{
  "source_prefix": "ACS 2024 P6 Prelim",
  "models": [
    {
      "n": 3,
      "answer": "\\(\\dfrac{3}{5}\\)",
      "answer_matches_key": true,
      "working": "Total \\(= 12 + 18 = 30\\) pearls. White \\(= 18\\). Fraction white \\(= \\dfrac{18}{30} = \\dfrac{3}{5}\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Pink", "segments": [
              {"u": 1, "text": "12", "fill": "#ffd6e0"}]},
          {"label": "White", "segments": [
              {"u": 1, "text": "18", "fill": "#dbe9ff"}]}
        ],
        "total": {"text": "30 pearls", "bars": [0, 1]}
      }
    },
    {
      "n": 14,
      "answer": "\\(\\dfrac{3}{4}\\)",
      "answer_matches_key": true,
      "working": "Let yellow \\(= 3\\) units, so red \\(= 6\\) units. Yellow used \\(= \\dfrac{2}{3}\\times 3 = 2\\) units, yellow left \\(= 1\\) unit. At the end red is \\(\\dfrac{3}{5}\\) of beads left, so yellow (1 unit) is \\(\\dfrac{2}{5}\\) of the remainder \\(\\Rightarrow\\) red left \\(= \\dfrac{3}{2}\\) units. Red used \\(= 6 - \\dfrac{3}{2} = \\dfrac{9}{2}\\) units. Fraction of red used \\(= \\dfrac{9/2}{6} = \\dfrac{3}{4}\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Red", "segments": [
              {"u": 1, "fill": "#ffd6d6"}, {"u": 1, "fill": "#ffd6d6"}, {"u": 1, "fill": "#ffd6d6"},
              {"u": 1, "fill": "#ffd6d6"}, {"u": 1, "fill": "#ffd6d6"}, {"u": 1, "fill": "#ffd6d6"}],
           "brace": {"text": "6 units"}},
          {"label": "Yellow", "segments": [
              {"u": 1, "fill": "#fff2c2"}, {"u": 1, "fill": "#fff2c2"}, {"u": 1, "fill": "#fff2c2"}],
           "brace": {"text": "3 units"}}
        ]
      }
    },
    {
      "n": 15,
      "answer": "40%",
      "answer_matches_key": true,
      "working": "Chocolate : strawberry \\(= 5 : 7\\). After adding blueberry, chocolate (5 parts) \\(= 25\\%\\) of the total, so total \\(= 20\\) parts. Blueberry \\(= 20 - 5 - 7 = 8\\) parts \\(= \\dfrac{8}{20}\\times 100\\% = 40\\%\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Total", "segments": [
              {"u": 5, "text": "Choc (25%)", "fill": "#d8b08c"},
              {"u": 7, "text": "Strawberry", "fill": "#ffd6e0"},
              {"u": 8, "text": "Blueberry = ?", "fill": "#cdd6ff"}],
           "brace": {"text": "20 parts = 100%"}}
        ]
      }
    },
    {
      "n": 35,
      "answer": "$4000",
      "answer_matches_key": true,
      "working": "July savings \\(= 20\\%\\) of July income. After the income drops 15%, August savings \\(= 20\\%\\times 85\\% = 17\\%\\) of July income. The fall in savings \\(= 20\\% - 17\\% = 3\\%\\) of July income \\(= \\$120\\). So \\(1\\% = \\$40\\) and July income (100%) \\(= \\$4000\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "July savings", "segments": [
              {"u": 17, "fill": "#cfe8d0"}, {"u": 3, "text": "$120", "fill": "#ffd6d6"}],
           "brace": {"text": "20% of income"}},
          {"label": "Aug savings", "segments": [
              {"u": 17, "fill": "#cfe8d0"}],
           "brace": {"text": "17% of July income"}}
        ]
      }
    },
    {
      "n": 37,
      "answer": "160",
      "answer_matches_key": true,
      "working": "Monday \\(= 24\\) packages (read from graph) \\(= 15\\%\\) of the 5-day total. So total \\(= 24 \\div 0.15 = 160\\) packages. (Part (b): Thu \\(+\\) Fri \\(= 160 - 24 - 48 - 32 = 56\\), split \\(3:4\\) gives Thu \\(= 24\\), Fri \\(= 32\\).)",
      "spec": {
        "unit_px": 64,
        "bars": [
          {"label": "Monday", "segments": [
              {"u": 3, "text": "24", "fill": "#dbe9ff"}],
           "brace": {"text": "15%"}},
          {"label": "5-day total", "segments": [
              {"u": 3, "text": "24", "fill": "#dbe9ff"},
              {"u": 17, "text": "remaining 85% = ?", "fill": "#eef2f7"}],
