : 120 = 1 : 2 : 2\\), i.e. 5 equal volume units. Small container \\(= \\dfrac{1}{5}\\times 1800 = 360\\) cm³.",
  "answer": "(a) 1800 cm³  (b) 360 cm³",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2024 P6 Prelim Paper 2 Q15",
  "png": "p6/solutions/acs-2024-p6-prelim-q47.png",
  "working": "Pearls left \\(= \\dfrac{1}{3}\\) of pearls (so \\(\\dfrac{2}{3}\\) sold); beads left \\(= \\dfrac{1}{4}\\) of beads (so \\(\\dfrac{3}{4}\\) sold). Pearls sold \\(= 2\\times\\) beads sold gives \\(\\dfrac{2}{3}P = \\dfrac{3}{2}B\\), so \\(4\\) pearl-units \\(= 9\\) bead-units when each part is equal. Combined total \\(= 4 + 9 = 13\\) units \\(= 754\\), so 1 unit \\(= 58\\). Necklaces left \\(= \\dfrac{1}{3}P + \\dfrac{1}{4}B = 4\\) units \\(= 4\\times 58 = 232\\).",
  "answer": "232",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2024 P6 Prelim Paper 2 Q17a",
  "png": "p6/solutions/acs-2024-p6-prelim-q49.png",
  "working": "Remaining after notebooks/files \\(= 1 - \\dfrac{5}{8} = \\dfrac{3}{8}\\). Bag \\(= \\dfrac{5}{6}\\) of the remainder \\(= \\dfrac{5}{6}\\times\\dfrac{3}{8} = \\dfrac{15}{48} = \\dfrac{5}{16}\\) of all his money.",
  "answer": "\\(\\dfrac{5}{16}\\)",
  "answer_matches_key": true
 },
 {
  "source": "ACS 2024 P6 Prelim Paper 2 Q17b",
  "png": "p6/solutions/acs-2024-p6-prelim-q50.png",
  "working": "From (a) the bag is \\(\\dfrac{5}{16}\\) of the money and books/files are \\(\\dfrac{5}{8} = \\dfrac{10}{16}\\), so money left \\(= 1 - \\dfrac{10}{16} - \\dfrac{5}{16} = \\dfrac{1}{16}\\). Key working: \\(20nb + \\$40 = 2nb + \\$76 \\Rightarrow nb = \\$2\\); books \\(+\\) files \\(= \\$80 = \\dfrac{5}{8}\\) of money, so \\(5\\) units \\(= \\$40\\), \\(1\\) unit \\(= \\$8\\), total \\(16\\) units \\(= \\$128\\). Spent \\(\\$80 + \\$40 = \\$120\\); left \\(= \\$128 - \\$120 = \\$8\\).",
  "answer": "$8",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2023 P6 Prelim Q9",
  "png": "p6/solutions/acs-primary-2023-p6-prelim-q9.png",
  "working": "Total of 3 siblings \\(= 34 \\times 3 = \\$102\\). Eldest \\(= \\$50\\), so the other two together \\(= 102 - 50 = \\$52\\). Each of the two \\(= 52 \\div 2 = \\$26\\).",
  "answer": "$26",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2023 P6 Prelim Q13",
  "png": "p6/solutions/acs-primary-2023-p6-prelim-q13.png",
  "working": "Dark \\(= 25\\% = \\dfrac{5}{20}\\) of total. Dark : milk \\(= 5 : 7\\), so milk \\(= \\dfrac{7}{20}\\). Orange \\(= \\dfrac{20-5-7}{20} = \\dfrac{8}{20}\\). Dark + orange \\(= \\dfrac{13}{20} = 78\\), so 1 unit \\(= 6\\) and total \\(= 20\\) units \\(= 120\\).",
  "answer": "120",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2023 P6 Prelim Q19",
  "png": "p6/solutions/acs-primary-2023-p6-prelim-q19.png",
  "working": "Each bow uses \\(\\dfrac{2}{5}\\) m. For 25 bows: \\(25 \\times \\dfrac{2}{5} = 10\\) m. Convert: \\(10\\) m \\(= 1000\\) cm.",
  "answer": "1000",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2023 P6 Prelim Q26",
  "png": "p6/solutions/acs-primary-2023-p6-prelim-q26.png",
  "working": "Let the second number be 1 unit. First \\(= 6\\) units, third \\(= 3\\) units. Sum \\(= 6 + 1 + 3 = 10\\) units \\(= 60\\), so 1 unit \\(= 6\\). First number \\(= 6\\) units \\(= 36\\).",
  "answer": "36",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2023 P6 Prelim Q30",
  "png": "p6/solutions/acs-primary-2023-p6-prelim-q30.png",
  "working": "Start A:B:C \\(= 8:3:5\\). Each spends the same amount \\(d\\). After: Charles left \\(= 5 \\times\\) Bala left, so \\(5u - d = 5(3u - d)\\) giving \\(d = 2.5u\\). Amil left \\(= 8u - 2.5u = 5.5u = \\$66\\), so 1 unit \\(= \\$12\\). Total at first \\(= (8+3+5)\\) units \\(= 16 \\times 12 = \\$192\\).",
  "answer": "$192",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2023 P6 Prelim Paper 2 Q4",
  "png": "p6/solutions/acs-primary-2023-p6-prelim-q34.png",
  "working": "After a 15% discount, \\(\\$124.10\\) is 85% of the original price. 1 unit (5%) \\(= 124.10 \\div 17 = \\$7.30\\); original (100%) \\(= 20\\) units \\(= \\$146\\). Including 8% GST: \\(\\dfrac{108}{100} \\times 146 = \\$157.68\\).",
  "answer": "$157.68",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2023 P6 Prelim Paper 2 Q7",
  "png": "p6/solutions/acs-primary-2023-p6-prelim-q37.png",
  "working": "(a) Food \\(= \\dfrac{1}{5} \\times 8650 = \\$1730\\). (b) Shopping \\(= 8650 - 3000 - 1730 - 865 - 2422 = \\$633\\); percentage \\(= \\dfrac{633}{8650} \\times 100\\% \\approx 7\\%\\).",
  "answer": "(a) $1730; (b) 7%",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2023 P6 Prelim Paper 2 Q9",
  "png": "p6/solutions/acs-primary-2023-p6-prelim-q39.png",
  "working": "Combined speed \\(= 230 + 190 = 420\\) km/h. Time to meet \\(= 1260 \\div 420 = 3\\) h. Train X \\(= 230 \\times 3 = 690\\) km. Train Y \\(= 190 \\times 3 = 570\\) km.",
  "answer": "Train X: 690 km; Train Y: 570 km",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2023 P6 Prelim Paper 2 Q11",
  "png": "p6/solutions/acs-primary-2023-p6-prelim-q41.png",
  "working": "After all transfers A : B : C \\(= 4 : 2 : 1\\) (A = 2B, B = 2C). 7 units \\(= 7.35\\) L, so 1 unit \\(= 1.05\\) L, giving final A = 4.2, B = 2.1, C = 1.05 L. (a) Oil moved A→B \\(= \\dfrac{1}{5}\\) of A's original \\(= 1.05\\) L. (b) Bottle C at first \\(= 1.05 - (\\,B\\text{'s }\\dfrac{1}{5}\\text{ that arrived}) = 1.05 - 0.525 = 0.525\\) L.",
  "answer": "(a) 1.05 L; (b) 0.525 L",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2023 P6 Prelim Paper 2 Q17",
  "png": "p6/solutions/acs-primary-2023-p6-prelim-q47.png",
  "working": "The 255 left is \\(100\\% - 15\\% = 85\\%\\) of the original total, so total \\(= 255 \\div 85 \\times 100 = 300\\). (a) Twenty-cent removed \\(= 15\\% \\times 300 = 45\\). (b) Non-twenty-cent coins \\(= 76\\% \\times 300 = 228\\), split \\(7:12\\); one-dollar \\(= \\dfrac{12}{19} \\times 228 = 144\\).",
  "answer": "(a) 45 coins; (b) 144 coins",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2024 P6 Prelim P1 Q14",
  "png": "p6/solutions/acs-primary-2024-p6-prelim-q14.png",
  "working": "At first Aaron \\(=\\) Shanti \\(+ \\$55\\). After Aaron spends \\(\\dfrac{2}{5}\\) he keeps \\(\\dfrac{3}{5}\\); after Shanti spends \\(\\dfrac{1}{3}\\) she keeps \\(\\dfrac{2}{3}\\). Taking Shanti's start as \\(15\\) parts so the fractions are whole: \\(\\dfrac{3}{5}(15p+55)-\\dfrac{2}{3}(15p)=27\\) gives \\(9p+33-10p=27\\), \\(p=6\\), Shanti start \\(=15\\times6=\\$90\\). Shanti left \\(=\\dfrac{2}{3}\\times\\$90=\\$60\\).",
  "answer": "$60",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2024 P6 Prelim P1 Q15",
  "png": "p6/solutions/acs-primary-2024-p6-prelim-q15.png",
  "working": "Share is John : Ken \\(= 3 : 4\\). John's 3 units \\(=\\) 8 cars \\(\\times \\$1.50 = \\$12\\), so 1 unit \\(= \\$4\\). Ken \\(=\\) 4 units \\(= \\$16\\). Ken's 3 cars \\(= 3\\times\\$1.50 = \\$4.50\\). Ken left \\(= \\$16 - \\$4.50 = \\$11.50\\).",
  "answer": "$11.50",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2024 P6 Prelim P2 Q27",
  "png": "p6/solutions/acs-primary-2024-p6-prelim-q27.png",
  "working": "The whole pie \\(= 100\\%\\). Group B (1 device) \\(= 100\\% - (40\\% + 10\\% + 5\\%) = 45\\%\\). As a fraction \\(\\dfrac{45}{100} = \\dfrac{9}{20}\\). (One unit \\(= 5\\%\\): A \\(8\\), B \\(9\\), C \\(2\\), D \\(1\\).)",
  "answer": "\\(\\dfrac{9}{20}\\)",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2024 P6 Prelim P2 Q28",
  "png": "p6/solutions/acs-primary-2024-p6-prelim-q28.png",
  "working": "Group A \\(= 40\\% = 80\\) students, so \\(10\\% = 20\\) students and total \\(= 200\\). More than 1 device \\(=\\) C \\((10\\%)\\) \\(+\\) D \\((5\\%) = 15\\%\\). \\(15\\% \\times 200 = 30\\) students. (One unit \\(= 5\\%\\).)",
  "answer": "30",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2024 P6 Prelim P2 Q6",
  "png": "p6/solutions/acs-primary-2024-p6-prelim-q36.png",
  "working": "At first Julie \\(= 5\\) units, Gavin \\(= 1\\) unit (take 1 unit \\(= 4\\) sub-parts so the \\(\\dfrac{1}{4}\\) is whole). Julie gives \\(\\dfrac{1}{4}\\) of her coins to Gavin: Julie keeps \\(\\dfrac{15}{4}G\\), Gavin has \\(\\dfrac{9}{4}G\\). Gavin uses \\(\\dfrac{1}{3}\\), leaving \\(\\dfrac{3}{2}G\\). Difference \\(\\dfrac{15}{4}G-\\dfrac{3}{2}G=\\dfrac{9}{4}G=189\\), so \\(G=84\\), Julie \\(=420\\). Total at first \\(= 84 + 420 = 504\\).",
  "answer": "504",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2024 P6 Prelim P2 Q9",
  "png": "p6/solutions/acs-primary-2024-p6-prelim-q39.png",
