{
  "source_prefix": "ACS 2023 P6 Prelim",
  "models": [
    {
      "n": 14,
      "answer": "160",
      "answer_matches_key": true,
      "working": "Adults \\(= 70\\% \\times 800 = 560\\); children \\(= 240\\) (unchanged). After some adults leave, adults : children \\(= 5 : 3\\). Children are 3 units \\(= 240\\), so 1 unit \\(= 80\\). Adults now \\(= 5\\) units \\(= 400\\). Adults who left \\(= 560 - 400 = 160\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Adults (after)", "segments": [
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}],
           "brace": {"text": "400"}},
          {"label": "Children", "segments": [
              {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}],
           "brace": {"text": "240"}},
          {"label": "Left", "segments": [{"u": 2, "text": "?", "fill": "#f5d0d0"}]}
        ]
      }
    },
    {
      "n": 15,
      "answer": "54",
      "answer_matches_key": true,
      "working": "Tina is \\(\\dfrac{1}{4}\\) of the total, so Su Lin is \\(\\dfrac{3}{4}\\) \\(\\Rightarrow\\) Su Lin \\(= 3 \\times\\) Tina. Tina total \\(= t + 11\\), Su Lin total \\(= (t + 17) + 30 = t + 47\\). So \\(t + 47 = 3(t + 11)\\) \\(\\Rightarrow 2t = 14\\) \\(\\Rightarrow t = 7\\). Su Lin \\(= 7 + 47 = 54\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Tina", "segments": [{"u": 1, "fill": "#ffe7c2"}]},
          {"label": "Su Lin", "segments": [
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}],
           "brace": {"text": "?"}},
          {"label": "Total", "segments": [
              {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}]}
        ]
      }
    },
    {
      "n": 24,
      "answer": "\\(0.027\\) kg",
      "answer_matches_key": true,
      "working": "Both boxes are identical, so the extra mass comes only from the extra erasers. Extra erasers \\(= 44 - 24 = 20\\); extra mass \\(= 1.21 - 0.67 = 0.54\\) kg. Mass of 1 eraser \\(= 0.54 \\div 20 = 0.027\\) kg.",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Box + 24", "segments": [
              {"u": 3, "text": "box + 24 erasers", "fill": "#dbe9ff"}],
           "brace": {"text": "0.67 kg"}},
          {"label": "Box + 44", "segments": [
              {"u": 3, "text": "box + 24 erasers", "fill": "#dbe9ff"}, {"u": 2, "text": "+20 erasers = 0.54 kg", "fill": "#ffe7c2"}],
           "brace": {"text": "1.21 kg"}}
        ]
      }
    },
    {
      "n": 31,
      "answer": "360",
      "answer_matches_key": true,
      "working": "After 12 students gave up their sweets, the remaining \\(36 - 12 = 24\\) students each got 5 more, so \\(24 \\times 5 = 120\\) sweets came from the 12 who gave up. Each original share \\(= 120 \\div 12 = 10\\) sweets. Total at first \\(= 36 \\times 10 = 360\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "12 gave up", "segments": [
              {"u": 1, "text": "10", "fill": "#f5d0d0"}, {"u": 1, "text": "...", "fill": "#f5d0d0"}],
           "brace": {"text": "12 \\u00d7 10 = 120"}},
          {"label": "24 got more", "segments": [
              {"u": 1, "text": "10", "fill": "#dbe9ff"}, {"u": 1, "text": "+5", "fill": "#ffe7c2"}],
           "brace": {"text": "24 \\u00d7 5 = 120 extra"}}
        ]
      }
    },
    {
      "n": 32,
      "answer": "72",
      "answer_matches_key": true,
      "working": "Red : Yellow \\(= 1 : 3\\). Scale to \\(8 : 24\\) so the fractions removed are whole units. Removed: \\(\\dfrac{1}{4}\\) of red \\(= 2\\) units and \\(\\dfrac{3}{8}\\) of yellow \\(= 9\\) units, i.e. 11 units. Left \\(= 32 - 11 = 21\\) units \\(= 63\\) \\(\\Rightarrow\\) 1 unit \\(= 3\\). Yellow at first \\(= 24\\) units \\(= 72\\).",
      "spec": {
        "unit_px": 30,
        "bars": [
          {"label": "Red", "segments": [
              {"u": 8, "text": "8 units", "fill": "#f5d0d0"}]},