           "brace": {"text": "100% = 160"}}
        ]
      }
    },
    {
      "n": 45,
      "answer": "(a) 1800 cm³  (b) 360 cm³",
      "answer_matches_key": true,
      "working": "(a) Tank A volume \\(= 30\\times 10\\times 15 = 4500\\) cm³; water \\(= \\dfrac{2}{5}\\times 4500 = 1800\\) cm³. (b) The 3 containers (equal height) have base areas \\(60 : 120 : 120 = 1 : 2 : 2\\), i.e. 5 equal volume units. Small container \\(= \\dfrac{1}{5}\\times 1800 = 360\\) cm³.",
      "spec": {
        "unit_px": 90,
        "bars": [
          {"label": "Small", "segments": [
              {"u": 1, "text": "?", "fill": "#dbe9ff"}]},
          {"label": "Large 1", "segments": [
              {"u": 1, "fill": "#cfe8d0"}, {"u": 1, "fill": "#cfe8d0"}]},
          {"label": "Large 2", "segments": [
              {"u": 1, "fill": "#cfe8d0"}, {"u": 1, "fill": "#cfe8d0"}]}
        ],
        "total": {"text": "1800 cm³", "bars": [0, 1, 2]}
      }
    },
    {
      "n": 47,
      "answer": "232",
      "answer_matches_key": true,
      "working": "Pearls left \\(= \\dfrac{1}{3}\\) of pearls (so \\(\\dfrac{2}{3}\\) sold); beads left \\(= \\dfrac{1}{4}\\) of beads (so \\(\\dfrac{3}{4}\\) sold). Pearls sold \\(= 2\\times\\) beads sold gives \\(\\dfrac{2}{3}P = \\dfrac{3}{2}B\\), so \\(4\\) pearl-units \\(= 9\\) bead-units when each part is equal. Combined total \\(= 4 + 9 = 13\\) units \\(= 754\\), so 1 unit \\(= 58\\). Necklaces left \\(= \\dfrac{1}{3}P + \\dfrac{1}{4}B = 4\\) units \\(= 4\\times 58 = 232\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Pearl", "segments": [
              {"u": 3, "text": "sold \\(\\frac{2}{3}\\)", "fill": "#ffe7c2"},
              {"u": 1, "text": "left", "fill": "#ffd089"}],
           "brace": {"text": "9 units"}},
          {"label": "Bead", "segments": [
              {"u": 3, "text": "sold \\(\\frac{3}{4}\\)", "fill": "#cdd6ff"},
              {"u": 1, "text": "left", "fill": "#9fb4ff"}],
           "brace": {"text": "4 units"}}
        ],
        "total": {"text": "13 units = 754", "bars": [0, 1]}
      }
    },
    {
      "n": 49,
      "answer": "\\(\\dfrac{5}{16}\\)",
      "answer_matches_key": true,
      "working": "Remaining after notebooks/files \\(= 1 - \\dfrac{5}{8} = \\dfrac{3}{8}\\). Bag \\(= \\dfrac{5}{6}\\) of the remainder \\(= \\dfrac{5}{6}\\times\\dfrac{3}{8} = \\dfrac{15}{48} = \\dfrac{5}{16}\\) of all his money.",
      "spec": {
        "unit_px": 48,
        "bars": [
          {"label": "All money", "segments": [
              {"u": 10, "text": "books + files \\(\\frac{5}{8}\\)", "fill": "#dbe9ff"},
              {"u": 5, "text": "bag = ?", "fill": "#ffd089"},
              {"u": 1, "text": "left", "fill": "#cfe8d0"}],
           "brace": {"text": "16 sixteenths"}}
        ]
      }
    },
    {
      "n": 50,
      "answer": "$8",
      "answer_matches_key": true,
      "working": "From (a) the bag is \\(\\dfrac{5}{16}\\) of the money and books/files are \\(\\dfrac{5}{8} = \\dfrac{10}{16}\\), so money left \\(= 1 - \\dfrac{10}{16} - \\dfrac{5}{16} = \\dfrac{1}{16}\\). Key working: \\(20nb + \\$40 = 2nb + \\$76 \\Rightarrow nb = \\$2\\); books \\(+\\) files \\(= \\$80 = \\dfrac{5}{8}\\) of money, so \\(5\\) units \\(= \\$40\\), \\(1\\) unit \\(= \\$8\\), total \\(16\\) units \\(= \\$128\\). Spent \\(\\$80 + \\$40 = \\$120\\); left \\(= \\$128 - \\$120 = \\$8\\).",
      "spec": {
        "unit_px": 48,
        "bars": [
          {"label": "Total $128", "segments": [
              {"u": 10, "text": "books+files $80", "fill": "#dbe9ff"},
              {"u": 5, "text": "bag $40", "fill": "#ffd089"},
              {"u": 1, "text": "$8", "fill": "#cfe8d0"}],
           "brace": {"text": "16 units = $128"}}
        ]
      }
    }
  ]
}