{
  "source_prefix": "Nanyang 2022 P6 Prelim",
  "models": [
    {
      "n": 8,
      "answer": "$2.50",
      "answer_matches_key": true,
      "working": "Huiling \\(= z\\), Ravi \\(= 2z\\), Jas \\(= 2z + 5\\). Jas \\(= \\$10\\), so \\(2z + 5 = 10\\), \\(2z = 5\\), \\(z = 2.5\\). Huiling had \\(\\$2.50\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Huiling", "segments": [{"u": 1, "text": "z", "fill": "#dbe9ff"}]},
          {"label": "Ravi", "segments": [{"u": 1, "text": "z", "fill": "#dbe9ff"}, {"u": 1, "text": "z", "fill": "#dbe9ff"}]},
          {"label": "Jas", "segments": [{"u": 1, "text": "z", "fill": "#dbe9ff"}, {"u": 1, "text": "z", "fill": "#dbe9ff"}, {"u": 1, "text": "$5", "fill": "#ffe7c2"}], "brace": {"text": "$10"}}
        ]
      }
    },
    {
      "n": 14,
      "answer": "\\(63 : 24 : 28\\)",
      "answer_matches_key": true,
      "working": "Each girl used the SAME number of beads. Make the used parts equal at 21 units. Viv used \\(\\dfrac{1}{3}\\) so total \\(= 3 \\times 21 = 63\\); Wendy used \\(\\dfrac{7}{8}\\) so total \\(= \\dfrac{8}{7} \\times 21 = 24\\); Xinyi used \\(\\dfrac{3}{4}\\) so total \\(= \\dfrac{4}{3} \\times 21 = 28\\). Ratio \\(= 63 : 24 : 28\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Viv", "segments": [{"u": 1, "text": "used", "fill": "#ffe7c2"}, {"u": 2, "fill": "#dbe9ff"}], "brace": {"text": "63"}},
          {"label": "Wendy", "segments": [{"u": 7, "text": "used", "fill": "#ffe7c2"}, {"u": 1, "fill": "#dbe9ff"}], "brace": {"text": "24"}},
          {"label": "Xinyi", "segments": [{"u": 3, "text": "used", "fill": "#ffe7c2"}, {"u": 1, "fill": "#dbe9ff"}], "brace": {"text": "28"}}
        ]
      }
    },
    {
      "n": 16,
      "answer": "\\(\\dfrac{8}{15}\\)",
      "answer_matches_key": true,
      "working": "Red \\(= 6 + 2 = 8\\). Total \\(= 6 + 3 + 2 + 4 = 15\\). Fraction red \\(= \\dfrac{8}{15}\\).",
      "spec": {
        "unit_px": 38,
        "bars": [
          {"label": "All marbles", "segments": [
              {"u": 1, "fill": "#f8b4b4"}, {"u": 1, "fill": "#f8b4b4"}, {"u": 1, "fill": "#f8b4b4"}, {"u": 1, "fill": "#f8b4b4"},
              {"u": 1, "fill": "#f8b4b4"}, {"u": 1, "fill": "#f8b4b4"}, {"u": 1, "fill": "#f8b4b4"}, {"u": 1, "fill": "#f8b4b4"},
              {"u": 1, "fill": "#ffffff"}, {"u": 1, "fill": "#ffffff"}, {"u": 1, "fill": "#ffffff"}, {"u": 1, "fill": "#ffffff"},
              {"u": 1, "fill": "#ffffff"}, {"u": 1, "fill": "#ffffff"}, {"u": 1, "fill": "#ffffff"}],
           "brace": {"text": "15 marbles total"}}
        ],
        "total": {"text": "8 red", "bars": [0]}
      }
    },
    {
      "n": 24,
      "answer": "30",
      "answer_matches_key": true,
      "working": "2021 monthly savings \\(= 20\\% \\times \\$3000 = \\$600\\). The increase is \\(\\$180\\). Percentage increase \\(= \\dfrac{180}{600} \\times 100\\% = 30\\%\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "2021 saved", "segments": [{"u": 3, "text": "$600", "fill": "#dbe9ff"}]},
