{
  "source_prefix": "MGS 2024 P6 Prelim",
  "models": [
    {
      "n": 15,
      "answer": "175",
      "answer_matches_key": true,
      "working": "Girl winners \\(= \\dfrac{4}{9}\\times 45 = 20\\); boy winners \\(= 45 - 20 = 25\\). \\(\\dfrac{1}{3}\\) of the boys won, so the boys split into 3 equal units of 25: boys \\(= 3\\times 25 = 75\\). \\(\\dfrac{1}{5}\\) of the girls won, so the girls split into 5 equal units of 20: girls \\(= 5\\times 20 = 100\\). Total children \\(= 75 + 100 = 175\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Boys", "segments": [
              {"u": 1, "text": "25", "fill": "#dbe9ff"},
              {"u": 1, "fill": "#dbe9ff"},
              {"u": 1, "fill": "#dbe9ff"}],
           "brace": {"text": "?"}},
          {"label": "Girls", "segments": [
              {"u": 1, "text": "20", "fill": "#ffe7c2"},
              {"u": 1, "fill": "#ffe7c2"},
              {"u": 1, "fill": "#ffe7c2"},
              {"u": 1, "fill": "#ffe7c2"},
              {"u": 1, "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "? children", "bars": [0, 1]}
      }
    },
    {
      "n": 28,
      "answer": "\\(9 : 23\\)",
      "answer_matches_key": true,
      "working": "Both teams have the same number of students. Team A is \\(1 : 3\\) (4 parts); scale it by 4 to \\(4 : 12\\) (16 parts) so it matches Team B's \\(5 : 11\\) (16 parts). Total boys \\(= 4 + 5 = 9\\); total girls \\(= 12 + 11 = 23\\). Ratio of total boys to total girls \\(= 9 : 23\\).",
      "spec": {
        "unit_px": 36,
        "bars": [
          {"label": "Team A", "segments": [
              {"u": 4, "text": "boys 4", "fill": "#dbe9ff"},
              {"u": 12, "text": "girls 12", "fill": "#ffe7c2"}]},
          {"label": "Team B", "segments": [
              {"u": 5, "text": "boys 5", "fill": "#dbe9ff"},
              {"u": 11, "text": "girls 11", "fill": "#ffe7c2"}]}
        ]
      }
    },
    {
      "n": 33,
      "answer": "\\((25 + 8n)\\) years",
      "answer_matches_key": true,
      "working": "Mr Lim is 3 times as old as his son, so split his age into 3 equal units: \\(3\\) units \\(= 12n\\), \\(1\\) unit \\(= 4n\\). The son is \\(4n\\) now and will be 25, which is \\((25 - 4n)\\) years later. Both age by the same amount, so Mr Lim \\(= 12n + (25 - 4n) = 25 + 8n\\).",
      "spec": {
        "unit_px": 90,
        "bars": [
          {"label": "Mr Lim", "segments": [
              {"u": 1, "text": "4n", "fill": "#dbe9ff"},
              {"u": 1, "text": "4n", "fill": "#dbe9ff"},
              {"u": 1, "text": "4n", "fill": "#dbe9ff"}],
           "brace": {"text": "12n"}},
          {"label": "Son", "segments": [{"u": 1, "text": "4n", "fill": "#ffe7c2"}]}
        ]
      }
    },
    {
      "n": 36,
      "answer": "57",
      "answer_matches_key": true,
      "working": "Cookies and Cream \\(= 60\\%\\), so Chocolate + Vanilla + Strawberry \\(= 40\\%\\) of \\(380 = 152\\) pupils. Let Vanilla \\(= 3\\) units, Strawberry \\(= 1\\) unit, and Chocolate \\(= 4\\) units (Strawberry + Vanilla). Total \\(= 8\\) units \\(= 152\\), so \\(1\\) unit \\(= 19\\). Vanilla \\(= 3\\times 19 = 57\\).",
      "spec": {
        "unit_px": 50,
        "bars": [
          {"label": "Strawberry", "segments": [{"u": 1, "fill": "#ffd9d9"}]},
          {"label": "Vanilla", "segments": [
              {"u": 1, "fill": "#ffe7c2"},
              {"u": 1, "fill": "#ffe7c2"},
              {"u": 1, "fill": "#ffe7c2"}],
           "brace": {"text": "?"}},
          {"label": "Chocolate", "segments": [
              {"u": 1, "fill": "#dbe9ff"},
              {"u": 1, "fill": "#dbe9ff"},
              {"u": 1, "fill": "#dbe9ff"},
              {"u": 1, "fill": "#dbe9ff"}]}
        ],
        "total": {"text": "152 (40%)", "bars": [0, 1, 2]}
      }
    },
    {
      "n": 39,
      "answer": "\\(\\dfrac{9}{20}\\)",
      "answer_matches_key": true,