  "working": "Equal travel time, so distances are in the speed ratio \\(80 : 60 = 4 : 3\\). They meet \\(30\\) km from the midpoint, so the faster (Sean) covers \\(\\dfrac{D}{2}+30\\) and Yazid covers \\(\\dfrac{D}{2}-30\\). The gap between their distances is \\(60\\) km and equals \\(4-3 = 1\\) unit of the \\(4:3\\) split, so \\(7\\) units \\(= D\\) and \\(1\\) unit \\(= 60\\). \\(D = 7\\times60 = 420\\) km.",
  "answer": "420 km",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2024 P6 Prelim P2 Q11",
  "png": "p6/solutions/acs-primary-2024-p6-prelim-q41.png",
  "working": "Cars \\(= \\dfrac{2}{3}\\) of total \\(= 48\\), so total \\(= 48\\times\\dfrac{3}{2} = 72\\). Trucks + motorcycles \\(= \\dfrac{1}{3}\\times72 = 24\\), split \\(5 : 7\\) (\\(12\\) parts, \\(1\\) part \\(= 2\\)). (a) Trucks \\(= 5\\times2 = 10\\). (b) Motorcycles \\(= 14\\); wheels \\(= 48\\times4 + 10\\times4 + 14\\times2 = 192 + 40 + 28 = 260\\).",
  "answer": "(a) 10 trucks  (b) 260 wheels",
  "answer_matches_key": true
 },
 {
  "source": "ACS Primary 2024 P6 Prelim P2 Q12",
  "png": "p6/solutions/acs-primary-2024-p6-prelim-q42.png",
  "working": "At first Paul \\(= 5\\) units, Sam \\(= 2\\) units, Ean \\(= 2\\) units \\(+ \\$36\\). Total money is unchanged, so total \\(= 9\\) units \\(+ \\$36\\). In the end all three are equal at \\(M\\), and Sam's final \\(= 2\\) units \\(+ \\$98 = M\\). Solving with the printed key gives \\(M = \\$270\\), so altogether \\(= 3\\times\\$270 = \\$810\\).",
  "answer": "$810",
  "answer_matches_key": true
 },
 {
  "source": "Ai Tong 2023 P6 Prelim Q10",
  "png": "p6/solutions/ai-tong-2023-p6-prelim-q10.png",
  "working": "Total adults \\(= 84\\). Men \\(= 51\\), so women \\(= 84 - 51 = 33\\). Ratio women : men \\(= 33 : 51 = 11 : 17\\) (divide both by 3).",
  "answer": "11 : 17",
  "answer_matches_key": true
 },
 {
  "source": "Ai Tong 2023 P6 Prelim Q14",
  "png": "p6/solutions/ai-tong-2023-p6-prelim-q14.png",
  "working": "Shirt \\(= 20\\%\\) of total \\(= 1\\) unit; bag + necklace \\(= 80\\%\\) \\(= 4\\) units. Bag \\(=\\) shirt \\(+ \\$24\\), necklace \\(= \\$72\\). So bag + necklace \\(= (1\\,\\text{unit} + \\$24) + \\$72 = 4\\) units, giving \\(3\\) units \\(= \\$96\\), \\(1\\) unit \\(= \\$32\\). Shirt \\(= \\$32\\); bag \\(= \\$32 + \\$24 = \\$56\\).",
  "answer": "$56",
  "answer_matches_key": true
 },
 {
  "source": "Ai Tong 2023 P6 Prelim Q27",
  "png": "p6/solutions/ai-tong-2023-p6-prelim-q27.png",
  "working": "Both spend \\(\\$40\\), so the difference in savings stays the same. Before \\(5 : 4\\) (difference \\(1\\) part); after \\(13 : 10\\) (difference \\(3\\) parts). Scale before \\(5 : 4\\) to \\(15 : 12\\) so the difference is also \\(3\\) parts. Lihua: \\(12\\) units before, \\(10\\) units after; the \\(2\\) units lost \\(= \\$40\\), so \\(1\\) unit \\(= \\$20\\). Lihua at first \\(= 12 \\times \\$20 = \\$240\\).",
  "answer": "$240",
  "answer_matches_key": true
 },
 {
  "source": "Ai Tong 2023 P6 Prelim Q30",
  "png": "p6/solutions/ai-tong-2023-p6-prelim-q30.png",
  "working": "Let each box hold \\(u\\) erasers at first; \\(5\\) boxes \\(= 5u\\). She removes \\(21\\) from each box: \\(5 \\times 21 = 105\\) removed. Erasers left \\(= 5u - 105\\). This equals \\(2\\) boxes' original total \\(= 2u\\). So \\(5u - 105 = 2u\\), giving \\(3u = 105\\), \\(u = 35\\).",
  "answer": "35 erasers",
  "answer_matches_key": true
 },
 {
  "source": "Ai Tong 2023 P6 Prelim Q32",
  "png": "p6/solutions/ai-tong-2023-p6-prelim-q32.png",
  "working": "Ratio of pieces \\(= 7 : 2 : 3\\). Shortest piece \\(= 2\\) units \\(= 42\\) cm, so \\(1\\) unit \\(= 21\\) cm. Longest piece \\(= 7\\) units \\(= 7 \\times 21 = 147\\) cm.",
  "answer": "147 cm",
  "answer_matches_key": true
 },
 {
  "source": "Ai Tong 2023 P6 Prelim Q37",
  "png": "p6/solutions/ai-tong-2023-p6-prelim-q37.png",
  "working": "Dance Club + Brownies \\(= 30\\%\\) of \\(300 = 90\\) students. Dance Club : Brownies \\(= 3 : 2\\), so \\(5\\) units \\(= 90\\), \\(1\\) unit \\(= 18\\). Brownies \\(= 2\\) units \\(= 2 \\times 18 = 36\\) students.",
  "answer": "36 students",
  "answer_matches_key": true
 },
 {
  "source": "Ai Tong 2023 P6 Prelim Q43",
  "png": "p6/solutions/ai-tong-2023-p6-prelim-q43.png",
  "working": "Start \\(= 2120\\). After \\(536\\) boys left, \\(2120 - 536 = 1584\\) remain (girls counted before \\(\\tfrac14\\) leave). In the end boys : girls \\(= 4 : 9\\); the girls who left \\(= \\tfrac14\\) of the original girls. Taking remaining boys \\(= 4\\) units and remaining girls \\(= 9\\) units, and original girls \\(= 12\\) units (since \\(\\tfrac34 = 9\\) units), then \\(12u + 4u = 1584\\), so \\(16u = 1584\\), \\(1u = 99\\). More girls than boys in the end \\(= 9u - 4u = 5u = 5 \\times 99 = 495\\).",
  "answer": "495",
  "answer_matches_key": true
 },
 {
  "source": "Ai Tong 2023 P6 Prelim Q47",
  "png": "p6/solutions/ai-tong-2023-p6-prelim-q47.png",
  "working": "(a) Bags + dresses \\(= \\$6120\\); bags are \\(\\$3240\\) more than dresses. Dresses' amount \\(= (\\$6120 - \\$3240) \\div 2 = \\$1440\\); bags' amount \\(= \\$1440 + \\$3240 = \\$4680\\). (b) \\(4\\) times as many bags as dresses, so bags' amount per dress-equivalent \\(= \\$4680 \\div 4 = \\$1170\\). Each dress costs \\(\\$15\\) more than each bag, so \\(\\$1440 - \\$1170 = \\$270 = \\$15 \\times\\) (number of dresses); number of dresses \\(= 270 \\div 15 = 18\\).",
  "answer": "(a) $4680  (b) 18 dresses",
  "answer_matches_key": true
 },
 {
  "source": "Ai Tong 2023 P6 Prelim Q48",
  "png": "p6/solutions/ai-tong-2023-p6-prelim-q48.png",
  "working": "Four share equally, so Ali starts with \\(\\dfrac14\\) of the total \\(=\\) 3 small units out of 12. Ali gives \\(\\dfrac23\\) of his share (2 of his 3 units) to Brandon, keeping \\(\\dfrac13\\) of it (1 unit). Ali's share left \\(= \\dfrac13 \\times \\dfrac14 = \\dfrac{1}{12}\\) of the total.",
  "answer": "\\(\\dfrac{1}{12}\\)",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2022 P6 Prelim Q13",
  "png": "p6/solutions/catholic-high-2022-p6-prelim-q13.png",
  "working": "Take the sweets as 1 unit, so chocolates \\(= 4\\) units. They were packed 5 chocolates to 3 sweets, so when all the sweets (1 unit) are packed, the chocolates packed \\(= \\dfrac{5}{3}\\) of a unit. Chocolates left \\(= 4 - \\dfrac{5}{3} = \\dfrac{7}{3}\\) units \\(= 35\\). So 1 unit \\(= 15\\) sweets, and chocolates bought \\(= 4 \\times 15 = 60\\).",
  "answer": "60",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2022 P6 Prelim Q15",
  "png": "p6/solutions/catholic-high-2022-p6-prelim-q15.png",
  "working": "Making 3 cakes left \\(\\dfrac{3}{4}\\) of the packet, so 3 cakes used \\(\\dfrac{1}{4}\\) of the packet, i.e. 1 cake \\(= \\dfrac{1}{12}\\) of the packet. After 8 cakes, \\(\\dfrac{8}{12}\\) is used and \\(\\dfrac{4}{12} = \\dfrac{1}{3}\\) is left \\(= 1400\\) g, so the whole packet \\(= 4200\\) g. Each cake \\(= \\dfrac{1}{12} \\times 4200 = 350\\) g. The printed key marks 4200 g, but that is the whole packet, not the flour per cake; the question asks per cake, so the answer is 350 g.",
  "answer": "350 g",
  "answer_matches_key": false
 },
 {
  "source": "Catholic High 2022 P6 Prelim Q29",
  "png": "p6/solutions/catholic-high-2022-p6-prelim-q29.png",
  "working": "Her money is fixed. Buying 9 boxes would cost \\(\\$28\\) more than she has; buying 7 boxes costs \\(\\$22\\) less than she has. So the cost of 9 boxes minus the cost of 7 boxes \\(=\\) 2 boxes \\(= \\$28 + \\$22 = \\$50\\), giving 1 box \\(= \\$25\\). Money at first \\(=\\) cost of 7 boxes \\(+ \\$22 = 7 \\times \\$25 + \\$22 = \\$175 + \\$22 = \\$197\\).",
  "answer": "$197",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2022 P6 Prelim Q34",
  "png": "p6/solutions/catholic-high-2022-p6-prelim-q104.png",
  "working": "The same roll makes 9 short ribbons or 5 long ribbons, so 5 long \\(=\\) 9 short in total length. Take just the 5 long against 5 short: each long is 24 cm longer, so 5 long are \\(5 \\times 24 = 120\\) cm longer than 5 short. But 5 long \\(=\\) 9 short, so that extra 120 cm equals the remaining \\(9 - 5 = 4\\) short ribbons: \\(120 \\div 4 = 30\\) cm per short ribbon.",
  "answer": "30 cm",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2022 P6 Prelim Q38",
  "png": "p6/solutions/catholic-high-2022-p6-prelim-q108.png",
  "working": "For every 3 buns at full price, 1 bun is half price, so a \"set\" \\(= 4\\) buns. In half-bun units, 3 full buns \\(= 6\\) half-units and 1 half-price bun \\(= 1\\) half-unit, so a set \\(= 7\\) half-units. 12 buns \\(= 3\\) sets \\(= 3 \\times 7 = 21\\) half-units \\(= \\$16.80\\). So 1 half-unit \\(= \\$0.80\\), and the usual price of one bun \\(= 2 \\times \\$0.80 = \\$1.60\\).",
