{
  "source_prefix": "Henry Park 2022 P6 Prelim",
  "models": [
    {
      "n": 11,
      "answer": "3",
      "answer_matches_key": true,
      "working": "Green \\(=\\dfrac13\\) of total \\(=20\\), so 1 third \\(=20\\) and total \\(=60\\). Purple \\(+\\) red \\(=\\dfrac14\\) of \\(60=15\\). Split \\(15\\) in the ratio purple : red \\(=4:1\\), so \\(5\\) parts \\(=15\\), \\(1\\) part \\(=3\\). Red \\(=3\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Purple", "segments": [
              {"u": 1, "fill": "#e6ccff"}, {"u": 1, "fill": "#e6ccff"}, {"u": 1, "fill": "#e6ccff"}, {"u": 1, "fill": "#e6ccff"}]},
          {"label": "Red", "segments": [{"u": 1, "text": "?", "fill": "#ffd2d2"}]}
        ],
        "total": {"text": "15", "bars": [0, 1]}
      }
    },
    {
      "n": 12,
      "answer": "\\(\\dfrac{11}{24}\\)",
      "answer_matches_key": true,
      "working": "Make both boxes hold the same total. Box 1 is black : white \\(=2:1\\) (3 parts); Box 2 is \\(5:7\\) (12 parts). Scale Box 1 to 12 parts: \\(8\\) black : \\(4\\) white. Box 2: \\(5\\) black : \\(7\\) white. Grand total \\(=24\\) parts; white \\(=4+7=11\\). Fraction white \\(=\\dfrac{11}{24}\\).",
      "spec": {
        "unit_px": 50,
        "bars": [
          {"label": "Box 1", "segments": [
              {"u": 8, "text": "8 black", "fill": "#cccccc"}, {"u": 4, "text": "4 white", "fill": "#ffffff"}]},
          {"label": "Box 2", "segments": [
              {"u": 5, "text": "5 black", "fill": "#cccccc"}, {"u": 7, "text": "7 white", "fill": "#ffffff"}]}
        ],
        "total": {"text": "24 units", "bars": [0, 1]}
      }
    },
    {
      "n": 15,
      "answer": "51",
      "answer_matches_key": true,
      "working": "Each had 60; total stays \\(180\\). Joan gave \\(\\dfrac25\\) of \\(60=24\\), keeping \\(36\\). Let Siti give \\(s\\); Joan \\(+\\) Siti left \\(=36+(60-s)=96-s\\). Xiuli \\(=60+24+s=84+s\\) and Xiuli \\(=3\\times(96-s)\\). So \\(84+s=288-3s\\Rightarrow 4s=204\\Rightarrow s=51\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Joan + Siti left", "segments": [{"u": 1, "text": "96 − s", "fill": "#dbe9ff"}]},
          {"label": "Xiuli", "segments": [
              {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}],
           "brace": {"text": "3 × (96 − s) = 84 + s"}}
        ]
      }
    },
    {
      "n": 23,
      "answer": "4",
      "answer_matches_key": true,
      "working": "Two sisters got \\(2\\times\\dfrac16=\\dfrac13\\). Remaining \\(=1-\\dfrac13=\\dfrac23\\). Put aside for brothers \\(=\\dfrac23\\) of \\(\\dfrac23=\\dfrac49\\). Each brother got \\(\\dfrac19\\), so number of brothers \\(=\\dfrac49\\div\\dfrac19=4\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "All stickers", "segments": [
              {"u": 1, "text": "sis", "fill": "#ffd2d2"}, {"u": 1, "text": "sis", "fill": "#ffd2d2"},
              {"u": 1, "text": "1/9", "fill": "#c2f0c2"}, {"u": 1, "text": "1/9", "fill": "#c2f0c2"},
              {"u": 1, "text": "1/9", "fill": "#c2f0c2"}, {"u": 1, "text": "1/9", "fill": "#c2f0c2"}]}
        ],
        "total": {"text": "9 ninths = whole", "bars": [0]}
      }
    },
    {
      "n": 31,
      "answer": "$14",
      "answer_matches_key": true,
      "working": "All the money buys exactly \\(10\\) notebooks. Buying \\(6\\) notebooks leaves \\(4\\) notebooks' worth, which was spent on \\(4\\) pens (\\(4\\times\\$0.80=\\$3.20\\)) plus \\(\\$2.40\\) left. So \\(4\\) notebooks \\(=\\$3.20+\\$2.40=\\$5.60\\); \\(1\\) notebook \\(=\\$1.40\\); all money \\(=10\\times\\$1.40=\\$14\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "10 notebooks", "segments": [
              {"u": 6, "text": "6 bought", "fill": "#dbe9ff"},
              {"u": 4, "text": "4 = $3.20 + $2.40", "fill": "#ffe7c2"}],
           "brace": {"text": "all money = ?"}}
        ]
      }
    },
    {
      "n": 36,
      "answer": "744",
      "answer_matches_key": true,
