{
  "source_prefix": "Nan Hua 2024 P6 Prelim",
  "models": [
    {
      "n": 10,
      "answer": "3 : 5",
      "answer_matches_key": true,
      "working": "Split the audience into 9 equal units: adults \\(= 5\\) units, children \\(= 4\\) units. The children split into \\(\\dfrac{1}{4}\\) boys (1 unit) and \\(\\dfrac{3}{4}\\) girls (3 units). So girls \\(= 3\\) units, adults \\(= 5\\) units, giving girls : adults \\(= 3 : 5\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Adults", "segments": [
              {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}]},
          {"label": "Children", "segments": [
              {"u": 1, "text": "boys", "fill": "#cfe9d4"}, {"u": 1, "text": "girls", "fill": "#dbe9ff"}, {"u": 1, "text": "girls", "fill": "#dbe9ff"}, {"u": 1, "text": "girls", "fill": "#dbe9ff"}]}
        ],
        "total": {"text": "audience = 9 units", "bars": [0, 1]}
      }
    },
    {
      "n": 11,
      "answer": "$75",
      "answer_matches_key": true,
      "working": "Total money \\(= 5\\) units (cap is 20% \\(= 1\\) unit; the rest is 4 units). The rest (4 units) buys the bag and the shirt. Bag \\(= \\text{cap} + \\$15 = 1\\) unit \\(+ \\$15\\); shirt \\(= \\$165\\). So 1 unit \\(+ \\$15 + \\$165 = 4\\) units, giving 3 units \\(= \\$180\\), 1 unit \\(= \\$60\\). Bag \\(= \\$60 + \\$15 = \\$75\\).",
      "spec": {
        "unit_px": 90,
        "bars": [
          {"label": "Total", "segments": [
              {"u": 1, "text": "cap", "fill": "#ffe7c2"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}]},
          {"label": "Bag + Shirt", "segments": [
              {"u": 1, "text": "cap +$15", "fill": "#cfe9d4"}, {"u": 3, "text": "shirt $165", "fill": "#f6d6e0"}]}
        ]
      }
    },
    {
      "n": 12,
      "answer": "50",
      "answer_matches_key": true,
      "working": "Split the ingredients into 3 equal units. 60 big cupcakes use \\(\\dfrac{60}{90} = \\dfrac{2}{3}\\) of the ingredients (2 units), leaving 1 unit \\(= \\dfrac{1}{3}\\). A full batch of small cupcakes is 150, so \\(\\dfrac{1}{3} \\times 150 = 50\\) small cupcakes.",
      "spec": {
        "unit_px": 90,
        "bars": [
          {"label": "Ingredients", "segments": [
              {"u": 1, "text": "used", "fill": "#f6d6e0"}, {"u": 1, "text": "used", "fill": "#f6d6e0"}, {"u": 1, "text": "left", "fill": "#cfe9d4"}],
           "brace": {"text": "60 big = full batch of 90"}},
          {"label": "Small (150)", "segments": [
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "text": "? = 50", "fill": "#dbe9ff"}]}
        ]
      }
    },
    {
      "n": 24,
      "answer": "$12",
      "answer_matches_key": true,
      "working": "The same total meal cost is split two ways. Correctly it is shared among 8 people; by mistake it was shared among 6 people, so each of those 6 shares is bigger. Each person paid \\$4 more, so the total over-collected \\(= \\$4 \\times 6 = \\$24\\). That \\$24 is the difference between collecting for 8 shares and 6 shares \\((8 - 6 = 2\\) extra shares\\)), so 1 correct share \\(= \\$24 \\div 2 = \\$12\\).",
      "spec": {
        "unit_px": 90,
        "bars": [
          {"label": "Correct (÷8)", "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"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "text": "?", "fill": "#dbe9ff"}]},
          {"label": "Wrong (÷6)", "segments": [
              {"u": 1, "text": "+$4", "fill": "#ffe7c2"}, {"u": 1, "text": "+$4", "fill": "#ffe7c2"}, {"u": 1, "text": "+$4", "fill": "#ffe7c2"}, {"u": 1, "text": "+$4", "fill": "#ffe7c2"}, {"u": 1, "text": "+$4", "fill": "#ffe7c2"}, {"u": 1, "text": "+$4", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "same total cost", "bars": [0, 1]}
      }
    },
    {
      "n": 26,
      "answer": "3",
      "answer_matches_key": true,