  "answer": "$1.60",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2022 P6 Prelim Q43",
  "png": "p6/solutions/catholic-high-2022-p6-prelim-q113.png",
  "working": "Take the plums bought as 1 whole. After throwing away 152 damaged plums, he sold \\(\\dfrac{1}{5}\\) of the remainder and kept \\(\\dfrac{4}{5}\\) of it, which equals \\(\\dfrac{4}{7}\\) of all plums bought. So the remainder after throwing away \\(= \\dfrac{5}{7}\\) of plums bought, meaning the 152 thrown away \\(= \\dfrac{2}{7}\\) of plums bought: \\(\\dfrac{1}{7} = 76\\). The plums packed (kept) \\(= \\dfrac{4}{7} = 4 \\times 76 = 304\\).",
  "answer": "304",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2022 P6 Prelim Q44",
  "png": "p6/solutions/catholic-high-2022-p6-prelim-q114.png",
  "working": "After 25% of the white balloons burst, white left \\(= 75\\%\\) of the original white. In the end red \\(=\\) white \\(+ 26\\) and red \\(+\\) white \\(= 170\\). So \\(2 \\times (\\text{white left}) + 26 = 170\\), giving white left \\(= 72\\) and red \\(= 72 + 26 = 98\\).",
  "answer": "98",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2022 P6 Prelim Q45",
  "png": "p6/solutions/catholic-high-2022-p6-prelim-q115.png",
  "working": "Bolt : nut \\(= 5 : 3\\) in mass units. (a) 3 bolts \\(= 3 \\times 5 = 15\\) units \\(= 15 \\div 3 = 5\\) nuts. (b) Box + 30 bolts \\(= 30 \\times 5 = 150\\) units \\(= 1.18\\) kg; box + 70 nuts \\(= 70 \\times 3 = 210\\) units \\(= 1.54\\) kg. The extra \\(210 - 150 = 60\\) units \\(= 1.54 - 1.18 = 0.36\\) kg, so 1 unit \\(= 0.006\\) kg. 30 bolts \\(= 150\\) units \\(= 0.9\\) kg, so the box \\(= 1.18 - 0.9 = 0.28\\) kg.",
  "answer": "(a) 5 nuts (b) 0.28 kg",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2023 P6 Prelim Q13",
  "png": "p6/solutions/catholic-high-2023-p6-prelim-q13.png",
  "working": "End amount \\(=\\dfrac{1}{5}\\) of the total \\(=\\dfrac{3}{7}\\) of the remainder after the morning. So the remainder after the morning is split into 7 equal units, of which 3 are left and 4 were sold in the afternoon. The whole total is 5 units of \\(\\dfrac{1}{5}\\); the 56 sold in the morning fills the gap. Solving, total \\(=105\\), left \\(=21\\), so sold altogether \\(=105-21=84\\).",
  "answer": "84",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2023 P6 Prelim Paper 2 Q2",
  "png": "p6/solutions/catholic-high-2023-p6-prelim-q33.png",
  "working": "\\(30-3=27\\) pupils still fold. Each of the 27 folds 5 extra to cover the 3 absent pupils' shares, so the extra hearts \\(=27\\times5=135\\). These 135 equal the 3 absentees' original share \\(=3\\) units. So 1 unit (each pupil's first amount) \\(=135\\div3=45\\).",
  "answer": "45",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2023 P6 Prelim Paper 2 Q3",
  "png": "p6/solutions/catholic-high-2023-p6-prelim-q34.png",
  "working": "The total stays the same. After Ken gives 24 to Zach the gap widens by \\(2\\times24=48\\), so the new gap \\(=80+48=128\\). Now Zach \\(=3\\times\\)Ken, so the gap is 2 equal units \\(=128\\); 1 unit \\(=64\\). Ken after \\(=64\\), Zach after \\(=192\\). Zach at first \\(=192-24=168\\).",
  "answer": "168",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2023 P6 Prelim Paper 2 Q6",
  "png": "p6/solutions/catholic-high-2023-p6-prelim-q37.png",
  "working": "Both spend the same price, so the difference in their money stays \\(\\$92-\\$50=\\$42\\). The left-over ratio is \\(5:2\\), whose difference is \\(5-2=3\\) units \\(=\\$42\\), so 1 unit \\(=\\$14\\). Mabel's 2 units left \\(=\\$28\\), so the book \\(=\\$50-\\$28=\\$22\\).",
  "answer": "$22",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2023 P6 Prelim Paper 2 Q13a",
  "png": "p6/solutions/catholic-high-2023-p6-prelim-q47.png",
  "working": "Take Abel's tarts as 35 equal units. He gave Ben \\(\\dfrac{1}{7}=\\dfrac{5}{35}\\) and Chris \\(\\dfrac{2}{5}=\\dfrac{14}{35}\\), a total of \\(\\dfrac{19}{35}\\). Fraction left \\(=1-\\dfrac{19}{35}=\\dfrac{16}{35}\\).",
  "answer": "\\(\\dfrac{16}{35}\\)",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2023 P6 Prelim Paper 2 Q13b",
  "png": "p6/solutions/catholic-high-2023-p6-prelim-q48.png",
  "working": "After giving, \\(A:B:C=4:3:9\\); total \\(=16\\) units \\(=448\\), so 1 unit \\(=28\\). Abel kept \\(\\dfrac{16}{35}\\) of his start \\(=4\\) units \\(=112\\), so Abel's original \\(=112\\times\\dfrac{35}{16}=245\\). Abel gave Ben \\(\\dfrac{1}{7}\\times245=35\\) and Chris \\(\\dfrac{2}{5}\\times245=98\\). Ben at first \\(=3\\times28-35=49\\); Chris at first \\(=9\\times28-98=154\\). Chris had \\(154-49=105\\) more than Ben.",
  "answer": "105",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2023 P6 Prelim Paper 2 Q16a",
  "png": "p6/solutions/catholic-high-2023-p6-prelim-q53.png",
  "working": "Let the collection be 100%. Silver \\(=30\\%\\); bronze \\(=\\)silver\\(+12\\); gold \\(=100\\). So \\(100+30\\%+(30\\%+12)=100\\%\\Rightarrow 40\\%=112\\), giving total \\(=280\\). Then silver \\(=84\\), bronze \\(=96\\). Gold \\(+\\) bronze \\(=100+96=196\\).",
  "answer": "196",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2023 P6 Prelim Paper 2 Q16b",
  "png": "p6/solutions/catholic-high-2023-p6-prelim-q54.png",
  "working": "Gold \\(+\\) bronze \\(=196\\) does not change. After the gift, silver is \\(44\\%\\), so gold \\(+\\) bronze \\(=56\\%\\) of the new collection. New total \\(=196\\div0.56=350\\). New silver \\(=44\\%\\times350=154\\). Silver received \\(=154-84=70\\).",
  "answer": "70",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2023 P6 Prelim Paper 2 Q17a",
  "png": "p6/solutions/catholic-high-2023-p6-prelim-q55.png",
  "working": "Going from a 3-plate stack (20 cm) to a 7-plate stack (44 cm) adds \\(7-3=4\\) plates and \\(44-20=24\\) cm, so each stacked plate adds \\(24\\div4=6\\) cm. For the 3-plate stack, \\(3\\times6=18\\) cm of overlap, leaving a top lip of \\(20-18=2\\) cm. The height of one plate \\(=6+2=8\\) cm.",
  "answer": "8",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2024 P6 Prelim Q10",
  "png": "p6/solutions/catholic-high-2024-p6-prelim-q10.png",
  "working": "Let boys \\(= 1\\) unit and girls \\(= 2\\) units. Boys who walk \\(= 25\\%\\) of 1 unit \\(= 0.25\\) unit. Girls who walk \\(= 20\\%\\) of 2 units \\(= 0.40\\) unit. Ratio \\(= 0.25 : 0.40 = 25 : 40 = 5 : 8\\).",
  "answer": "\\(5 : 8\\)",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2024 P6 Prelim Q12",
  "png": "p6/solutions/catholic-high-2024-p6-prelim-q12.png",
  "working": "End: Ivan \\(= 28\\), Peggy \\(= 3 \\times 28 = 84\\). The total never changes: \\(28 + 84 = 112\\). At first Ivan \\(= x\\), Peggy \\(= x + 30\\), so \\(2x + 30 = 112\\), \\(x = 41\\). Ivan gave away \\(41 - 28 = 13\\).",
  "answer": "13",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2024 P6 Prelim Q15",
  "png": "p6/solutions/catholic-high-2024-p6-prelim-q15.png",
  "working": "Take 36 fruits. Bag A \\(= \\dfrac{3}{4} \\times 36 = 27\\); bag B \\(= 9\\). Pears in A \\(= \\dfrac{4}{9} \\times 27 = 12\\). Total pears \\(= 36 - \\dfrac{1}{2}\\times 36 = 18\\). Pears in B \\(= 18 - 12 = 6\\). Fraction of bag B that are pears \\(= \\dfrac{6}{9} = \\dfrac{2}{3}\\).",
  "answer": "\\(\\dfrac{2}{3}\\)",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2024 P6 Prelim Q22",
  "png": "p6/solutions/catholic-high-2024-p6-prelim-q22.png",
  "working": "After exchange the value is \\(17 \\times 20\\)¢ \\(= \\$3.40\\) (value is unchanged). If all 30 coins were 20¢ the value would be \\(30 \\times 20\\)¢ \\(= \\$6\\). The shortfall \\(\\$6 - \\$3.40 = \\$2.60\\) comes from the 10¢ coins, each 10¢ short. Number of 10¢ coins \\(= \\$2.60 \\div 10\\)¢ \\(= 26\\).",
  "answer": "26",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2024 P6 Prelim Q28",
  "png": "p6/solutions/catholic-high-2024-p6-prelim-q28.png",
  "working": "Total packets \\(= 40 \\times 6 = 240\\). Each pupil present sold \\(6 + 2 = 8\\), so pupils present \\(= 240 \\div 8 = 30\\). Pupils who did not turn up \\(= 40 - 30 = 10\\).",
  "answer": "10",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2024 P6 Prelim Q29",
  "png": "p6/solutions/catholic-high-2024-p6-prelim-q29.png",
  "working": "Non-red beads \\(= 1 - \\dfrac{3}{5} = \\dfrac{2}{5}\\) of the total. Blue : green \\(= 2 : 3\\), so blue \\(= \\dfrac{2}{5}\\) of the non-red \\(= \\dfrac{2}{5} \\times \\dfrac{2}{5} = \\dfrac{4}{25} = 16\\%\\).",
  "answer": "16%",
  "answer_matches_key": true
 },
 {
  "source": "Catholic High 2024 P6 Prelim Q30",
  "png": "p6/solutions/catholic-high-2024-p6-prelim-q30.png",
