{
  "source_prefix": "Ai Tong 2023 P6 Prelim",
  "models": [
    {
      "n": 10,
      "answer": "11 : 17",
      "answer_matches_key": true,
      "working": "Total adults \\(= 84\\). Men \\(= 51\\), so women \\(= 84 - 51 = 33\\). Ratio women : men \\(= 33 : 51 = 11 : 17\\) (divide both by 3).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Women", "segments": [{"u": 1, "text": "33", "fill": "#ffe7c2"}]},
          {"label": "Men", "segments": [{"u": 1, "text": "51", "fill": "#dbe9ff"}]}
        ],
        "total": {"text": "84", "bars": [0, 1]}
      }
    },
    {
      "n": 14,
      "answer": "$56",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Shirt (20%)", "segments": [{"u": 1, "text": "1u", "fill": "#dbe9ff"}]},
          {"label": "Bag + Necklace (80%)", "segments": [
              {"u": 1, "text": "1u", "fill": "#ffe7c2"},
              {"u": 1, "text": "$24", "fill": "#ffd1d1"},
              {"u": 1, "text": "$72", "fill": "#d6f5d6"}]}
        ]
      }
    },
    {
      "n": 27,
      "answer": "$240",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 36,
        "bars": [
          {"label": "Weijie (before)", "segments": [{"u": 15, "fill": "#dbe9ff"}], "brace": {"text": "15 units"}},
          {"label": "Lihua (before)", "segments": [
              {"u": 10, "fill": "#ffe7c2"},
              {"u": 2, "text": "$40", "fill": "#ffd1d1"}],
           "brace": {"text": "Lihua = 12 units = ?"}}
        ]
      }
    },
    {
      "n": 30,
      "answer": "35 erasers",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 48,
        "bars": [
          {"label": "5 boxes at first", "segments": [
              {"u": 2, "text": "left = 2u", "fill": "#d6f5d6"},
              {"u": 3, "text": "removed = 105", "fill": "#ffd1d1"}],
           "brace": {"text": "5 units"}}
        ]
      }
    },
    {
      "n": 32,
      "answer": "147 cm",
      "answer_matches_key": true,
      "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.",
      "spec": {
        "unit_px": 32,
        "bars": [
          {"label": "Longest (7u)", "segments": [{"u": 7, "text": "?", "fill": "#dbe9ff"}]},
          {"label": "Shortest (2u)", "segments": [{"u": 2, "text": "42 cm", "fill": "#ffe7c2"}]},
          {"label": "Middle (3u)", "segments": [{"u": 3, "fill": "#d6f5d6"}]}
        ]
      }
    },
    {
      "n": 37,
      "answer": "36 students",
      "answer_matches_key": true,
      "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.",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Dance Club (3u)", "segments": [{"u": 3, "fill": "#dbe9ff"}]},
          {"label": "Brownies (2u)", "segments": [{"u": 2, "text": "?", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "30% of 300 = 90", "bars": [0, 1]}
      }
    },
    {
      "n": 43,
      "answer": "495",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 30,
        "bars": [
          {"label": "Boys (end)", "segments": [{"u": 4, "fill": "#dbe9ff"}]},
          {"label": "Girls (end)", "segments": [
              {"u": 4, "fill": "#ffe7c2"},
              {"u": 5, "text": "5u = ?", "fill": "#ffd1d1"}]}
        ],
        "total": {"text": "diff = 5u", "bars": [1]}
      }
    },
    {
      "n": 47,
      "answer": "(a) $4680  (b) 18 dresses",
      "answer_matches_key": true,
      "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\\).",
      "spec": {
        "unit_px": 30,
        "bars": [
          {"label": "Bags", "segments": [
              {"u": 8, "fill": "#dbe9ff"},
              {"u": 9, "text": "$3240 more", "fill": "#ffd1d1"}],
           "brace": {"text": "$4680"}},
          {"label": "Dresses", "segments": [{"u": 8, "fill": "#ffe7c2"}], "brace": {"text": "$1440"}}
        ],
        "total": {"text": "$6120", "bars": [0, 1]}
      }
    },
    {
      "n": 48,
      "answer": "\\(\\dfrac{1}{12}\\)",
      "answer_matches_key": true,
      "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.",
      "spec": {
        "unit_px": 44,
        "bars": [
          {"label": "Ali (1/4)", "segments": [
              {"u": 1, "text": "kept", "fill": "#d6f5d6"},
              {"u": 1, "text": "given", "fill": "#ffd1d1"},
              {"u": 1, "text": "given", "fill": "#ffd1d1"}]},
          {"label": "Brandon (1/4)", "segments": [{"u": 3, "fill": "#dbe9ff"}]},
          {"label": "Carrie (1/4)", "segments": [{"u": 3, "fill": "#dbe9ff"}]},
          {"label": "Devi (1/4)", "segments": [{"u": 3, "fill": "#dbe9ff"}]}
        ],
        "total": {"text": "whole = 12 units", "bars": [0, 1, 2, 3]}
      }
    }
  ]
}