          {"label": "2022 saved", "segments": [{"u": 3, "text": "$600", "fill": "#dbe9ff"}, {"u": 1, "text": "+$180", "fill": "#ffe7c2"}]}
        ]
      }
    },
    {
      "n": 26,
      "answer": "63",
      "answer_matches_key": true,
      "working": "After Monday, \\(\\dfrac{2}{3}\\) of the apples remain. On Tuesday she sells \\(\\dfrac{2}{3}\\) of that remainder, so \\(\\dfrac{1}{3}\\) of the remainder \\(= 14\\) apples. Monday remainder \\(= 14 \\times 3 = 42\\), which is \\(\\dfrac{2}{3}\\) of the original. Original \\(= 42 \\div \\dfrac{2}{3} = 42 \\times \\dfrac{3}{2} = 63\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "At first", "segments": [
              {"u": 1, "text": "Mon", "fill": "#ffe7c2"}, {"u": 1, "text": "14", "fill": "#dbe9ff"}, {"u": 1, "text": "Tue", "fill": "#ffe7c2"}],
           "brace": {"text": "? (63)"}}
        ]
      }
    },
    {
      "n": 27,
      "answer": "6",
      "answer_matches_key": true,
      "working": "Remaining \\(= 1 - \\dfrac{1}{5} = \\dfrac{4}{5}\\) m. Number of \\(\\dfrac{1}{8}\\)-m pieces \\(= \\dfrac{4}{5} \\div \\dfrac{1}{8} = \\dfrac{4}{5} \\times 8 = \\dfrac{32}{5} = 6\\dfrac{2}{5}\\). At most 6 whole pieces.",
      "spec": {
        "unit_px": 96,
        "bars": [
          {"label": "1 m roll", "segments": [{"u": 1, "text": "cut off 1/5 m", "fill": "#ffe7c2"}, {"u": 4, "text": "remaining 4/5 m → ÷ 1/8 m", "fill": "#dbe9ff"}]}
        ]
      }
    },
    {
      "n": 30,
      "answer": "2040",
      "answer_matches_key": true,
      "working": "Pam \\(= m = 100\\) kg/month. Sweet \\(= m - 30 = 70\\) kg/month. Combined per month \\(= 100 + 70 = 170\\) kg. One year \\(= 170 \\times 12 = 2040\\) kg.",
      "spec": {
        "unit_px": 28,
        "bars": [
          {"label": "Pam", "segments": [{"u": 7, "text": "70", "fill": "#dbe9ff"}, {"u": 3, "text": "30", "fill": "#ffe7c2"}], "brace": {"text": "100 kg"}},
          {"label": "Sweet", "segments": [{"u": 7, "text": "70", "fill": "#dbe9ff"}], "brace": {"text": "70 kg"}}
        ]
      }
    },
    {
      "n": 31,
      "answer": "$8",
      "answer_matches_key": true,
      "working": "15 books at \\(\\$k\\) each, less a \\(\\$10\\) discount, cost \\(\\$110\\). So \\(15k - 10 = 110\\), \\(15k = 120\\), \\(k = 8\\). The original price of one book was \\(\\$8\\).",
      "spec": {
        "unit_px": 36,
        "bars": [
          {"label": "15 books", "segments": [
              {"u": 1, "text": "k", "fill": "#dbe9ff"}, {"u": 1, "text": "k", "fill": "#dbe9ff"}, {"u": 1, "text": "k", "fill": "#dbe9ff"},
              {"u": 1, "text": "k", "fill": "#dbe9ff"}, {"u": 1, "text": "k", "fill": "#dbe9ff"}, {"u": 1, "text": "k", "fill": "#dbe9ff"},
              {"u": 1, "text": "k", "fill": "#dbe9ff"}, {"u": 1, "text": "k", "fill": "#dbe9ff"}, {"u": 1, "text": "k", "fill": "#dbe9ff"},
              {"u": 1, "text": "k", "fill": "#dbe9ff"}, {"u": 1, "text": "k", "fill": "#dbe9ff"}, {"u": 1, "text": "k", "fill": "#dbe9ff"},
              {"u": 1, "text": "k", "fill": "#dbe9ff"}, {"u": 1, "text": "k", "fill": "#dbe9ff"}, {"u": 1, "text": "k", "fill": "#dbe9ff"}],