      "working": "Split Aaron's marbles into 5 equal units. He gives \\(\\dfrac{2}{5}\\) (2 units) to Benny, leaving \\(\\dfrac{3}{5}\\) (3 units). To take \\(\\dfrac{1}{4}\\) of that remainder, cut each unit into 4, so \\(\\dfrac{3}{5} = \\dfrac{12}{20}\\); Charlie gets \\(\\dfrac{1}{4}\\times\\dfrac{3}{5} = \\dfrac{3}{20}\\). Aaron has \\(\\dfrac{12}{20} - \\dfrac{3}{20} = \\dfrac{9}{20}\\) left.",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Aaron's start", "segments": [
              {"u": 4, "text": "to Benny \\(\\dfrac{2}{5}\\)", "fill": "#ffe7c2"},
              {"u": 4, "text": "to Benny \\(\\dfrac{2}{5}\\)", "fill": "#ffe7c2"},
              {"u": 3, "text": "to C", "fill": "#ffd9d9"},
              {"u": 9, "text": "left \\(\\dfrac{9}{20}\\)", "fill": "#dbe9ff"}],
           "brace": {"text": "20 twentieths"}}
        ]
      }
    },
    {
      "n": 40,
      "answer": "\\(20 : 1 : 6\\)",
      "answer_matches_key": true,
      "working": "Let Aaron start with 20 units. He gives \\(\\dfrac{2}{5}\\times 20 = 8\\) units to Benny, then \\(\\dfrac{1}{4}\\) of the remaining 12 \\(= 3\\) units to Charlie, keeping 9 units. All three end equal, so each ends with 9 units. Benny started with \\(9 - 8 = 1\\) unit; Charlie started with \\(9 - 3 = 6\\) units. At first, Aaron : Benny : Charlie \\(= 20 : 1 : 6\\).",
      "spec": {
        "unit_px": 24,
        "bars": [
          {"label": "Aaron", "segments": [
              {"u": 9, "text": "keeps 9", "fill": "#dbe9ff"},
              {"u": 8, "text": "to Benny 8", "fill": "#ffe7c2"},
              {"u": 3, "text": "to C 3", "fill": "#ffd9d9"}],
           "brace": {"text": "20"}},
          {"label": "Benny", "segments": [{"u": 1, "text": "1", "fill": "#dbe9ff"}]},
          {"label": "Charlie", "segments": [{"u": 6, "text": "6", "fill": "#dbe9ff"}]}
        ]
      }
    },
    {
      "n": 41,
      "answer": "420",
      "answer_matches_key": true,
      "working": "Each pupil first got an equal share of \\(u\\) cards. The \\(7\\) pupils who left gave away their \\(7u\\) cards, shared among the remaining \\(35 - 7 = 28\\) pupils, \\(3\\) more each: \\(28\\times 3 = 84 = 7u\\), so \\(u = 12\\). Total cards \\(= 35\\times 12 = 420\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "7 leavers' cards", "segments": [
              {"u": 1, "text": "u", "fill": "#ffe7c2"},
              {"u": 1, "text": "u", "fill": "#ffe7c2"},
              {"u": 1, "text": "u", "fill": "#ffe7c2"},
              {"u": 1, "text": "u", "fill": "#ffe7c2"},
              {"u": 1, "text": "u", "fill": "#ffe7c2"},
              {"u": 1, "text": "u", "fill": "#ffe7c2"},
              {"u": 1, "text": "u", "fill": "#ffe7c2"}],
           "brace": {"text": "7u = 28 \\times 3 = 84"}},
          {"label": "28 stayers gain", "segments": [{"u": 4, "text": "3 each \\times 28 = 84"}]}
        ]
      }
    },
    {
      "n": 45,
      "answer": "(a) $360, (b) 8%",
      "answer_matches_key": true,
      "working": "(a) Shop X \\(= \\$450\\). Split into 5 equal units: \\(1\\) unit \\(= \\$90\\). Shop Y \\(= 80\\% = 4\\) units \\(= 4\\times \\$90 = \\$360\\). (b) Before the sale Shop X \\(-\\) Shop Y \\(= \\$450 - \\$360 = \\$90\\). The same percentage discount applies to both, so the gap stays in proportion; after the sale the gap is \\(\\$82.80\\): \\(\\dfrac{82.80}{90}\\times 100 = 92\\%\\) of the price remains, so the discount \\(= 100\\% - 92\\% = 8\\%\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Shop X", "segments": [
              {"u": 1, "text": "$90", "fill": "#dbe9ff"},
              {"u": 1, "text": "$90", "fill": "#dbe9ff"},
              {"u": 1, "text": "$90", "fill": "#dbe9ff"},
              {"u": 1, "text": "$90", "fill": "#dbe9ff"},
              {"u": 1, "text": "$90", "fill": "#dbe9ff"}],
           "brace": {"text": "$450"}},
          {"label": "Shop Y", "segments": [
              {"u": 1, "text": "$90", "fill": "#ffe7c2"},
              {"u": 1, "text": "$90", "fill": "#ffe7c2"},
              {"u": 1, "text": "$90", "fill": "#ffe7c2"},
              {"u": 1, "text": "$90", "fill": "#ffe7c2"}],
           "brace": {"text": "? = $360"}}
        ]
      }
    }
  ]
}