{
  "source_prefix": "Catholic High 2024 P6 Prelim",
  "models": [
    {
      "n": 10,
      "answer": "\\(5 : 8\\)",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Boys", "segments": [
              {"u": 1, "text": "1 unit", "fill": "#dbe9ff"}],
           "brace": {"text": "walk = 25% × 1 = 0.25"}},
          {"label": "Girls", "segments": [
              {"u": 1, "text": "1 unit", "fill": "#ffe7c2"}, {"u": 1, "text": "1 unit", "fill": "#ffe7c2"}],
           "brace": {"text": "walk = 20% × 2 = 0.40"}}
        ]
      }
    },
    {
      "n": 12,
      "answer": "13",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 26,
        "bars": [
          {"label": "Ivan (start)", "segments": [
              {"u": 41, "text": "41", "fill": "#dbe9ff"}]},
          {"label": "Peggy (start)", "segments": [
              {"u": 41, "text": "41", "fill": "#ffe7c2"}, {"u": 30, "text": "30", "fill": "#ffd0d0"}]},
          {"label": "Ivan (end)", "segments": [
              {"u": 28, "text": "28", "fill": "#dbe9ff"}]},
          {"label": "Peggy (end)", "segments": [
              {"u": 28, "text": "28", "fill": "#ffe7c2"}, {"u": 28, "text": "28", "fill": "#ffe7c2"}, {"u": 28, "text": "84", "fill": "#ffe7c2"}]}
        ]
      }
    },
    {
      "n": 15,
      "answer": "\\(\\dfrac{2}{3}\\)",
      "answer_matches_key": true,
      "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}\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Bag A (27)", "segments": [
              {"u": 1, "text": "P 12", "fill": "#dbe9ff"}, {"u": 1, "text": "15 oranges", "fill": "#ffe7c2"}]},
          {"label": "Bag B (9)", "segments": [
              {"u": 1, "text": "P 6", "fill": "#dbe9ff"}, {"u": 1, "text": "3", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "36 fruits (18 pears, 18 oranges)", "bars": [0, 1]}
      }
    },
    {
      "n": 22,
      "answer": "26",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 22,
        "bars": [
          {"label": "If all 20¢", "segments": [
              {"u": 30, "text": "30 × 20¢ = $6", "fill": "#dbe9ff"}]},
          {"label": "Actual value", "segments": [
              {"u": 17, "text": "$3.40", "fill": "#ffe7c2"}, {"u": 13, "text": "shortfall $2.60", "fill": "#ffd0d0"}]}
        ]
      }
    },
    {
      "n": 28,
      "answer": "10",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 26,
        "bars": [
          {"label": "Planned (40 × 6)", "segments": [
              {"u": 40, "text": "240 packets", "fill": "#dbe9ff"}]},
          {"label": "Present (× 8)", "segments": [
              {"u": 30, "text": "30 pupils × 8 = 240", "fill": "#ffe7c2"}, {"u": 10, "text": "10 absent", "fill": "#ffd0d0"}]}
        ]
      }
    },
    {
      "n": 29,
      "answer": "16%",
      "answer_matches_key": true,
      "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\\%\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "All beads", "segments": [
              {"u": 3, "text": "red (3/5)", "fill": "#ffd0d0"}, {"u": 1, "text": "blue", "fill": "#dbe9ff"}, {"u": 1, "text": "green", "fill": "#c8f0c8"}],
           "brace": {"text": "non-red = 2/5; blue:green = 2:3"}}
        ]
      }
    },
    {
      "n": 30,
      "answer": "35",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 40,
        "bars": [
          {"label": "Original (15u)", "segments": [
              {"u": 6, "text": "42 sold (morning)", "fill": "#ffd0d0"}, {"u": 4, "text": "afternoon (4/9)", "fill": "#ffe7c2"}, {"u": 5, "text": "left (1/3)", "fill": "#c8f0c8"}]},
          {"label": "Remainder (9u)", "segments": [
              {"u": 4, "text": "4/9 sold", "fill": "#ffe7c2"}, {"u": 5, "text": "5/9 left = 35", "fill": "#c8f0c8"}]}
        ]
      }
    }
  ]
}