           "brace": {"text": "$120 (= $110 paid + $10 discount)"}}
        ]
      }
    },
    {
      "n": 34,
      "answer": "850",
      "answer_matches_key": true,
      "working": "Dana paid \\(85\\% \\times \\$800 = \\$680\\) (15% off). Hailey paid the same \\(\\$680\\), which is \\(80\\%\\) of Shop B's price. Shop B price \\(= 680 \\div 80\\% = 680 \\times \\dfrac{100}{80} = \\$850\\).",
      "spec": {
        "unit_px": 36,
        "bars": [
          {"label": "Shop A $800", "segments": [
              {"u": 17, "text": "paid $680 (85%)", "fill": "#dbe9ff"}, {"u": 3, "text": "15% off", "fill": "#ffe7c2"}]},
          {"label": "Shop B ?", "segments": [
              {"u": 17, "text": "paid $680 (80%)", "fill": "#dbe9ff"}, {"u": 3, "fill": "#dbe9ff"}, {"u": 1, "text": "20% off", "fill": "#ffe7c2"}],
           "brace": {"text": "? = $850"}}
        ]
      }
    },
    {
      "n": 37,
      "answer": "$23.70",
      "answer_matches_key": true,
      "working": "At first Peter is \\(\\$18.20\\) behind Jane; afterwards he is \\(\\$29.20\\) ahead. The gap swings by \\(18.20 + 29.20 = \\$47.40\\). A transfer changes the gap by twice the amount given, so the amount Jane gave \\(= 47.40 \\div 2 = \\$23.70\\).",
      "spec": {
        "unit_px": 40,
        "bars": [
          {"label": "Before · Peter", "segments": [{"u": 4, "fill": "#dbe9ff"}]},
          {"label": "Before · Jane", "segments": [{"u": 4, "fill": "#ffe7c2"}, {"u": 2, "text": "$18.20 more", "fill": "#ffe7c2"}]},
          {"label": "After · Peter", "segments": [{"u": 4, "fill": "#dbe9ff"}, {"u": 3, "text": "$29.20 more", "fill": "#dbe9ff"}]},
          {"label": "After · Jane", "segments": [{"u": 4, "fill": "#ffe7c2"}]}
        ]
      }
    },
    {
      "n": 38,
      "answer": "\\(\\dfrac{21}{111}\\)",
      "answer_matches_key": true,
      "working": "Blue length is \\(\\dfrac{1}{2}\\) the red length; use red \\(= 126\\) cm, blue \\(= 63\\) cm. Blue: \\(63 \\div 9 = 7\\) parts \\(\\times 3 = 21\\) stars. Red: \\(126 \\div 7 = 18\\) parts \\(\\times 5 = 90\\) hearts. Star fraction \\(= \\dfrac{21}{21 + 90} = \\dfrac{21}{111}\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Blue → stars", "segments": [{"u": 1, "text": "21 stars", "fill": "#dbe9ff"}]},
          {"label": "Red → hearts", "segments": [{"u": 2, "text": "90 hearts", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "111 shapes", "bars": [0, 1]}
      }
    },
    {
      "n": 39,
      "answer": "240",
      "answer_matches_key": true,
      "working": "\\(A + B = 2 \\times 324 = 648\\). \\(B + C + D = 3 \\times 344 = 1032\\). Adding these counts B twice, so total \\(A + B + C + D = 648 + 1032 - B = 1680 - B\\). This equals \\(6B\\): \\(1680 - B = 6B\\), \\(7B = 1680\\), \\(B = 240\\).",
      "spec": {
        "unit_px": 90,
        "bars": [
          {"label": "A + B", "segments": [{"u": 1, "text": "648", "fill": "#dbe9ff"}]},
          {"label": "B + C + D", "segments": [{"u": 1, "text": "1032", "fill": "#ffe7c2"}]},