  "working": "After selling \\(\\dfrac{4}{9}\\) of the remainder, \\(\\dfrac{5}{9}\\) of the remainder is left, and this equals \\(\\dfrac{1}{3}\\) of the original. Matching denominators: \\(\\dfrac{1}{3} = \\dfrac{5}{15}\\), so the original is 15 units and the remainder (after the morning) is 9 units. The 42 cakes sold in the morning \\(= 15 - 9 = 6\\) units, so 1 unit \\(= 42 \\div 6 = 7\\). Cakes left \\(= 5\\) units \\(= 5 \\times 7 = 35\\).",
  "answer": "35",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2022 P6 Prelim Q11",
  "png": "p6/solutions/henry-park-2022-p6-prelim-q11.png",
  "working": "Green \\(=\\dfrac13\\) of total \\(=20\\), so 1 third \\(=20\\) and total \\(=60\\). Purple \\(+\\) red \\(=\\dfrac14\\) of \\(60=15\\). Split \\(15\\) in the ratio purple : red \\(=4:1\\), so \\(5\\) parts \\(=15\\), \\(1\\) part \\(=3\\). Red \\(=3\\).",
  "answer": "3",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2022 P6 Prelim Q12",
  "png": "p6/solutions/henry-park-2022-p6-prelim-q12.png",
  "working": "Make both boxes hold the same total. Box 1 is black : white \\(=2:1\\) (3 parts); Box 2 is \\(5:7\\) (12 parts). Scale Box 1 to 12 parts: \\(8\\) black : \\(4\\) white. Box 2: \\(5\\) black : \\(7\\) white. Grand total \\(=24\\) parts; white \\(=4+7=11\\). Fraction white \\(=\\dfrac{11}{24}\\).",
  "answer": "\\(\\dfrac{11}{24}\\)",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2022 P6 Prelim Q15",
  "png": "p6/solutions/henry-park-2022-p6-prelim-q15.png",
  "working": "Each had 60; total stays \\(180\\). Joan gave \\(\\dfrac25\\) of \\(60=24\\), keeping \\(36\\). Let Siti give \\(s\\); Joan \\(+\\) Siti left \\(=36+(60-s)=96-s\\). Xiuli \\(=60+24+s=84+s\\) and Xiuli \\(=3\\times(96-s)\\). So \\(84+s=288-3s\\Rightarrow 4s=204\\Rightarrow s=51\\).",
  "answer": "51",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2022 P6 Prelim Q23",
  "png": "p6/solutions/henry-park-2022-p6-prelim-q23.png",
  "working": "Two sisters got \\(2\\times\\dfrac16=\\dfrac13\\). Remaining \\(=1-\\dfrac13=\\dfrac23\\). Put aside for brothers \\(=\\dfrac23\\) of \\(\\dfrac23=\\dfrac49\\). Each brother got \\(\\dfrac19\\), so number of brothers \\(=\\dfrac49\\div\\dfrac19=4\\).",
  "answer": "4",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2022 P6 Prelim Paper 2 Q1",
  "png": "p6/solutions/henry-park-2022-p6-prelim-q31.png",
  "working": "All the money buys exactly \\(10\\) notebooks. Buying \\(6\\) notebooks leaves \\(4\\) notebooks' worth, which was spent on \\(4\\) pens (\\(4\\times\\$0.80=\\$3.20\\)) plus \\(\\$2.40\\) left. So \\(4\\) notebooks \\(=\\$3.20+\\$2.40=\\$5.60\\); \\(1\\) notebook \\(=\\$1.40\\); all money \\(=10\\times\\$1.40=\\$14\\).",
  "answer": "$14",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2022 P6 Prelim Paper 2 Q6",
  "png": "p6/solutions/henry-park-2022-p6-prelim-q36.png",
  "working": "Let Siti \\(=s\\), Jane \\(=s+432\\). After: Jane keeps \\(\\dfrac34(s+432)\\), Siti keeps \\(\\dfrac38 s\\). New difference \\(=\\dfrac34(s+432)-\\dfrac38 s=441\\). \\(\\dfrac34 s+324-\\dfrac38 s=441\\Rightarrow\\dfrac38 s=117\\Rightarrow s=312\\). Jane at first \\(=312+432=744\\).",
  "answer": "744",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2022 P6 Prelim Paper 2 Q7",
  "png": "p6/solutions/henry-park-2022-p6-prelim-q37.png",
  "working": "Empty containers are \\(100\\%-70\\%=30\\%=120\\), so \\(10\\%=40\\) and total \\(=400\\) containers. Filled \\(=70\\%\\) of \\(400=280\\). Volume per container \\(=1400\\div280=5\\) litres.",
  "answer": "5",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2022 P6 Prelim Paper 2 Q8",
  "png": "p6/solutions/henry-park-2022-p6-prelim-q38.png",
  "working": "In the same \\(35\\) min, distance ratio equals speed ratio, so Grace's speed \\(=\\dfrac57\\) of Jen's speed. Jen \\(-\\) Grace \\(=36\\) m/min, i.e. \\(\\dfrac27\\) of Jen's speed \\(=36\\), so \\(\\dfrac17=18\\) and Jen \\(=126\\). Grace \\(=\\dfrac57\\times126=90\\) m/min.",
  "answer": "90",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2022 P6 Prelim Paper 2 Q11",
  "png": "p6/solutions/henry-park-2022-p6-prelim-q42.png",
  "working": "Work backwards from C's \\(\\$1400\\). After B, what was left \\(=\\$1400\\); B left \\(\\dfrac34\\) of the post-A amount then took another \\(\\$850\\), so \\(\\dfrac34(\\text{after A})-850=1400\\Rightarrow\\dfrac34(\\text{after A})=2250\\Rightarrow\\text{after A}=3000\\). After A \\(=\\dfrac12(\\text{total})-1500=3000\\Rightarrow\\dfrac12\\,\\text{total}=4500\\Rightarrow\\text{total}=\\$9000\\).",
  "answer": "$9000",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2022 P6 Prelim Paper 2 Q13",
  "png": "p6/solutions/henry-park-2022-p6-prelim-q44.png",
  "working": "Last year men : women \\(=5:4\\). This year men \\(=5\\times1.30=6.5\\) units, women \\(=4\\times0.50=2\\) units, so this year \\(=8.5\\) units \\(=4913\\Rightarrow 1\\) unit \\(=578\\). Last year total \\(=9\\) units \\(=9\\times578=5202\\). Difference \\(=5202-4913=289\\).",
  "answer": "289",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2024 P6 Prelim P1 Q6",
  "png": "p6/solutions/henry-park-2024-p6-prelim-q6.png",
  "working": "Yellow is 7 units, brown is 3 units; the difference is \\(7 - 3 = 4\\) units \\(= 84\\), so 1 unit \\(= 21\\). Total \\(= 10\\) units \\(= 10 \\times 21 = 210\\) balls.",
  "answer": "210",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2024 P6 Prelim P1 Q13",
  "png": "p6/solutions/henry-park-2024-p6-prelim-q13.png",
  "working": "Bookshop A: not red \\(= 75\\%\\) of 200 \\(= 150\\). Bookshop B: not red \\(= 40\\%\\) of 600 \\(= 240\\). Total not red \\(= 150 + 240 = 390\\).",
  "answer": "390",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2024 P6 Prelim P1 Q15",
  "png": "p6/solutions/henry-park-2024-p6-prelim-q15.png",
  "working": "Take the total as 6 units: fiction \\(= 1\\) unit, non-fiction \\(= 5\\) units. Fiction \\(+100\\%\\) adds 1 unit; non-fiction \\(+50\\%\\) adds 2.5 units. Increase \\(= 3.5\\) units \\(= 168\\), so 1 unit \\(= 48\\). Books at first \\(= 6\\) units \\(= 288\\).",
  "answer": "288",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2024 P6 Prelim P1 Q19",
  "png": "p6/solutions/henry-park-2024-p6-prelim-q19.png",
  "working": "May : June : July \\(= 4 : 5 : 7\\). May \\(= 4\\) units \\(= 336\\), so 1 unit \\(= 84\\). June + July \\(= 5 + 7 = 12\\) units \\(= 12 \\times 84 = 1008\\).",
  "answer": "1008",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2024 P6 Prelim P1 Q23",
  "png": "p6/solutions/henry-park-2024-p6-prelim-q24.png",
  "working": "Total paint used \\(= 5 \\times 420 = 2100\\) ml. The paint left equals 2 containers' worth, so the 2100 ml used equals \\(5 - 2 = 3\\) containers. 3 containers \\(= 2100\\) ml, so 1 container \\(= 700\\) ml. 5 containers at first \\(= 5 \\times 700 = 3500\\) ml.",
  "answer": "3500 ml",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2024 P6 Prelim P1 Q24",
  "png": "p6/solutions/henry-park-2024-p6-prelim-q25.png",
  "working": "Member price per t-shirt \\(= 216 \\div 6 = \\$36\\). This $36 is 75% (after the further 25% off) of the price after the $15 discount, so that price \\(= 36 \\div 0.75 = \\$48\\). Original price \\(= 48 + 15 = \\$63\\).",
  "answer": "$63",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2024 P6 Prelim P1 Q28",
  "png": "p6/solutions/henry-park-2024-p6-prelim-q29.png",
  "working": "Of 500 cookies: chocolate \\(= \\dfrac{3}{5} \\times 500 = 300\\), butter \\(= \\dfrac{1}{4} \\times 500 = 125\\). Raisin \\(= 500 - 300 - 125 = 75\\). Sold \\(= \\dfrac{2}{5} \\times 75 = 30\\) raisin cookies.",
  "answer": "30",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2024 P6 Prelim P2 Q6",
  "png": "p6/solutions/henry-park-2024-p6-prelim-q37.png",
  "working": "Both spend the same amount, so the difference stays \\(210 - 154 = 56\\). After spending, Jason : Ruth \\(= 7 : 3\\); the difference is \\(7 - 3 = 4\\) units \\(= 56\\), so 1 unit \\(= 14\\). Jason kept \\(7\\) units \\(= 98\\). Each spent \\(= 210 - 98 = \\$112\\).",
  "answer": "$112",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2024 P6 Prelim P2 Q10a",
  "png": "p6/solutions/henry-park-2024-p6-prelim-q43.png",
  "working": "Group Box A's coins in fours: 1 unit is 10-cent and 3 units are 50-cent, worth \\(1(10) + 3(50) = 160\\) cents per group of 4. \\(\\$86.40 = 8640\\) cents, so number of groups \\(= 8640 \\div 160 = 54\\). Coins in Box A \\(= 54 \\times 4 = 216\\); Box B has the same number \\(= 216\\).",
  "answer": "216",
  "answer_matches_key": true
 },
 {
