{
  "source_prefix": "Catholic High 2022 P6 Prelim",
  "models": [
    {
      "n": 13,
      "answer": "60",
      "answer_matches_key": true,
      "working": "Take the sweets as 1 unit, so chocolates \\(= 4\\) units. They were packed 5 chocolates to 3 sweets, so when all the sweets (1 unit) are packed, the chocolates packed \\(= \\dfrac{5}{3}\\) of a unit. Chocolates left \\(= 4 - \\dfrac{5}{3} = \\dfrac{7}{3}\\) units \\(= 35\\). So 1 unit \\(= 15\\) sweets, and chocolates bought \\(= 4 \\times 15 = 60\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Chocolates", "segments": [
            {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}],
           "brace": {"text": "?"}},
          {"label": "Sweets", "segments": [{"u": 1, "fill": "#ffe7c2"}]}
        ]
      }
    },
    {
      "n": 15,
      "answer": "350 g",
      "answer_matches_key": false,
      "working": "Making 3 cakes left \\(\\dfrac{3}{4}\\) of the packet, so 3 cakes used \\(\\dfrac{1}{4}\\) of the packet, i.e. 1 cake \\(= \\dfrac{1}{12}\\) of the packet. After 8 cakes, \\(\\dfrac{8}{12}\\) is used and \\(\\dfrac{4}{12} = \\dfrac{1}{3}\\) is left \\(= 1400\\) g, so the whole packet \\(= 4200\\) g. Each cake \\(= \\dfrac{1}{12} \\times 4200 = 350\\) g. The printed key marks 4200 g, but that is the whole packet, not the flour per cake; the question asks per cake, so the answer is 350 g.",
      "spec": {
        "unit_px": 130,
        "bars": [
          {"label": "Packet", "segments": [
            {"u": 1, "text": "4 cakes", "fill": "#cfe8cf"},
            {"u": 1, "text": "4 cakes", "fill": "#cfe8cf"},
            {"u": 1, "text": "\\u2153 left = 1400 g", "fill": "#ffe7c2"}],
           "brace": {"text": "? g per packet"}}
        ]
      }
    },
    {
      "n": 29,
      "answer": "$197",
      "answer_matches_key": true,
      "working": "Her money is fixed. Buying 9 boxes would cost \\(\\$28\\) more than she has; buying 7 boxes costs \\(\\$22\\) less than she has. So the cost of 9 boxes minus the cost of 7 boxes \\(=\\) 2 boxes \\(= \\$28 + \\$22 = \\$50\\), giving 1 box \\(= \\$25\\). Money at first \\(=\\) cost of 7 boxes \\(+ \\$22 = 7 \\times \\$25 + \\$22 = \\$175 + \\$22 = \\$197\\).",
      "spec": {
        "unit_px": 90,
        "bars": [
          {"label": "Cost of 9 boxes", "segments": [
            {"u": 5, "text": "her money", "fill": "#dbe9ff"}, {"u": 1, "text": "$28 short", "fill": "#ffd9d9"}]},
          {"label": "Cost of 7 boxes", "segments": [
            {"u": 4, "text": "money spent", "fill": "#cfe8cf"}, {"u": 1, "text": "$22 left", "fill": "#fff2b8"}]}
        ],
        "total": {"text": "diff = 2 boxes = $50", "bars": [0, 1]}
      }
    },
    {
      "n": 104,
      "answer": "30 cm",
      "answer_matches_key": true,
      "working": "The same roll makes 9 short ribbons or 5 long ribbons, so 5 long \\(=\\) 9 short in total length. Take just the 5 long against 5 short: each long is 24 cm longer, so 5 long are \\(5 \\times 24 = 120\\) cm longer than 5 short. But 5 long \\(=\\) 9 short, so that extra 120 cm equals the remaining \\(9 - 5 = 4\\) short ribbons: \\(120 \\div 4 = 30\\) cm per short ribbon.",
      "spec": {
        "unit_px": 95,
        "bars": [
          {"label": "5 long", "segments": [
            {"u": 5, "text": "5 short", "fill": "#ffe7c2"}, {"u": 1, "text": "120 cm", "fill": "#fff2b8"}]},
          {"label": "5 short", "segments": [
            {"u": 5, "text": "= 4 short of the roll", "fill": "#dbe9ff"}]}
        ]
      }
    },
    {
      "n": 108,
      "answer": "$1.60",
      "answer_matches_key": true,