          {"label": "Total = 6B", "segments": [
              {"u": 1, "text": "B", "fill": "#d7f0d7"}, {"u": 1, "text": "B", "fill": "#d7f0d7"}, {"u": 1, "text": "B", "fill": "#d7f0d7"},
              {"u": 1, "text": "B", "fill": "#d7f0d7"}, {"u": 1, "text": "B", "fill": "#d7f0d7"}, {"u": 1, "text": "B", "fill": "#d7f0d7"}],
           "brace": {"text": "648 + 1032 − B = 1680 − B"}}
        ]
      }
    },
    {
      "n": 41,
      "answer": "(a) 42000  (b) 12.5",
      "answer_matches_key": true,
      "working": "(a) Volume in Y \\(= 40 \\times 25 \\times 42 = 42\\,000\\) cm³. (b) Total water stays \\(42\\,000\\) cm³, shared Y : Z \\(= 2 : 5\\). So 7 units \\(= 42\\,000\\), 1 unit \\(= 6000\\); Z has 5 units \\(= 30\\,000\\) cm³. Height in Z \\(= 30\\,000 \\div (80 \\times 30) = 30\\,000 \\div 2400 = 12.5\\) cm.",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Tank Y", "segments": [{"u": 1, "text": "6000", "fill": "#dbe9ff"}, {"u": 1, "text": "6000", "fill": "#dbe9ff"}]},
          {"label": "Tank Z", "segments": [
              {"u": 1, "text": "6000", "fill": "#ffe7c2"}, {"u": 1, "text": "6000", "fill": "#ffe7c2"}, {"u": 1, "text": "6000", "fill": "#ffe7c2"},
              {"u": 1, "text": "6000", "fill": "#ffe7c2"}, {"u": 1, "text": "6000", "fill": "#ffe7c2"}],
           "brace": {"text": "30 000 cm³ → 12.5 cm"}}
        ],
        "total": {"text": "42 000 cm³", "bars": [0, 1]}
      }
    },
    {
      "n": 44,
      "answer": "\\(40 : 27 : 9\\)",
      "answer_matches_key": true,
      "working": "Gold : (silver + copper) \\(= 10 : 9\\). Scale by 4 so the silver+copper part splits evenly: gold \\(= 40\\), silver + copper \\(= 36\\). Silver : copper \\(= 3 : 1\\), so of the 36, silver \\(= 27\\) and copper \\(= 9\\). Therefore gold : silver : copper \\(= 40 : 27 : 9\\).",
      "spec": {
        "unit_px": 18,
        "bars": [
          {"label": "Gold", "segments": [{"u": 40, "text": "40", "fill": "#ffe9a8"}]},
          {"label": "Silver + Copper", "segments": [{"u": 27, "text": "silver 27", "fill": "#dbe9ff"}, {"u": 9, "text": "copper 9", "fill": "#ffd9b3"}]}
        ]
      }
    },
    {
      "n": 45,
      "answer": "45",
      "answer_matches_key": true,
      "working": "Copper is unchanged. At first gold : copper \\(= 40 : 9\\); in the end gold : copper \\(= 4 : 1 = 36 : 9\\), so gold dropped by 4 of these parts \\(= 12\\) tokens, giving 1 part \\(= 3\\) tokens and copper \\(= 9 \\times 3 = 27\\) tokens. End silver : copper \\(= 4 : 3\\), so end silver \\(= \\dfrac{4}{3} \\times 27 = 36\\). At first silver \\(= 27 \\times 3 = 81\\). Silver exchanged \\(= 81 - 36 = 45\\).",
      "spec": {
        "unit_px": 24,
        "bars": [
          {"label": "Silver at first", "segments": [{"u": 9, "text": "81", "fill": "#dbe9ff"}], "brace": {"text": "81 tokens"}},
          {"label": "Silver at end", "segments": [{"u": 4, "text": "36", "fill": "#dbe9ff"}, {"u": 5, "text": "exchanged = 45", "fill": "#ffe7c2"}]}
        ]
      }
    }
  ]
}