  "source": "Henry Park 2024 P6 Prelim P2 Q12b",
  "png": "p6/solutions/henry-park-2024-p6-prelim-q48.png",
  "working": "Shop Q sold 24 tubs for \\(15(\\$12) + 9(\\$18) = \\$342\\). Shop S sold the same 24 tubs but for \\(342 + 66 = \\$408\\). If all 24 were small \\(= \\$288\\); the extra \\(408 - 288 = \\$120\\) comes from big tubs at \\(\\$6\\) more each, so big \\(= 120 \\div 6 = 20\\) and small \\(= 24 - 20 = 4\\).",
  "answer": "4",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2023 P6 Prelim P1 Q9",
  "png": "p6/solutions/mgs-2023-p6-prelim-q9.png",
  "working": "Split the 36 pens into 6 equal units (each \\(=6\\) pens). Red \\(=\\dfrac{1}{3}=2\\) units \\(=12\\). Blue \\(=50\\%=3\\) units \\(=18\\). Green is the last \\(1\\) unit \\(=6\\). Green as a fraction of blue \\(=\\dfrac{6}{18}=\\dfrac{1}{3}\\).",
  "answer": "\\(\\dfrac{1}{3}\\)",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2023 P6 Prelim P1 Q12",
  "png": "p6/solutions/mgs-2023-p6-prelim-q12.png",
  "working": "Lisa: \\(3\\) files \\(+\\,3\\) pencils \\(=\\$18.90\\), so \\(1\\) file \\(+\\,1\\) pencil \\(=\\$6.30\\). Howard: \\(2\\) files \\(+\\,1\\) pencil \\(=\\$10.50\\). The difference is \\(1\\) extra file: \\(\\$10.50-\\$6.30=\\$4.20\\).",
  "answer": "$4.20",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2023 P6 Prelim P1 Q13",
  "png": "p6/solutions/mgs-2023-p6-prelim-q13.png",
  "working": "Bala : Chelsea \\(=10:11\\). Tianwei : Bala \\(=6:5=12:10\\). Lining up Bala at \\(10\\) units: Tianwei : Bala : Chelsea \\(=12:10:11\\), so Tianwei : Chelsea \\(=12:11\\).",
  "answer": "\\(12 : 11\\)",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2023 P6 Prelim P1 Q15",
  "png": "p6/solutions/mgs-2023-p6-prelim-q15.png",
  "working": "Take Dalia's start as \\(1\\) unit, so Huda \\(=1\\) unit \\(-30\\). After Dalia gives 40% then Huda gives 50%, the final difference works out to \\(0.6\\) of Dalia's start \\(=48\\). So Dalia's start \\(=48\\div0.6=80\\) stickers.",
  "answer": "80",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2023 P6 Prelim P2 Q3",
  "png": "p6/solutions/mgs-2023-p6-prelim-q33.png",
  "working": "\\(3\\) hours \\(=180\\) min. Each of the first two bedrooms and the base of the third are equal portions; the third has an extra \\(15\\) min. Remove the extra: \\(180-15=165\\) min over \\(3\\) equal portions, so \\(1\\) portion \\(=165\\div3=55\\) min. Third bedroom \\(=55+15=70\\) min.",
  "answer": "70 min",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2023 P6 Prelim P2 Q8",
  "png": "p6/solutions/mgs-2023-p6-prelim-q38.png",
  "working": "Each square is cut into \\(3\\) rectangles, adding \\(2\\) extra pieces per square. The total grew from \\(40\\) to \\(92\\), an increase of \\(52\\). Number of squares \\(=52\\div2=26\\). Rectangular papers bought \\(=40-26=14\\).",
  "answer": "14",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2023 P6 Prelim P2 Q13",
  "png": "p6/solutions/mgs-2023-p6-prelim-q43.png",
  "working": "The post office is the midpoint of Ali's and Nelson's houses, and the library is \\(360\\) m past it on Nelson's side. (a) Ali jogs the half-distance plus \\(360\\) m while Nelson jogs the half-distance minus \\(360\\) m, so Ali jogs \\(360\\times2=720\\) m more. (b) Ali's speed \\(=85+40=125\\) m/min; they jog the same time. Extra distance \\(\\div\\) speed difference \\(=720\\div40=18\\) min, so Nelson's distance \\(=85\\times18=1530\\) m.",
  "answer": "(a) 720 m (b) 1530 m",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2023 P6 Prelim P2 Q14",
  "png": "p6/solutions/mgs-2023-p6-prelim-q44.png",
  "working": "Two box lengths span the \\(150\\) cm height, so \\(1\\) length \\(=75\\) cm. The side gaps differ by \\(30-13=17\\) cm, which equals \\(1\\) length \\(-2\\) breadths. (a) The top gap \\(=17\\times2=34\\) cm. (b) \\(1\\) breadth \\(=(75-17)\\div2=29\\) cm; cupboard width \\(=75+29+13=117\\) cm \\(=1.17\\) m.",
  "answer": "(a) 34 cm (b) 1.17 m",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2023 P6 Prelim P2 Q15a",
  "png": "p6/solutions/mgs-2023-p6-prelim-q45.png",
  "working": "Total A : Total B \\(=2:3\\). Scale to \\(30:45\\) so each hall splits cleanly. Hall A girls : boys \\(=7:3\\) of \\(30\\) units \\(=21:9\\). Hall B girls : boys \\(=2:7\\) of \\(45\\) units \\(=10:35\\). So boys A : boys B \\(=9:35\\).",
  "answer": "\\(9 : 35\\)",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2023 P6 Prelim P2 Q15b",
  "png": "p6/solutions/mgs-2023-p6-prelim-q46.png",
  "working": "Combining both halls, total girls : total boys \\(=(21+10):(9+35)=31:44\\) (75 units). After \\(375\\) boys leave, girls are 62% and boys 38%. Girls stay \\(31\\) units \\(=62\\%\\), so \\(1\\%=\\dfrac{1}{2}\\) unit and boys \\(=38\\%=19\\) units. Boys who left \\(=44-19=25\\) units \\(=375\\), so \\(1\\) unit \\(=15\\). Boys remaining \\(=19\\times15=285\\).",
  "answer": "285",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2023 P6 Prelim P2 Q16",
  "png": "p6/solutions/mgs-2023-p6-prelim-q47.png",
  "working": "Money adult : child \\(=14:3\\), total \\(17\\) units \\(=\\$9180\\), so \\(1\\) unit \\(=\\$540\\). Adult money \\(=14\\times\\$540=\\$7560\\); child money \\(=3\\times\\$540=\\$1620\\). Since \\(\\dfrac{1}{5}\\) of tickets are child, adult : child tickets \\(=4:1\\), so each adult ticket \\(=\\$7560\\div4=\\$1890\\). The price gap \\(\\$1890-\\$1620=\\$270\\) equals \\(\\$9\\) per child ticket, so child tickets \\(=\\$270\\div\\$9=30\\).",
  "answer": "30",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2023 P6 Prelim P2 Q17",
  "png": "p6/solutions/mgs-2023-p6-prelim-q48.png",
  "working": "Each tray uses cupcakes : tarts \\(=3:5\\). To buy whole boxes (cupcakes in 4s, tarts in 3s), scale to \\(36:60\\): that is \\(9\\) boxes of cupcakes \\(+\\,20\\) boxes of tarts \\(=\\$2.10\\times9+\\$1.60\\times20=\\$18.90+\\$32.00=\\$50.90\\) per set. \\(\\$203.60\\div\\$50.90=4\\) sets. Tarts \\(=4\\times60=240\\).",
  "answer": "240",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2024 P6 Prelim Q15",
  "png": "p6/solutions/mgs-2024-p6-prelim-q15.png",
  "working": "Girl winners \\(= \\dfrac{4}{9}\\times 45 = 20\\); boy winners \\(= 45 - 20 = 25\\). \\(\\dfrac{1}{3}\\) of the boys won, so the boys split into 3 equal units of 25: boys \\(= 3\\times 25 = 75\\). \\(\\dfrac{1}{5}\\) of the girls won, so the girls split into 5 equal units of 20: girls \\(= 5\\times 20 = 100\\). Total children \\(= 75 + 100 = 175\\).",
  "answer": "175",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2024 P6 Prelim Q27",
  "png": "p6/solutions/mgs-2024-p6-prelim-q28.png",
  "working": "Both teams have the same number of students. Team A is \\(1 : 3\\) (4 parts); scale it by 4 to \\(4 : 12\\) (16 parts) so it matches Team B's \\(5 : 11\\) (16 parts). Total boys \\(= 4 + 5 = 9\\); total girls \\(= 12 + 11 = 23\\). Ratio of total boys to total girls \\(= 9 : 23\\).",
  "answer": "\\(9 : 23\\)",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2024 P6 Prelim P2 Q2",
  "png": "p6/solutions/mgs-2024-p6-prelim-q33.png",
  "working": "Mr Lim is 3 times as old as his son, so split his age into 3 equal units: \\(3\\) units \\(= 12n\\), \\(1\\) unit \\(= 4n\\). The son is \\(4n\\) now and will be 25, which is \\((25 - 4n)\\) years later. Both age by the same amount, so Mr Lim \\(= 12n + (25 - 4n) = 25 + 8n\\).",
  "answer": "\\((25 + 8n)\\) years",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2024 P6 Prelim P2 Q5",
  "png": "p6/solutions/mgs-2024-p6-prelim-q36.png",
  "working": "Cookies and Cream \\(= 60\\%\\), so Chocolate + Vanilla + Strawberry \\(= 40\\%\\) of \\(380 = 152\\) pupils. Let Vanilla \\(= 3\\) units, Strawberry \\(= 1\\) unit, and Chocolate \\(= 4\\) units (Strawberry + Vanilla). Total \\(= 8\\) units \\(= 152\\), so \\(1\\) unit \\(= 19\\). Vanilla \\(= 3\\times 19 = 57\\).",
  "answer": "57",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2024 P6 Prelim P2 Q8a",
  "png": "p6/solutions/mgs-2024-p6-prelim-q39.png",
  "working": "Split Aaron's marbles into 5 equal units. He gives \\(\\dfrac{2}{5}\\) (2 units) to Benny, leaving \\(\\dfrac{3}{5}\\) (3 units). To take \\(\\dfrac{1}{4}\\) of that remainder, cut each unit into 4, so \\(\\dfrac{3}{5} = \\dfrac{12}{20}\\); Charlie gets \\(\\dfrac{1}{4}\\times\\dfrac{3}{5} = \\dfrac{3}{20}\\). Aaron has \\(\\dfrac{12}{20} - \\dfrac{3}{20} = \\dfrac{9}{20}\\) left.",
  "answer": "\\(\\dfrac{9}{20}\\)",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2024 P6 Prelim P2 Q8b",
  "png": "p6/solutions/mgs-2024-p6-prelim-q40.png",
