{
  "source_prefix": "Nan Hua 2023 P6 Prelim",
  "models": [
    {
      "n": 14,
      "answer": "15",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Blue + Red", "segments": [{"u": 2, "text": "20", "fill": "#dbe9ff"}]},
          {"label": "Red + White", "segments": [{"u": 4, "text": "45", "fill": "#ffe7c2"}], "brace": {"text": "overlap = Red"}},
          {"label": "Total", "segments": [{"u": 5, "text": "50"}]}
        ]
      }
    },
    {
      "n": 15,
      "answer": "14%",
      "answer_matches_key": true,
      "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\\%\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Boys", "segments": [{"u": 2, "text": "40% (walk 2%)", "fill": "#dbe9ff"}]},
          {"label": "Girls", "segments": [{"u": 3, "text": "60% (walk 12%)", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "100% pupils", "bars": [0, 1]}
      }
    },
    {
      "n": 26,
      "answer": "24 km",
      "answer_matches_key": true,
      "working": "Jack: \\(7 \\times 2 = 14\\) km. Keith: \\(5 \\times 2 = 10\\) km. They go opposite ways, so distance apart \\(= 14 + 10 = 24\\) km.",
      "spec": {
        "unit_px": 50,
        "bars": [
          {"label": "Jack", "segments": [{"u": 7, "text": "14 km", "fill": "#dbe9ff"}]},
          {"label": "Keith", "segments": [{"u": 5, "text": "10 km", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "? km apart", "bars": [0, 1]}
      }
    },
    {
      "n": 28,
      "answer": "24",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "X", "segments": [{"u": 3, "text": "3u", "fill": "#dbe9ff"}, {"u": 1, "text": "+4", "fill": "#cfe9cf"}]},
          {"label": "half of Y", "segments": [{"u": 3, "text": "3u", "fill": "#ffe7c2"}, {"u": 1, "text": "0.5u", "fill": "#ffe7c2"}], "brace": {"text": "X+4 = half Y (equal): 0.5u = 4"}}
        ]
      }
    },
    {
      "n": 29,
      "answer": "18",
      "answer_matches_key": true,
      "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.",
      "spec": {
        "unit_px": 55,
        "bars": [
          {"label": "Option 1", "segments": [{"u": 6, "text": "6 rulers", "fill": "#dbe9ff"}, {"u": 3, "text": "3 erasers", "fill": "#ffe7c2"}]},
          {"label": "Option 2", "segments": [{"u": 4, "text": "4 rulers", "fill": "#dbe9ff"}, {"u": 8, "text": "8 erasers", "fill": "#ffe7c2"}], "brace": {"text": "2 rulers = 5 erasers"}}
        ]
      }
    },
    {
      "n": 31,
      "answer": "31.8 kg",
      "answer_matches_key": true,
      "working": "Lighter boy \\(=\\) (total \\(-\\) difference) \\(\\div 2 = (72.9 - 9.3) \\div 2 = 63.6 \\div 2 = 31.8\\) kg.",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Heavier", "segments": [{"u": 3, "fill": "#dbe9ff"}, {"u": 1, "text": "9.3", "fill": "#cfe9cf"}]},
          {"label": "Lighter", "segments": [{"u": 3, "text": "?", "fill": "#dbe9ff"}], "brace": {"text": "(72.9 - 9.3) / 2 = 31.8"}}
        ],
        "total": {"text": "72.9 kg", "bars": [0, 1]}
      }
    },
    {
      "n": 36,
      "answer": "(a) 7 pieces; (b) \\(0.2\\) m",
      "answer_matches_key": true,
      "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.",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Fabric", "segments": [
            {"u": 1, "text": "2/5", "fill": "#dbe9ff"},
            {"u": 1, "text": "2/5", "fill": "#dbe9ff"},
            {"u": 1, "text": "2/5", "fill": "#dbe9ff"},
            {"u": 1, "text": "2/5", "fill": "#dbe9ff"},
            {"u": 1, "text": "2/5", "fill": "#dbe9ff"},
            {"u": 1, "text": "2/5", "fill": "#dbe9ff"},
            {"u": 1, "text": "2/5", "fill": "#dbe9ff"},
            {"u": 1, "text": "0.2", "fill": "#ffe7c2"}],
           "brace": {"text": "3 m total (7 pieces + last 0.2 m)"}}
        ]
      }
    },
    {
      "n": 37,
      "answer": "387",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 26,
        "bars": [
          {"label": "Original", "segments": [{"u": 7, "text": "Boys 70%", "fill": "#dbe9ff"}, {"u": 3, "text": "Girls 30%", "fill": "#ffe7c2"}], "brace": {"text": "100%"}},
          {"label": "End", "segments": [{"u": 8, "text": "Boys 84%", "fill": "#dbe9ff"}, {"u": 4, "text": "Girls 45%", "fill": "#ffe7c2"}, {"u": 1, "text": "+29% = 87", "fill": "#cfe9cf"}], "brace": {"text": "129% = 387"}}
        ]
      }
    },
    {
      "n": 43,
      "answer": "$180",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 40,
        "bars": [
          {"label": "Tim (before)", "segments": [{"u": 6, "text": "6 parts = $180", "fill": "#dbe9ff"}]},
          {"label": "Peter (before)", "segments": [{"u": 9, "text": "9 parts", "fill": "#ffe7c2"}]},
          {"label": "Tim (after)", "segments": [{"u": 6, "fill": "#dbe9ff"}, {"u": 4, "text": "+$120", "fill": "#cfe9cf"}]},
          {"label": "Peter (after)", "segments": [{"u": 5, "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "4 parts = $120, 1 part = $30", "bars": [2, 3]}
      }
    }
  ]
}