      "working": "For every 3 buns at full price, 1 bun is half price, so a \"set\" \\(= 4\\) buns. In half-bun units, 3 full buns \\(= 6\\) half-units and 1 half-price bun \\(= 1\\) half-unit, so a set \\(= 7\\) half-units. 12 buns \\(= 3\\) sets \\(= 3 \\times 7 = 21\\) half-units \\(= \\$16.80\\). So 1 half-unit \\(= \\$0.80\\), and the usual price of one bun \\(= 2 \\times \\$0.80 = \\$1.60\\).",
      "spec": {
        "unit_px": 110,
        "bars": [
          {"label": "1 set (4 buns)", "segments": [
            {"u": 2, "text": "bun", "fill": "#dbe9ff"}, {"u": 2, "text": "bun", "fill": "#dbe9ff"}, {"u": 2, "text": "bun", "fill": "#dbe9ff"},
            {"u": 1, "text": "\\u00bd bun", "fill": "#ffe7c2"}],
           "brace": {"text": "7 half-units"}},
          {"label": "Paid (3 sets)", "segments": [
            {"u": 3, "text": "21 half-units = $16.80", "fill": "#cfe8cf"}]}
        ]
      }
    },
    {
      "n": 113,
      "answer": "304",
      "answer_matches_key": true,
      "working": "Take the plums bought as 1 whole. After throwing away 152 damaged plums, he sold \\(\\dfrac{1}{5}\\) of the remainder and kept \\(\\dfrac{4}{5}\\) of it, which equals \\(\\dfrac{4}{7}\\) of all plums bought. So the remainder after throwing away \\(= \\dfrac{5}{7}\\) of plums bought, meaning the 152 thrown away \\(= \\dfrac{2}{7}\\) of plums bought: \\(\\dfrac{1}{7} = 76\\). The plums packed (kept) \\(= \\dfrac{4}{7} = 4 \\times 76 = 304\\).",
      "spec": {
        "unit_px": 75,
        "bars": [
          {"label": "Plums bought", "segments": [
            {"u": 1, "text": "152", "fill": "#ffd9d9"}, {"u": 1, "fill": "#ffd9d9"},
            {"u": 1, "text": "sold", "fill": "#fff2b8"},
            {"u": 1, "fill": "#cfe8cf"}, {"u": 1, "fill": "#cfe8cf"}, {"u": 1, "fill": "#cfe8cf"}, {"u": 1, "fill": "#cfe8cf"}],
           "brace": {"text": "7 units"}}
        ],
        "total": {"text": "kept = 4 units = ?", "bars": [0]}
      }
    },
    {
      "n": 114,
      "answer": "98",
      "answer_matches_key": true,
      "working": "After 25% of the white balloons burst, white left \\(= 75\\%\\) of the original white. In the end red \\(=\\) white \\(+ 26\\) and red \\(+\\) white \\(= 170\\). So \\(2 \\times (\\text{white left}) + 26 = 170\\), giving white left \\(= 72\\) and red \\(= 72 + 26 = 98\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Red (end)", "segments": [
            {"u": 3, "text": "= white", "fill": "#ffd9d9"}, {"u": 1, "text": "26", "fill": "#ffb3b3"}],
           "brace": {"text": "?"}},
          {"label": "White (end)", "segments": [
            {"u": 3, "text": "72", "fill": "#eef3ff"}]}
        ],
        "total": {"text": "170", "bars": [0, 1]}
      }
    },
    {
      "n": 115,
      "answer": "(a) 5 nuts (b) 0.28 kg",
      "answer_matches_key": true,
      "working": "Bolt : nut \\(= 5 : 3\\) in mass units. (a) 3 bolts \\(= 3 \\times 5 = 15\\) units \\(= 15 \\div 3 = 5\\) nuts. (b) Box + 30 bolts \\(= 30 \\times 5 = 150\\) units \\(= 1.18\\) kg; box + 70 nuts \\(= 70 \\times 3 = 210\\) units \\(= 1.54\\) kg. The extra \\(210 - 150 = 60\\) units \\(= 1.54 - 1.18 = 0.36\\) kg, so 1 unit \\(= 0.006\\) kg. 30 bolts \\(= 150\\) units \\(= 0.9\\) kg, so the box \\(= 1.18 - 0.9 = 0.28\\) kg.",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Box + 30 bolts", "segments": [
            {"u": 1, "text": "box", "fill": "#d9d9d9"}, {"u": 5, "text": "150 units", "fill": "#dbe9ff"}],
           "brace": {"text": "1.18 kg"}},
          {"label": "Box + 70 nuts", "segments": [
            {"u": 1, "text": "box", "fill": "#d9d9d9"}, {"u": 7, "text": "210 units", "fill": "#ffe7c2"}],
           "brace": {"text": "1.54 kg"}}
        ]
      }
    }
  ]
}