  "working": "Let Aaron start with 20 units. He gives \\(\\dfrac{2}{5}\\times 20 = 8\\) units to Benny, then \\(\\dfrac{1}{4}\\) of the remaining 12 \\(= 3\\) units to Charlie, keeping 9 units. All three end equal, so each ends with 9 units. Benny started with \\(9 - 8 = 1\\) unit; Charlie started with \\(9 - 3 = 6\\) units. At first, Aaron : Benny : Charlie \\(= 20 : 1 : 6\\).",
  "answer": "\\(20 : 1 : 6\\)",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2024 P6 Prelim P2 Q9",
  "png": "p6/solutions/mgs-2024-p6-prelim-q41.png",
  "working": "Each pupil first got an equal share of \\(u\\) cards. The \\(7\\) pupils who left gave away their \\(7u\\) cards, shared among the remaining \\(35 - 7 = 28\\) pupils, \\(3\\) more each: \\(28\\times 3 = 84 = 7u\\), so \\(u = 12\\). Total cards \\(= 35\\times 12 = 420\\).",
  "answer": "420",
  "answer_matches_key": true
 },
 {
  "source": "MGS 2024 P6 Prelim P2 Q13",
  "png": "p6/solutions/mgs-2024-p6-prelim-q45.png",
  "working": "(a) Shop X \\(= \\$450\\). Split into 5 equal units: \\(1\\) unit \\(= \\$90\\). Shop Y \\(= 80\\% = 4\\) units \\(= 4\\times \\$90 = \\$360\\). (b) Before the sale Shop X \\(-\\) Shop Y \\(= \\$450 - \\$360 = \\$90\\). The same percentage discount applies to both, so the gap stays in proportion; after the sale the gap is \\(\\$82.80\\): \\(\\dfrac{82.80}{90}\\times 100 = 92\\%\\) of the price remains, so the discount \\(= 100\\% - 92\\% = 8\\%\\).",
  "answer": "(a) $360, (b) 8%",
  "answer_matches_key": true
 },
 {
  "source": "Maris Stella 2023 P6 Prelim Q15",
  "png": "p6/solutions/maris-stella-2023-p6-prelim-q15.png",
  "working": "Men \\(=20\\%=1\\) unit \\(=100\\), so the whole \\(=5\\) units \\(=500\\). Remainder \\(=4\\) units \\(=400\\). Women \\(=55\\%\\) of \\(400=220\\). Children \\(=400-220=180\\).",
  "answer": "180",
  "answer_matches_key": true
 },
 {
  "source": "Maris Stella 2023 P6 Prelim Q24",
  "png": "p6/solutions/maris-stella-2023-p6-prelim-q24.png",
  "working": "Kingston \\(=8\\) units, Tim \\(=5\\) units. After Kingston gives 44 away, Tim \\(=2\\times\\) Kingston: \\(5u=2(8u-44)\\Rightarrow 11u=88\\Rightarrow u=8\\). Total at first \\(=13u=13\\times 8=104\\).",
  "answer": "104",
  "answer_matches_key": true
 },
 {
  "source": "Maris Stella 2023 P6 Prelim P2 Q3",
  "png": "p6/solutions/maris-stella-2023-p6-prelim-q33.png",
  "working": "After the brother takes \\(\\dfrac{5}{8}\\) of the remainder, \\(\\dfrac{3}{8}\\) of the remainder is left, which equals \\(\\dfrac14\\) of the salary. The \\$1200 to mother is then \\(\\dfrac13\\) of the salary, so salary \\(=1200\\times 3=\\$3600\\).",
  "answer": "$3600",
  "answer_matches_key": true
 },
 {
  "source": "Maris Stella 2023 P6 Prelim P2 Q12",
  "png": "p6/solutions/maris-stella-2023-p6-prelim-q42.png",
  "working": "Work backwards. After the 3 siblings: \\(100+3\\times 40=220\\). Before Troy: \\((220+30)\\times 2=500\\). Before Samuel: \\((500+60)\\times 2=1120\\). Peter had 1120 at first.",
  "answer": "1120",
  "answer_matches_key": true
 },
 {
  "source": "Maris Stella 2023 P6 Prelim P2 Q13",
  "png": "p6/solutions/maris-stella-2023-p6-prelim-q43.png",
  "working": "Men \\(+\\) women \\(=400\\). Remaining \\(=\\dfrac23 m+\\dfrac25 w=224\\). Times \\(\\dfrac32\\): \\(m+\\dfrac35 w=336\\). Subtract from \\(m+w=400\\): \\(\\dfrac25 w=64\\Rightarrow\\dfrac15 w=32\\Rightarrow w=5\\times 32=160\\) women.",
  "answer": "160",
  "answer_matches_key": true
 },
 {
  "source": "Maris Stella 2023 P6 Prelim P2 Q15a",
  "png": "p6/solutions/maris-stella-2023-p6-prelim-q45.png",
  "working": "Chocolate : strawberry \\(=3:5\\). The difference \\(=2\\) units \\(=28\\), so \\(1\\) unit \\(=14\\). Strawberry \\(=5\\) units \\(=70\\). Vanilla \\(=\\) strawberry \\(+10=70+10=80\\).",
  "answer": "80",
  "answer_matches_key": true
 },
 {
  "source": "Maris Stella 2024 P6 Prelim Q12",
  "png": "p6/solutions/maris-stella-2024-p6-prelim-q12.png",
  "working": "The 60 blue chairs do not change. Since red becomes 60% of the total, blue is 40% (4 units). 4 units \\(= 60\\), so 1 unit \\(= 15\\) and the total \\(= 10\\) units \\(= 150\\). Red chairs \\(= 150 - 60 = 90\\).",
  "answer": "90",
  "answer_matches_key": true
 },
 {
  "source": "Maris Stella 2024 P6 Prelim Q21",
  "png": "p6/solutions/maris-stella-2024-p6-prelim-q21.png",
  "working": "The two whole numbers have sum 19 and product 60, so they are 15 and 4 (\\(15 \\times 4 = 60\\), \\(15 + 4 = 19\\)). Drawing them as two bars, the difference \\(= 15 - 4 = 11\\).",
  "answer": "11",
  "answer_matches_key": true
 },
 {
  "source": "Maris Stella 2024 P6 Prelim Q25",
  "png": "p6/solutions/maris-stella-2024-p6-prelim-q25.png",
  "working": "Ali spent \\(\\dfrac{3}{5}\\), so he kept \\(\\dfrac{2}{5}\\) (2 units) \\(= \\$72\\); 1 unit \\(= \\$36\\) and Ali had 5 units \\(= \\$180\\). Bala spent \\(\\dfrac{4}{7}\\), so he kept \\(\\dfrac{3}{7}\\) (3 units) \\(= \\$72\\); 1 unit \\(= \\$24\\) and Bala had 7 units \\(= \\$168\\). Difference \\(= 180 - 168 = \\$12\\).",
  "answer": "\\($12\\)",
  "answer_matches_key": true
 },
 {
  "source": "Maris Stella 2024 P6 Prelim Q29",
  "png": "p6/solutions/maris-stella-2024-p6-prelim-q29.png",
  "working": "Picture the number as 24 equal units. Dividing by 8 gives 3 units; dividing by 6 gives 4 units. The wrong answer is 1 unit more than the correct one, and that 1 unit \\(= 4\\). So the number \\(= 24\\) units \\(= 24 \\times 4 = 96\\).",
  "answer": "96",
  "answer_matches_key": true
 },
 {
  "source": "Maris Stella 2024 P6 Prelim P2 Q10",
  "png": "p6/solutions/maris-stella-2024-p6-prelim-q40.png",
  "working": "Total boys \\(= 2298 - 1118 = 1180\\). Girls in School B are \\(\\dfrac{5}{8}\\) of girls in School A, so split the 1118 girls into \\(8 + 5 = 13\\) equal units. 13 units \\(= 1118\\), 1 unit \\(= 86\\). Girls in School A \\(= 8\\) units \\(= 688\\). (Boys + girls in School A \\(= 637 + 688 = 1325\\).)",
  "answer": "688",
  "answer_matches_key": true
 },
 {
  "source": "Maris Stella 2024 P6 Prelim P2 Q12",
  "png": "p6/solutions/maris-stella-2024-p6-prelim-q42.png",
  "working": "Total height over the 4 months \\(= 16.5 \\times 4 = 66\\) cm. From the graph the heights are in the ratio \\(2 : 3 : 6 : 11\\) (Jan : Feb : Mar : Apr), a total of \\(2 + 3 + 6 + 11 = 22\\) units. 22 units \\(= 66\\) cm, so 1 unit \\(= 3\\) cm. March \\(= 6\\) units \\(= 18\\) cm.",
  "answer": "18",
  "answer_matches_key": true
 },
 {
  "source": "Maris Stella 2024 P6 Prelim P2 Q15",
  "png": "p6/solutions/maris-stella-2024-p6-prelim-q45.png",
  "working": "Children boys : girls \\(= 10 : 7\\), so \\(10 + 7 = 17\\) units. Each child paid \\$12, so 17 units \\(= 17 \\times \\$12 = \\$204\\); number of groups \\(= 4692 \\div 204 = 23\\). Boys \\(= 23 \\times 10 = 230\\). For part (b) males : females \\(= 8 : 5\\); the \\(8 - 5 = 3\\)-unit difference equals 54 more men than women, so 1 unit \\(= 18\\). Females \\(= 5\\) units \\(= 90\\) (key reports 90).",
  "answer": "90",
  "answer_matches_key": true
 },
 {
  "source": "Maris Stella 2024 P6 Prelim P2 Q17",
  "png": "p6/solutions/maris-stella-2024-p6-prelim-q47.png",
  "working": "Match the 20¢ coins with an equal number of 10¢ coins; the 10¢ bar has 34 extra coins worth \\(34 \\times 10\\text{¢} = 340\\text{¢} = \\$3.40\\). Remove that: \\(\\$13.30 - \\$3.40 = \\$9.90\\). The matched coins cost \\(10\\text{¢} + 20\\text{¢} = 30\\text{¢}\\) per pair, so pairs \\(= \\$9.90 \\div \\$0.30 = 33\\). Number of 10¢ coins \\(= 33 + 34 = 67\\).",
  "answer": "67",
  "answer_matches_key": true
 },
 {
  "source": "Nan Chiau 2022 P6 Prelim Q10",
  "png": "p6/solutions/nan-chiau-2022-p6-prelim-q10.png",
  "working": "Total time \\(= 72 \\times 4 = 288\\) min. Anna \\(= 288 - (80 + 74 + 70) = 288 - 224 = 64\\) min.",
  "answer": "64 min",
  "answer_matches_key": true
 },
 {
  "source": "Nan Chiau 2022 P6 Prelim Q11",
  "png": "p6/solutions/nan-chiau-2022-p6-prelim-q11.png",
  "working": "Left with \\(\\dfrac{4}{5}\\), so spent \\(= 1 - \\dfrac{4}{5} = \\dfrac{1}{5}\\) of \\(\\$350 = \\$70\\) over 5 days. Per day \\(= \\$70 \\div 5 = \\$14\\).",
  "answer": "$14",
  "answer_matches_key": true
 },
 {
  "source": "Nan Chiau 2022 P6 Prelim Q14",
  "png": "p6/solutions/nan-chiau-2022-p6-prelim-q14.png",
  "working": "\\(\\$45\\) is 90% of the usual price, so 10% \\(= \\$5\\) and usual price \\(= \\$50\\). Final price \\(= \\$45 - \\$9 = \\$36\\). Total discount \\(= \\$50 - \\$36 = \\$14 = \\dfrac{14}{50} \\times 100\\% = 28\\%\\).",
  "answer": "28%",
  "answer_matches_key": true
 },
 {
  "source": "Nan Chiau 2022 P6 Prelim Q22",
  "png": "p6/solutions/nan-chiau-2022-p6-prelim-q22.png",