      "working": "Let Siti \\(=s\\), Jane \\(=s+432\\). After: Jane keeps \\(\\dfrac34(s+432)\\), Siti keeps \\(\\dfrac38 s\\). New difference \\(=\\dfrac34(s+432)-\\dfrac38 s=441\\). \\(\\dfrac34 s+324-\\dfrac38 s=441\\Rightarrow\\dfrac38 s=117\\Rightarrow s=312\\). Jane at first \\(=312+432=744\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Jane", "segments": [
              {"u": 1, "text": "Siti's amount", "fill": "#dbe9ff"}, {"u": 1, "text": "432 more", "fill": "#ffe7c2"}],
           "brace": {"text": "Jane = ?"}},
          {"label": "Siti", "segments": [{"u": 1, "fill": "#dbe9ff"}]}
        ]
      }
    },
    {
      "n": 37,
      "answer": "5",
      "answer_matches_key": true,
      "working": "Empty containers are \\(100\\%-70\\%=30\\%=120\\), so \\(10\\%=40\\) and total \\(=400\\) containers. Filled \\(=70\\%\\) of \\(400=280\\). Volume per container \\(=1400\\div280=5\\) litres.",
      "spec": {
        "unit_px": 90,
        "bars": [
          {"label": "Filled 70%", "segments": [
              {"u": 7, "text": "280 containers = 1400 ℓ", "fill": "#cfe8ff"}]},
          {"label": "Empty 30%", "segments": [
              {"u": 3, "text": "120 containers", "fill": "#eeeeee"}]}
        ],
        "total": {"text": "100% = total", "bars": [0, 1]}
      }
    },
    {
      "n": 38,
      "answer": "90",
      "answer_matches_key": true,
      "working": "In the same \\(35\\) min, distance ratio equals speed ratio, so Grace's speed \\(=\\dfrac57\\) of Jen's speed. Jen \\(-\\) Grace \\(=36\\) m/min, i.e. \\(\\dfrac27\\) of Jen's speed \\(=36\\), so \\(\\dfrac17=18\\) and Jen \\(=126\\). Grace \\(=\\dfrac57\\times126=90\\) m/min.",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Jen", "segments": [
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"},
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"},
              {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}]},
          {"label": "Grace", "segments": [
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"},
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}],
           "brace": {"text": "?"}}
        ],
        "total": {"text": "2 units = 36 m/min", "bars": [0]}
      }
    },
    {
      "n": 42,
      "answer": "$9000",
      "answer_matches_key": true,
      "working": "Work backwards from C's \\(\\$1400\\). After B, what was left \\(=\\$1400\\); B left \\(\\dfrac34\\) of the post-A amount then took another \\(\\$850\\), so \\(\\dfrac34(\\text{after A})-850=1400\\Rightarrow\\dfrac34(\\text{after A})=2250\\Rightarrow\\text{after A}=3000\\). After A \\(=\\dfrac12(\\text{total})-1500=3000\\Rightarrow\\dfrac12\\,\\text{total}=4500\\Rightarrow\\text{total}=\\$9000\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "At first", "segments": [
              {"u": 2, "text": "A: ½", "fill": "#cfe8ff"}, {"u": 2, "text": "after A", "fill": "#ffe7c2"}],
           "brace": {"text": "total = ?"}},
          {"label": "After A took", "segments": [
              {"u": 1, "text": "B: ¼", "fill": "#e6ccff"}, {"u": 3, "text": "left = 3000", "fill": "#ffe7c2"}]}
        ]
      }
    },
    {
      "n": 44,
      "answer": "289",
      "answer_matches_key": true,
      "working": "Last year men : women \\(=5:4\\). This year men \\(=5\\times1.30=6.5\\) units, women \\(=4\\times0.50=2\\) units, so this year \\(=8.5\\) units \\(=4913\\Rightarrow 1\\) unit \\(=578\\). Last year total \\(=9\\) units \\(=9\\times578=5202\\). Difference \\(=5202-4913=289\\).",
      "spec": {
        "unit_px": 55,
        "bars": [
          {"label": "Men (last yr)", "segments": [
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"},
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}]},
          {"label": "Women (last yr)", "segments": [
              {"u": 1, "fill": "#ffd2d2"}, {"u": 1, "fill": "#ffd2d2"}, {"u": 1, "fill": "#ffd2d2"}, {"u": 1, "fill": "#ffd2d2"}]},
          {"label": "This yr total", "segments": [
              {"u": 1, "fill": "#c2f0c2"}, {"u": 1, "fill": "#c2f0c2"}, {"u": 1, "fill": "#c2f0c2"}, {"u": 1, "fill": "#c2f0c2"},
              {"u": 1, "fill": "#c2f0c2"}, {"u": 1, "fill": "#c2f0c2"}, {"u": 1, "fill": "#c2f0c2"}, {"u": 1, "fill": "#c2f0c2"}],
           "brace": {"text": "8.5 units = 4913"}}
        ]
      }
    }
  ]
}