{
  "source_prefix": "ACS 2022 P6 Prelim",
  "models": [
    {
      "n": 4,
      "answer": "38 cm",
      "answer_matches_key": true,
      "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.",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "A + B", "segments": [{"u": 2, "text": "96 cm", "fill": "#dbe9ff"}]},
          {"label": "C + D", "segments": [{"u": 2, "text": "56 cm", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "152 cm ÷ 4 = 38 cm each", "bars": [0, 1]}
      }
    },
    {
      "n": 14,
      "answer": "\\(\\dfrac{18}{55}\\)",
      "answer_matches_key": true,
      "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}\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Red", "segments": [{"u": 2, "text": "12", "fill": "#ffd6d6"}]},
          {"label": "Blue", "segments": [{"u": 3, "text": "18", "fill": "#dbe9ff"}]},
          {"label": "Yellow", "segments": [{"u": 5, "text": "25", "fill": "#fff3b0"}]}
        ],
        "total": {"text": "Total = 55", "bars": [0, 1, 2]}
      }
    },
    {
      "n": 22,
      "answer": "$35",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Bag", "segments": [
            {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}],
            "brace": {"text": "?"}},
          {"label": "Pencil case 1", "segments": [{"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}]},
          {"label": "Pencil case 2", "segments": [{"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "$63 (9 units)", "bars": [0, 1, 2]}
      }
    },
    {
      "n": 26,
      "answer": "145 books",
      "answer_matches_key": true,
      "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.",
      "spec": {
        "unit_px": 50,
        "bars": [
          {"label": "Mystery (38%)", "segments": [{"u": 8, "fill": "#dbe9ff"}, {"u": 30, "text": "+150", "fill": "#ffd6d6"}]},
          {"label": "Humour (8%)", "segments": [{"u": 8, "fill": "#dbe9ff"}]},
          {"label": "Adventure (29%)", "segments": [{"u": 29, "text": "? = 145", "fill": "#fff3b0"}]}
        ]
      }
    },
    {
      "n": 30,
      "answer": "$120",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "July", "segments": [{"u": 2, "text": "spent $80", "fill": "#ffd6d6"}, {"u": 1, "text": "saved $40", "fill": "#cdeccd"}], "brace": {"text": "pocket money = ?"}},
          {"label": "August", "segments": [{"u": 2, "text": "spent $72", "fill": "#ffd6d6"}, {"u": 1, "text": "saved $48 (+20%)", "fill": "#cdeccd"}]}
        ]
      }
    },
    {
      "n": 31,
      "answer": "286",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 55,
        "bars": [
          {"label": "Curry (first)", "segments": [
            {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "text": "26 sold", "fill": "#ffd6d6"}]},
          {"label": "Tuna", "segments": [{"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}]}
        ],
        "total": {"text": "11 units = 286", "bars": [0, 1]}
      }
    },
    {
      "n": 34,
      "answer": "4.8 kg",
      "answer_matches_key": true,
      "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.",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "At first (10 tenths)", "segments": [
            {"u": 1, "text": "d1-8", "fill": "#ffd6d6"}, {"u": 1, "text": "", "fill": "#ffd6d6"}, {"u": 1, "text": "", "fill": "#ffd6d6"}, {"u": 1, "text": "", "fill": "#ffd6d6"}, {"u": 1, "text": "", "fill": "#ffd6d6"}, {"u": 1, "text": "", "fill": "#ffd6d6"}, {"u": 1, "text": "d9-10", "fill": "#ffe7c2"}, {"u": 1, "text": "", "fill": "#ffe7c2"}, {"u": 1, "text": "1.2 kg left", "fill": "#cdeccd"}, {"u": 1, "text": "", "fill": "#cdeccd"}],
            "brace": {"text": "? = 4.8 kg"}}
        ]
      }
    },
    {
      "n": 35,
      "answer": "37",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 90,
        "bars": [
          {"label": "Game 1", "segments": [{"u": 1, "text": "31", "fill": "#dbe9ff"}]},
          {"label": "Game 3", "segments": [{"u": 1, "text": "28", "fill": "#dbe9ff"}]},
          {"label": "Game 4", "segments": [{"u": 1, "text": "? = 37", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "best 3 total = 96 (avg 32)", "bars": [0, 1, 2]}
      }
    },
    {
      "n": 36,
      "answer": "(a) \\(4y + 9\\) km   (b) \\(y = 11\\)",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Gerald", "segments": [{"u": 1, "text": "y", "fill": "#dbe9ff"}]},
          {"label": "Leon", "segments": [{"u": 1, "text": "y", "fill": "#dbe9ff"}, {"u": 1, "text": "3", "fill": "#ffe7c2"}]},
          {"label": "Ali", "segments": [{"u": 1, "text": "y", "fill": "#dbe9ff"}, {"u": 1, "text": "3", "fill": "#ffe7c2"}, {"u": 1, "text": "y", "fill": "#dbe9ff"}, {"u": 1, "text": "3", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "4y + 9 = 53 km", "bars": [0, 1, 2]}
      }
    },
    {
      "n": 39,
      "answer": "(a) 84 workers   (b) $108",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 50,
        "bars": [
          {"label": "Yellow cost (×2)", "segments": [{"u": 24, "text": "24", "fill": "#fff3b0"}]},
          {"label": "Blue cost (×1)", "segments": [{"u": 45, "text": "45", "fill": "#dbe9ff"}]},
          {"label": "Purple cost (×1)", "segments": [{"u": 27, "text": "27 → $108", "fill": "#e3d0f0"}]}
        ],
        "total": {"text": "96 units = $384 (1 unit = $4)", "bars": [0, 1, 2]}
      }
    },
    {
      "n": 45,
      "answer": "(a) 560 beads (printed key says 260)   (b) 308 blue beads",
      "answer_matches_key": false,
      "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\\).",
      "spec": {
        "unit_px": 55,
        "bars": [
          {"label": "Blue (30%)", "segments": [{"u": 3, "text": "30%", "fill": "#dbe9ff"}]},
          {"label": "Red", "segments": [{"u": 3, "text": "blue", "fill": "#ffd6d6"}, {"u": 1, "text": "-24", "fill": "#ffffff"}]},
          {"label": "Green", "segments": [{"u": 4, "text": "248", "fill": "#cdeccd"}]}
        ],
        "total": {"text": "40% = 224 → 10% = 56 → total = 560", "bars": [0, 1, 2]}
      }
    },
    {
      "n": 46,
      "answer": "$0.80",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "All key chains", "segments": [{"u": 4, "text": "47 key chains", "fill": "#dbe9ff"}]},
          {"label": "All pencil cases", "segments": [{"u": 1, "text": "= 9 key chains", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "difference 38 key chains = $30.40", "bars": [0, 1]}
      }
    },
    {
      "n": 47,
      "answer": "156 cm²",
      "answer_matches_key": true,
      "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.",
      "spec": {
        "unit_px": 55,
        "bars": [
          {"label": "Triangle ABC", "segments": [
            {"u": 7, "text": "new figure (7/12)", "fill": "#dbe9ff"}, {"u": 5, "text": "overlap ADE = 65 cm² (5/12)", "fill": "#ffd6d6"}],
            "brace": {"text": "? = 156 cm²"}}
        ]
      }
    }
  ]
}