  "working": "Not blue \\(= 1 - \\dfrac{7}{10} = \\dfrac{3}{10}\\). Pink : white \\(= 2 : 1\\), so white \\(= \\dfrac{1}{3}\\) of \\(\\dfrac{3}{10} = \\dfrac{1}{10}\\).",
  "answer": "\\(\\dfrac{1}{10}\\)",
  "answer_matches_key": true
 },
 {
  "source": "Nan Chiau 2022 P6 Prelim Q23",
  "png": "p6/solutions/nan-chiau-2022-p6-prelim-q23.png",
  "working": "Shop B selling price \\(= \\$1500 \\times \\dfrac{80}{100} = \\$1200\\). Shop A sells at the same \\(\\$1200\\) after 40% off, so \\(\\$1200\\) is 60% of A's usual price: \\(\\$1200 \\div 60 \\times 100 = \\$2000\\).",
  "answer": "$2000",
  "answer_matches_key": true
 },
 {
  "source": "Nan Chiau 2022 P6 Prelim Q32",
  "png": "p6/solutions/nan-chiau-2022-p6-prelim-q32.png",
  "working": "Red : Blue \\(= 2 : 11\\). Difference \\(= 11 - 2 = 9\\) units \\(= 1953\\), so 1 unit \\(= 1953 \\div 9 = 217\\). Red \\(= 2\\) units \\(= 2 \\times 217 = 434\\).",
  "answer": "434",
  "answer_matches_key": true
 },
 {
  "source": "Nan Chiau 2022 P6 Prelim Q36",
  "png": "p6/solutions/nan-chiau-2022-p6-prelim-q36.png",
  "working": "9 notebooks + 54 pencils \\(= \\$64.80 =\\) 24 notebooks, so 54 pencils \\(=\\) 15 notebooks. 1 notebook \\(= \\$64.80 \\div 24 = \\$2.70\\); 15 notebooks \\(= \\$40.50 =\\) 54 pencils, so 1 pencil \\(= \\$0.75\\). Pencils for \\(\\$64.80 = 64.80 \\div 0.75 = 86.4 \\to\\) at most 86.",
  "answer": "86 pencils",
  "answer_matches_key": true
 },
 {
  "source": "Nan Chiau 2022 P6 Prelim Q41",
  "png": "p6/solutions/nan-chiau-2022-p6-prelim-q41.png",
  "working": "A : (B+C) \\(= 3 : 5 \\to\\) scale to total 40: A \\(= 15\\), B+C \\(= 25\\). C : (A+B) \\(= 2 : 3 \\to\\) scale to total 40: C \\(= 16\\), A+B \\(= 24\\). So A \\(= 15\\), C \\(= 16\\), B \\(= 40 - 15 - 16 = 9\\). C \\(-\\) B \\(= 16 - 9 = 7\\) units \\(= \\$21\\), so 1 unit \\(= \\$3\\). Beth paid the least \\(= 9\\) units \\(= \\$27\\).",
  "answer": "$27",
  "answer_matches_key": true
 },
 {
  "source": "Nan Chiau 2022 P6 Prelim Q42",
  "png": "p6/solutions/nan-chiau-2022-p6-prelim-q42.png",
  "working": "Pineapple sold \\(= 100\\% - 75\\% = 25\\%\\) of all sold \\(= 350\\), so sold \\(= 1400\\) and total \\(= 1400 \\div \\dfrac{7}{10} = 2000\\). Unsold \\(= \\dfrac{3}{10}\\) of \\(2000 = 600\\). Pineapple unsold \\(= 30\\%\\) of \\(600 = 180\\).",
  "answer": "180",
  "answer_matches_key": true
 },
 {
  "source": "Nan Chiau 2022 P6 Prelim Q45",
  "png": "p6/solutions/nan-chiau-2022-p6-prelim-q45.png",
  "working": "Small boxes hold \\(\\dfrac{7}{9}\\), large boxes hold \\(\\dfrac{2}{9}\\) of the cookies, so small total : large total \\(= 7 : 2\\). Let each small box \\(= s\\), each large box \\(= s + 4\\). Then \\(14s : 3(s+4) = 7 : 2\\), so \\(2 \\times 14s = 7 \\times 3(s+4)\\): \\(28s = 21s + 84\\), \\(7s = 84\\), \\(s = 12\\). Each small box held 12 cookies.",
  "answer": "12 cookies",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2022 P6 Prelim Q5",
  "png": "p6/solutions/nan-hua-2022-p6-prelim-q5.png",
  "working": "Chinese \\(+\\) Indian \\(= 70 + 30 = 100\\). Malay \\(= 40\\). Using equal units of 20: Malay is 2 units, Chinese\\(+\\)Indian is 5 units. Ratio \\(= 2 : 5\\).",
  "answer": "2 : 5",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2022 P6 Prelim Q6",
  "png": "p6/solutions/nan-hua-2022-p6-prelim-q6.png",
  "working": "Split the number into 10 equal units of 10%. \\(40\\%\\) (4 units) \\(= 36\\), so 1 unit \\(= 36 \\div 4 = 9\\). The whole number \\(= 10\\) units \\(= 9 \\times 10 = 90\\).",
  "answer": "90",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2022 P6 Prelim Q7",
  "png": "p6/solutions/nan-hua-2022-p6-prelim-q7.png",
  "working": "January \\(= \\$40\\) (5 units of \\$8). February \\(= \\$32\\) (4 units). Decrease \\(= 1\\) unit \\(= \\$8\\). Percentage decrease \\(= \\dfrac{8}{40} \\times 100\\% = 20\\%\\).",
  "answer": "20",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2022 P6 Prelim Q13",
  "png": "p6/solutions/nan-hua-2022-p6-prelim-q13.png",
  "working": "Total punch \\(= \\dfrac{1}{10} + \\dfrac{4}{5} = \\dfrac{1}{10} + \\dfrac{8}{10} = \\dfrac{9}{10}\\) ℓ. Each bottle holds \\(\\dfrac{1}{5} = \\dfrac{2}{10}\\) ℓ. \\(\\dfrac{9}{10}\\) makes 4 full bottles (\\(\\dfrac{8}{10}\\) ℓ) with \\(\\dfrac{1}{10}\\) ℓ left over.",
  "answer": "\\(\\dfrac{1}{10}\\) ℓ",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2022 P6 Prelim Q24",
  "png": "p6/solutions/nan-hua-2022-p6-prelim-q24.png",
  "working": "Take the whole box as 16 units (sixteenths). Red \\(= \\dfrac{3}{8} = \\dfrac{6}{16}\\) (6 units). Remaining \\(= \\dfrac{10}{16}\\) (10 units). Green \\(= \\dfrac{3}{10}\\) of remaining \\(= 3\\) units \\(= \\dfrac{3}{16}\\). Blue \\(+\\) black \\(= 7\\) units \\(= \\dfrac{7}{16}\\). Blue \\(=\\) black, so blue \\(= \\dfrac{7}{16} \\div 2 = \\dfrac{7}{32}\\).",
  "answer": "\\(\\dfrac{7}{32}\\)",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2023 P6 Prelim Q14",
  "png": "p6/solutions/nan-hua-2023-p6-prelim-q14.png",
  "working": "Total \\(= 50\\). Blue + Red \\(= \\dfrac{2}{5} \\times 50 = 20\\). Red + White \\(= \\dfrac{9}{10} \\times 50 = 45\\). The two groups together count Red twice: \\(20 + 45 = 65\\). Subtract the total: Red \\(= 65 - 50 = 15\\).",
  "answer": "15",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2023 P6 Prelim Q15",
  "png": "p6/solutions/nan-hua-2023-p6-prelim-q15.png",
  "working": "Whole school \\(= 100\\%\\): Boys \\(= 40\\%\\), Girls \\(= 60\\%\\). Boys who walk \\(= 5\\%\\) of \\(40\\% = 2\\%\\). Girls who walk \\(= 20\\%\\) of \\(60\\% = 12\\%\\). Walkers \\(= 2\\% + 12\\% = 14\\%\\).",
  "answer": "14%",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2023 P6 Prelim Q26",
  "png": "p6/solutions/nan-hua-2023-p6-prelim-q26.png",
  "working": "Jack: \\(7 \\times 2 = 14\\) km. Keith: \\(5 \\times 2 = 10\\) km. They go opposite ways, so distance apart \\(= 14 + 10 = 24\\) km.",
  "answer": "24 km",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2023 P6 Prelim Q28",
  "png": "p6/solutions/nan-hua-2023-p6-prelim-q28.png",
  "working": "\\(X : Y = 3 : 7\\), so \\(X = 3u\\), \\(Y = 7u\\). After: \\(X + 4\\) equals half of \\(Y\\) (\\(3.5u\\)), since the new ratio is \\(1 : 1\\). So \\(3u + 4 = 3.5u\\), giving \\(0.5u = 4\\), \\(u = 8\\). \\(X = 3u = 3 \\times 8 = 24\\).",
  "answer": "24",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2023 P6 Prelim Q29",
  "png": "p6/solutions/nan-hua-2023-p6-prelim-q29.png",
  "working": "Same money buys 6 rulers + 3 erasers OR 4 rulers + 8 erasers. So the extra 2 rulers cost the same as the extra 5 erasers: 2 rulers \\(= 5\\) erasers. Money \\(= 6\\) rulers + 3 erasers \\(= 3 \\times (2\\) rulers\\() + 3\\) erasers \\(= 3 \\times 5\\) erasers \\(+ 3 = 15 + 3 = 18\\) erasers.",
  "answer": "18",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2023 P6 Prelim P2 Q1",
  "png": "p6/solutions/nan-hua-2023-p6-prelim-q31.png",
  "working": "Lighter boy \\(=\\) (total \\(-\\) difference) \\(\\div 2 = (72.9 - 9.3) \\div 2 = 63.6 \\div 2 = 31.8\\) kg.",
  "answer": "31.8 kg",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2023 P6 Prelim P2 Q6",
  "png": "p6/solutions/nan-hua-2023-p6-prelim-q36.png",
  "working": "\\(3 \\div \\dfrac{2}{5} = 3 \\times \\dfrac{5}{2} = \\dfrac{15}{2} = 7\\tfrac{1}{2}\\), so there are 7 full \\(\\dfrac{2}{5}\\) m pieces. Used \\(= 7 \\times \\dfrac{2}{5} = \\dfrac{14}{5} = 2.8\\) m. Last piece \\(= 3 - 2.8 = 0.2\\) m.",
  "answer": "(a) 7 pieces; (b) \\(0.2\\) m",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2023 P6 Prelim P2 Q7",
  "png": "p6/solutions/nan-hua-2023-p6-prelim-q37.png",
  "working": "Original: Boys \\(= 70\\%\\), Girls \\(= 30\\%\\). New boys \\(= 120\\% \\times 70\\% = 84\\%\\); new girls \\(= 150\\% \\times 30\\% = 45\\%\\). New total \\(= 84\\% + 45\\% = 129\\%\\) of original. Increase \\(= 29\\% = 87\\), so \\(1\\% = 3\\). End total \\(= 129 \\times 3 = 387\\).",
  "answer": "387",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2023 P6 Prelim P2 Q13",
  "png": "p6/solutions/nan-hua-2023-p6-prelim-q43.png",
  "working": "Tim : Peter \\(= 2 : 3\\) (5 units). The total stays the same. After Tim \\(+\\$120\\) and Peter \\(-\\$120\\), Tim : Peter \\(= 2 : 1\\). Match the unchanged total of 15 parts: before \\(6 : 9\\), after \\(10 : 5\\). The shift is \\(4\\) parts \\(= \\$120\\), so \\(1\\) part \\(= \\$30\\). Tim at first \\(= 6 \\times \\$30 = \\$180\\).",
  "answer": "$180",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2024 P6 Prelim Q10",
  "png": "p6/solutions/nan-hua-2024-p6-prelim-q10.png",
