{
  "source_prefix": "ACS Primary 2023 P6 Prelim",
  "models": [
    {
      "n": 9,
      "answer": "$26",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Eldest", "segments": [{"u": 2, "text": "$50", "fill": "#ffe7c2"}]},
          {"label": "Sibling 2", "segments": [{"u": 1, "text": "?", "fill": "#dbe9ff"}]},
          {"label": "Sibling 3", "segments": [{"u": 1, "text": "?", "fill": "#dbe9ff"}]}
        ],
        "total": {"text": "$102", "bars": [0, 1, 2]}
      }
    },
    {
      "n": 13,
      "answer": "120",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 30,
        "bars": [
          {"label": "Dark", "segments": [{"u": 5, "fill": "#cfd6dd"}]},
          {"label": "Milk", "segments": [{"u": 7, "fill": "#dbe9ff"}]},
          {"label": "Orange", "segments": [{"u": 8, "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "Total = ?", "bars": [0, 1, 2]}
      }
    },
    {
      "n": 19,
      "answer": "1000",
      "answer_matches_key": true,
      "working": "Each bow uses \\(\\dfrac{2}{5}\\) m. For 25 bows: \\(25 \\times \\dfrac{2}{5} = 10\\) m. Convert: \\(10\\) m \\(= 1000\\) cm.",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "1 bow", "segments": [{"u": 1, "text": "\\(\\frac{2}{5}\\) m", "fill": "#dbe9ff"}]},
          {"label": "25 bows", "segments": [
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"},
              {"u": 1, "text": "... 25 units ...", "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}],
           "brace": {"text": "? cm (= 10 m)"}}
        ]
      }
    },
    {
      "n": 26,
      "answer": "36",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "First", "segments": [
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"},
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}],
           "brace": {"text": "?"}},
          {"label": "Second", "segments": [{"u": 1, "fill": "#ffe7c2"}]},
          {"label": "Third", "segments": [{"u": 1, "fill": "#c8e6c9"}, {"u": 1, "fill": "#c8e6c9"}, {"u": 1, "fill": "#c8e6c9"}]}
        ],
        "total": {"text": "60", "bars": [0, 1, 2]}
      }
    },
    {
      "n": 30,
      "answer": "$192",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 36,
        "bars": [
          {"label": "Amil", "segments": [{"u": 8, "fill": "#dbe9ff"}]},
          {"label": "Bala", "segments": [{"u": 3, "fill": "#ffe7c2"}]},
          {"label": "Charles", "segments": [{"u": 5, "fill": "#c8e6c9"}]}
        ],
        "total": {"text": "Total = ? (16 units)", "bars": [0, 1, 2]}
      }
    },
    {
      "n": 34,
      "answer": "$157.68",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 38,
        "bars": [
          {"label": "Paid (85%)", "segments": [{"u": 17, "text": "$124.10", "fill": "#dbe9ff"}]},
          {"label": "Original (100%)", "segments": [{"u": 17, "fill": "#dbe9ff"}, {"u": 3, "fill": "#eef2f7"}], "brace": {"text": "$146"}}
        ]
      }
    },
    {
      "n": 37,
      "answer": "(a) $1730; (b) 7%",
      "answer_matches_key": true,
      "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\\%\\).",
      "spec": {
        "unit_px": 30,
        "bars": [
          {"label": "Salary", "segments": [
              {"u": 4, "text": "Savings $3000", "fill": "#cfd6dd"},
              {"u": 3, "text": "Food $1730", "fill": "#dbe9ff"},
              {"u": 4, "text": "Rent $2422", "fill": "#c8e6c9"},
              {"u": 1, "text": "Tpt $865", "fill": "#ffe7c2"},
              {"u": 1, "text": "Shop ?", "fill": "#f8c8d4"}],
           "brace": {"text": "$8650"}}
        ]
      }
    },
    {
      "n": 39,
      "answer": "Train X: 690 km; Train Y: 570 km",
      "answer_matches_key": true,
      "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.",
      "spec": {
        "unit_px": 8,
        "bars": [
          {"label": "Train X (230 km/h)", "segments": [{"u": 69, "text": "690 km", "fill": "#dbe9ff"}]},
          {"label": "Train Y (190 km/h)", "segments": [{"u": 57, "text": "570 km", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "1260 km", "bars": [0, 1]}
      }
    },
    {
      "n": 41,
      "answer": "(a) 1.05 L; (b) 0.525 L",
      "answer_matches_key": true,
      "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.",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Bottle A", "segments": [{"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}], "brace": {"text": "4.2 L"}},
          {"label": "Bottle B", "segments": [{"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}], "brace": {"text": "2.1 L"}},
          {"label": "Bottle C", "segments": [{"u": 1, "fill": "#c8e6c9"}], "brace": {"text": "1.05 L"}}
        ],
        "total": {"text": "7.35 L", "bars": [0, 1, 2]}
      }
    },
    {
      "n": 47,
      "answer": "(a) 45 coins; (b) 144 coins",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 30,
        "bars": [
          {"label": "20c (24%)", "segments": [{"u": 24, "text": "72", "fill": "#cfd6dd"}]},
          {"label": "50c : $1 = 7:12", "segments": [{"u": 28, "text": "50c (84)", "fill": "#dbe9ff"}, {"u": 48, "text": "$1 = ? (144)", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "Total = 300", "bars": [0, 1]}
      }
    }
  ]
}