      "working": "Take the 1 m of string as 20 twentieths. Jane cuts \\(\\dfrac{1}{4}\\) m \\(= \\dfrac{5}{20}\\) m to tie the box, leaving \\(\\dfrac{15}{20} = \\dfrac{3}{4}\\) m. Each piece is \\(\\dfrac{1}{5}\\) m \\(= \\dfrac{4}{20}\\) m, and \\(\\dfrac{15}{20} \\div \\dfrac{4}{20} = 3\\dfrac{3}{4}\\). Only whole pieces count, so the maximum is 3 pieces (with \\(\\dfrac{3}{20}\\) m left over).",
      "spec": {
        "unit_px": 44,
        "bars": [
          {"label": "1 m string", "segments": [
              {"u": 5, "text": "tied box (¼ m)", "fill": "#f6d6e0"},
              {"u": 4, "text": "⅕ m", "fill": "#dbe9ff"},
              {"u": 4, "text": "⅕ m", "fill": "#dbe9ff"},
              {"u": 4, "text": "⅕ m", "fill": "#dbe9ff"},
              {"u": 3, "text": "left", "fill": "#eeeeee"}]}
        ]
      }
    },
    {
      "n": 36,
      "answer": "$32.70",
      "answer_matches_key": true,
      "working": "Let Alice's money be 1 unit; then Diana's money is 4 units. Both want the same dress. Dress \\(= \\text{Alice} + \\$28.40 = 1\\) unit \\(+ \\$28.40\\); dress \\(= \\text{Diana} + \\$15.50 = 4\\) units \\(+ \\$15.50\\). Setting these equal: 3 units \\(= \\$28.40 - \\$15.50 = \\$12.90\\), so 1 unit \\(= \\$4.30\\). Dress \\(= \\$4.30 + \\$28.40 = \\$32.70\\).",
      "spec": {
        "unit_px": 90,
        "bars": [
          {"label": "Alice + short", "segments": [
              {"u": 1, "text": "1u", "fill": "#dbe9ff"}, {"u": 1, "text": "short $28.40", "fill": "#f6d6e0"}],
           "brace": {"text": "dress"}},
          {"label": "Diana + short", "segments": [
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "text": "short $15.50", "fill": "#ffe7c2"}],
           "brace": {"text": "dress"}}
        ]
      }
    },
    {
      "n": 39,
      "answer": "3640",
      "answer_matches_key": true,
      "working": "Non-fiction does not change, so make its share equal in both ratios. Before fiction : non-fiction \\(= 8 : 5 = 32 : 20\\); after \\(= 9 : 4 = 45 : 20\\). After, fiction \\(= 45\\) units \\(= 12\\,600\\), so 1 unit \\(= 280\\). Fiction before \\(= 32\\) units; fiction added \\(= 45 - 32 = 13\\) units \\(= 13 \\times 280 = 3640\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Fiction before", "segments": [
              {"u": 4, "text": "32 units", "fill": "#dbe9ff"}]},
          {"label": "Fiction after", "segments": [
              {"u": 4, "text": "32 units", "fill": "#dbe9ff"}, {"u": 2, "text": "+13 units = ?", "fill": "#cfe9d4"}],
           "brace": {"text": "45 units = 12 600"}},
          {"label": "Non-fiction", "segments": [
              {"u": 3, "text": "20 units (unchanged)", "fill": "#ffe7c2"}]}
        ]
      }
    },
    {
      "n": 45,
      "answer": "60",
      "answer_matches_key": true,
      "working": "Build the money ratio. Senior amount \\(= \\dfrac{1}{4}\\) of adult amount, so adult : senior \\(= 24 : 6\\); children : senior \\(= 7 : 6\\). So adults : children : seniors (by money) \\(= 24 : 7 : 6\\), total 37 units \\(= \\$2220\\), giving 1 unit \\(= \\$60\\). Adults' money \\(= 24\\) units \\(= \\$1440\\); number of adults \\(= \\$1440 \\div \\$48 = 30\\). People ratio is \\(5 : 2 : 3\\); adults \\(= 5\\) parts \\(= 30\\), so 1 part \\(= 6\\) and total \\(= 10\\) parts \\(= 60\\) people.",
      "spec": {
        "unit_px": 56,
        "bars": [
          {"label": "Adults' $", "segments": [
              {"u": 6, "text": "24 units = $1440", "fill": "#ffe7c2"}]},
          {"label": "Children's $", "segments": [
              {"u": 2, "text": "7 units", "fill": "#dbe9ff"}]},
          {"label": "Seniors' $", "segments": [
              {"u": 1, "text": "6 units", "fill": "#cfe9d4"}]}
        ],
        "total": {"text": "$2220 = 37 units", "bars": [0, 1, 2]}
      }
    }
  ]
}