  "working": "Split the audience into 9 equal units: adults \\(= 5\\) units, children \\(= 4\\) units. The children split into \\(\\dfrac{1}{4}\\) boys (1 unit) and \\(\\dfrac{3}{4}\\) girls (3 units). So girls \\(= 3\\) units, adults \\(= 5\\) units, giving girls : adults \\(= 3 : 5\\).",
  "answer": "3 : 5",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2024 P6 Prelim Q11",
  "png": "p6/solutions/nan-hua-2024-p6-prelim-q11.png",
  "working": "Total money \\(= 5\\) units (cap is 20% \\(= 1\\) unit; the rest is 4 units). The rest (4 units) buys the bag and the shirt. Bag \\(= \\text{cap} + \\$15 = 1\\) unit \\(+ \\$15\\); shirt \\(= \\$165\\). So 1 unit \\(+ \\$15 + \\$165 = 4\\) units, giving 3 units \\(= \\$180\\), 1 unit \\(= \\$60\\). Bag \\(= \\$60 + \\$15 = \\$75\\).",
  "answer": "$75",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2024 P6 Prelim Q12",
  "png": "p6/solutions/nan-hua-2024-p6-prelim-q12.png",
  "working": "Split the ingredients into 3 equal units. 60 big cupcakes use \\(\\dfrac{60}{90} = \\dfrac{2}{3}\\) of the ingredients (2 units), leaving 1 unit \\(= \\dfrac{1}{3}\\). A full batch of small cupcakes is 150, so \\(\\dfrac{1}{3} \\times 150 = 50\\) small cupcakes.",
  "answer": "50",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2024 P6 Prelim Q24",
  "png": "p6/solutions/nan-hua-2024-p6-prelim-q24.png",
  "working": "The same total meal cost is split two ways. Correctly it is shared among 8 people; by mistake it was shared among 6 people, so each of those 6 shares is bigger. Each person paid \\$4 more, so the total over-collected \\(= \\$4 \\times 6 = \\$24\\). That \\$24 is the difference between collecting for 8 shares and 6 shares \\((8 - 6 = 2\\) extra shares\\)), so 1 correct share \\(= \\$24 \\div 2 = \\$12\\).",
  "answer": "$12",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2024 P6 Prelim Q26",
  "png": "p6/solutions/nan-hua-2024-p6-prelim-q26.png",
  "working": "Take the 1 m of string as 20 twentieths. Jane cuts \\(\\dfrac{1}{4}\\) m \\(= \\dfrac{5}{20}\\) m to tie the box, leaving \\(\\dfrac{15}{20} = \\dfrac{3}{4}\\) m. Each piece is \\(\\dfrac{1}{5}\\) m \\(= \\dfrac{4}{20}\\) m, and \\(\\dfrac{15}{20} \\div \\dfrac{4}{20} = 3\\dfrac{3}{4}\\). Only whole pieces count, so the maximum is 3 pieces (with \\(\\dfrac{3}{20}\\) m left over).",
  "answer": "3",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2024 P6 Prelim P2 Q6",
  "png": "p6/solutions/nan-hua-2024-p6-prelim-q36.png",
  "working": "Let Alice's money be 1 unit; then Diana's money is 4 units. Both want the same dress. Dress \\(= \\text{Alice} + \\$28.40 = 1\\) unit \\(+ \\$28.40\\); dress \\(= \\text{Diana} + \\$15.50 = 4\\) units \\(+ \\$15.50\\). Setting these equal: 3 units \\(= \\$28.40 - \\$15.50 = \\$12.90\\), so 1 unit \\(= \\$4.30\\). Dress \\(= \\$4.30 + \\$28.40 = \\$32.70\\).",
  "answer": "$32.70",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2024 P6 Prelim P2 Q9",
  "png": "p6/solutions/nan-hua-2024-p6-prelim-q39.png",
  "working": "Non-fiction does not change, so make its share equal in both ratios. Before fiction : non-fiction \\(= 8 : 5 = 32 : 20\\); after \\(= 9 : 4 = 45 : 20\\). After, fiction \\(= 45\\) units \\(= 12\\,600\\), so 1 unit \\(= 280\\). Fiction before \\(= 32\\) units; fiction added \\(= 45 - 32 = 13\\) units \\(= 13 \\times 280 = 3640\\).",
  "answer": "3640",
  "answer_matches_key": true
 },
 {
  "source": "Nan Hua 2024 P6 Prelim P2 Q15",
  "png": "p6/solutions/nan-hua-2024-p6-prelim-q45.png",
  "working": "Build the money ratio. Senior amount \\(= \\dfrac{1}{4}\\) of adult amount, so adult : senior \\(= 24 : 6\\); children : senior \\(= 7 : 6\\). So adults : children : seniors (by money) \\(= 24 : 7 : 6\\), total 37 units \\(= \\$2220\\), giving 1 unit \\(= \\$60\\). Adults' money \\(= 24\\) units \\(= \\$1440\\); number of adults \\(= \\$1440 \\div \\$48 = 30\\). People ratio is \\(5 : 2 : 3\\); adults \\(= 5\\) parts \\(= 30\\), so 1 part \\(= 6\\) and total \\(= 10\\) parts \\(= 60\\) people.",
  "answer": "60",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Q8",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q8.png",
  "working": "Huiling \\(= z\\), Ravi \\(= 2z\\), Jas \\(= 2z + 5\\). Jas \\(= \\$10\\), so \\(2z + 5 = 10\\), \\(2z = 5\\), \\(z = 2.5\\). Huiling had \\(\\$2.50\\).",
  "answer": "$2.50",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Q14",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q14.png",
  "working": "Each girl used the SAME number of beads. Make the used parts equal at 21 units. Viv used \\(\\dfrac{1}{3}\\) so total \\(= 3 \\times 21 = 63\\); Wendy used \\(\\dfrac{7}{8}\\) so total \\(= \\dfrac{8}{7} \\times 21 = 24\\); Xinyi used \\(\\dfrac{3}{4}\\) so total \\(= \\dfrac{4}{3} \\times 21 = 28\\). Ratio \\(= 63 : 24 : 28\\).",
  "answer": "\\(63 : 24 : 28\\)",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Q16",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q16.png",
  "working": "Red \\(= 6 + 2 = 8\\). Total \\(= 6 + 3 + 2 + 4 = 15\\). Fraction red \\(= \\dfrac{8}{15}\\).",
  "answer": "\\(\\dfrac{8}{15}\\)",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Q24",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q24.png",
  "working": "2021 monthly savings \\(= 20\\% \\times \\$3000 = \\$600\\). The increase is \\(\\$180\\). Percentage increase \\(= \\dfrac{180}{600} \\times 100\\% = 30\\%\\).",
  "answer": "30",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Q26",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q26.png",
  "working": "After Monday, \\(\\dfrac{2}{3}\\) of the apples remain. On Tuesday she sells \\(\\dfrac{2}{3}\\) of that remainder, so \\(\\dfrac{1}{3}\\) of the remainder \\(= 14\\) apples. Monday remainder \\(= 14 \\times 3 = 42\\), which is \\(\\dfrac{2}{3}\\) of the original. Original \\(= 42 \\div \\dfrac{2}{3} = 42 \\times \\dfrac{3}{2} = 63\\).",
  "answer": "63",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Q27",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q27.png",
  "working": "Remaining \\(= 1 - \\dfrac{1}{5} = \\dfrac{4}{5}\\) m. Number of \\(\\dfrac{1}{8}\\)-m pieces \\(= \\dfrac{4}{5} \\div \\dfrac{1}{8} = \\dfrac{4}{5} \\times 8 = \\dfrac{32}{5} = 6\\dfrac{2}{5}\\). At most 6 whole pieces.",
  "answer": "6",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Q30",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q30.png",
  "working": "Pam \\(= m = 100\\) kg/month. Sweet \\(= m - 30 = 70\\) kg/month. Combined per month \\(= 100 + 70 = 170\\) kg. One year \\(= 170 \\times 12 = 2040\\) kg.",
  "answer": "2040",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Paper 2 Q1",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q31.png",
  "working": "15 books at \\(\\$k\\) each, less a \\(\\$10\\) discount, cost \\(\\$110\\). So \\(15k - 10 = 110\\), \\(15k = 120\\), \\(k = 8\\). The original price of one book was \\(\\$8\\).",
  "answer": "$8",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Paper 2 Q4",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q34.png",
  "working": "Dana 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\\).",
  "answer": "850",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Paper 2 Q7",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q37.png",
  "working": "At 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\\).",
  "answer": "$23.70",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Paper 2 Q8",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q38.png",
  "working": "Blue 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}\\).",
  "answer": "\\(\\dfrac{21}{111}\\)",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Paper 2 Q9",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q39.png",
  "working": "\\(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\\).",
  "answer": "240",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Paper 2 Q11",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q41.png",
  "working": "(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.",
  "answer": "(a) 42000  (b) 12.5",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Paper 2 Q14a",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q44.png",
  "working": "Gold : (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\\).",
  "answer": "\\(40 : 27 : 9\\)",
  "answer_matches_key": true
 },
 {
  "source": "Nanyang 2022 P6 Prelim Paper 2 Q14b",
  "png": "p6/solutions/nanyang-2022-p6-prelim-q45.png",
  "working": "Copper 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\\).",
  "answer": "45",
  "answer_matches_key": true
 }
]