          {"label": "Yellow", "segments": [
              {"u": 24, "text": "24 units = ?", "fill": "#ffe7c2"}]}
        ]
      }
    },
    {
      "n": 40,
      "answer": "(a) \\(7.5\\) km  (b) \\(10\\) km/h",
      "answer_matches_key": true,
      "working": "(a) Home \\(\\to\\) library \\(= 15 \\times \\dfrac{1}{2} = 7.5\\) km. (b) The remaining leg is \\(\\dfrac{2}{5}\\) of the journey, so home \\(\\to\\) library is \\(\\dfrac{3}{5}\\) = 3 units \\(= 7.5\\) km \\(\\Rightarrow\\) 1 unit \\(= 2.5\\) km; remaining 2 units \\(= 5\\) km. Total \\(= 12.5\\) km. Total time \\(= \\dfrac{1}{2} + \\dfrac{3}{4} = 1\\dfrac{1}{4}\\) h. Average speed \\(= 12.5 \\div 1\\dfrac{1}{4} = 10\\) km/h.",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Journey", "segments": [
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}]},
          {"label": "Home\\u2192Library", "segments": [
              {"u": 3, "text": "7.5 km", "fill": "#dbe9ff"}]},
          {"label": "Library\\u2192Park", "segments": [
              {"u": 2, "text": "?", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "12.5 km", "bars": [1, 2]}
      }
    },
    {
      "n": 42,
      "answer": "\\(\\dfrac{3}{8}\\)",
      "answer_matches_key": true,
      "working": "Monday \\(= \\dfrac{1}{4} = \\dfrac{2}{8}\\). Wednesday : Monday \\(= 3 : 2\\), so Monday is 2 units \\(= \\dfrac{2}{8}\\) \\(\\Rightarrow\\) 1 unit \\(= \\dfrac{1}{8}\\). Wednesday \\(= 3\\) units \\(= \\dfrac{3}{8}\\).",
      "spec": {
        "unit_px": 90,
        "bars": [
          {"label": "Monday", "segments": [
              {"u": 1, "text": "1/8", "fill": "#ffe7c2"}, {"u": 1, "text": "1/8", "fill": "#ffe7c2"}],
           "brace": {"text": "2/8 = 1/4"}},
          {"label": "Wednesday", "segments": [
              {"u": 1, "text": "1/8", "fill": "#dbe9ff"}, {"u": 1, "text": "1/8", "fill": "#dbe9ff"}, {"u": 1, "text": "1/8", "fill": "#dbe9ff"}],
           "brace": {"text": "? = 3/8"}}
        ]
      }
    },
    {
      "n": 43,
      "answer": "(a) 240  (b) 264",
      "answer_matches_key": true,
      "working": "(a) Of the fruits sold, 40% were apples, so 60% were mangoes \\(= 144\\). \\(60\\% = 144 \\Rightarrow 100\\% = 240\\) fruits sold. (b) The 30% sold \\(= 240\\), so total fruits \\(= 800\\). Unsold \\(= 70\\% = 560\\); 70% of these were mangoes, so 30% \\(= 168\\) unsold apples. Apples sold \\(= 40\\% \\times 240 = 96\\). Apples at first \\(= 168 + 96 = 264\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Sold (30%)", "segments": [
              {"u": 3, "text": "mangoes 144", "fill": "#dbe9ff"}, {"u": 2, "text": "apples 96", "fill": "#ffe7c2"}],
           "brace": {"text": "240"}},
          {"label": "Unsold (70%)", "segments": [
              {"u": 7, "text": "mangoes 392 (70%)", "fill": "#dbe9ff"}, {"u": 3, "text": "apples 168 (30%)", "fill": "#ffe7c2"}],
           "brace": {"text": "560"}}
        ],
        "total": {"text": "800 total", "bars": [0, 1]}
      }
    },
    {
      "n": 45,
      "answer": "$720",
      "answer_matches_key": true,
      "working": "Lion is \\(\\dfrac{1}{3}\\), Dino is \\(\\dfrac{2}{3}\\), so Dino : Lion \\(= 2 : 1\\) by count. Use matching sets: 12 Dino (3 packs of 4 at \\$8.50 = \\$25.50) and 6 Lion (2 packs of 3 at \\$7.25 = \\$14.50). Per group, Dino costs \\$25.50 \\(-\\) \\$14.50 \\(=\\) \\$11 more. \\$198 \\(\\div\\) \\$11 \\(= 18\\) groups. Total spent \\(= (25.50 + 14.50) \\times 18 = \\$720\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Dino (per group)", "segments": [
              {"u": 1, "text": "$14.50", "fill": "#dbe9ff"}, {"u": 1, "text": "+$11", "fill": "#ffe7c2"}],
           "brace": {"text": "$25.50"}},
          {"label": "Lion (per group)", "segments": [
              {"u": 1, "text": "$14.50", "fill": "#dbe9ff"}]}
        ]
      }
    }
  ]
}