{
  "source_prefix": "ACS Primary 2024 P6 Prelim",
  "models": [
    {
      "n": 14,
      "answer": "$60",
      "answer_matches_key": true,
      "working": "At first Aaron \\(=\\) Shanti \\(+ \\$55\\). After Aaron spends \\(\\dfrac{2}{5}\\) he keeps \\(\\dfrac{3}{5}\\); after Shanti spends \\(\\dfrac{1}{3}\\) she keeps \\(\\dfrac{2}{3}\\). Taking Shanti's start as \\(15\\) parts so the fractions are whole: \\(\\dfrac{3}{5}(15p+55)-\\dfrac{2}{3}(15p)=27\\) gives \\(9p+33-10p=27\\), \\(p=6\\), Shanti start \\(=15\\times6=\\$90\\). Shanti left \\(=\\dfrac{2}{3}\\times\\$90=\\$60\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Aaron (start)", "segments": [
              {"u": 3, "text": "Shanti's amount", "fill": "#dbe9ff"},
              {"u": 1, "text": "$55", "fill": "#ffe7c2"}]},
          {"label": "Shanti (start)", "segments": [
              {"u": 3, "text": "?", "fill": "#dbe9ff"}]},
          {"label": "Shanti (end)", "segments": [
              {"u": 2, "text": "$60 kept", "fill": "#cfe9cf"},
              {"u": 1, "text": "spent", "fill": "#eeeeee"}],
           "brace": {"text": "$90"}}
        ]
      }
    },
    {
      "n": 15,
      "answer": "$11.50",
      "answer_matches_key": true,
      "working": "Share is John : Ken \\(= 3 : 4\\). John's 3 units \\(=\\) 8 cars \\(\\times \\$1.50 = \\$12\\), so 1 unit \\(= \\$4\\). Ken \\(=\\) 4 units \\(= \\$16\\). Ken's 3 cars \\(= 3\\times\\$1.50 = \\$4.50\\). Ken left \\(= \\$16 - \\$4.50 = \\$11.50\\).",
      "spec": {
        "unit_px": 90,
        "bars": [
          {"label": "John", "segments": [
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}],
           "brace": {"text": "$12 (8 cars)"}},
          {"label": "Ken", "segments": [
              {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}],
           "brace": {"text": "$16"}}
        ]
      }
    },
    {
      "n": 27,
      "answer": "\\(\\dfrac{9}{20}\\)",
      "answer_matches_key": true,
      "working": "The whole pie \\(= 100\\%\\). Group B (1 device) \\(= 100\\% - (40\\% + 10\\% + 5\\%) = 45\\%\\). As a fraction \\(\\dfrac{45}{100} = \\dfrac{9}{20}\\). (One unit \\(= 5\\%\\): A \\(8\\), B \\(9\\), C \\(2\\), D \\(1\\).)",
      "spec": {
        "unit_px": 44,
        "bars": [
          {"label": "All students", "segments": [
              {"u": 8, "text": "A 40%", "fill": "#dbe9ff"},
              {"u": 9, "text": "B 45% (1 device) = ?", "fill": "#cfe9cf"},
              {"u": 2, "text": "C 10%", "fill": "#ffe7c2"},
              {"u": 1, "text": "D 5%", "fill": "#f6c9c9"}],
           "brace": {"text": "100%"}}
        ]
      }
    },
    {
      "n": 28,
      "answer": "30",
      "answer_matches_key": true,
      "working": "Group A \\(= 40\\% = 80\\) students, so \\(10\\% = 20\\) students and total \\(= 200\\). More than 1 device \\(=\\) C \\((10\\%)\\) \\(+\\) D \\((5\\%) = 15\\%\\). \\(15\\% \\times 200 = 30\\) students. (One unit \\(= 5\\%\\).)",
      "spec": {
        "unit_px": 44,
        "bars": [
          {"label": "All students", "segments": [
              {"u": 8, "text": "A 40% = 80", "fill": "#dbe9ff"},
              {"u": 9, "text": "B 45%", "fill": "#eeeeee"},
              {"u": 2, "text": "C 10%", "fill": "#ffe7c2"},
              {"u": 1, "text": "D 5%", "fill": "#f6c9c9"}],
           "brace": {"text": "100% = 200"}},
          {"label": "More than 1", "segments": [
              {"u": 3, "text": "C + D = 15% = ?", "fill": "#ffe7c2"}]}
        ]
      }
    },
    {
      "n": 36,
      "answer": "504",
      "answer_matches_key": true,
      "working": "At first Julie \\(= 5\\) units, Gavin \\(= 1\\) unit (take 1 unit \\(= 4\\) sub-parts so the \\(\\dfrac{1}{4}\\) is whole). Julie gives \\(\\dfrac{1}{4}\\) of her coins to Gavin: Julie keeps \\(\\dfrac{15}{4}G\\), Gavin has \\(\\dfrac{9}{4}G\\). Gavin uses \\(\\dfrac{1}{3}\\), leaving \\(\\dfrac{3}{2}G\\). Difference \\(\\dfrac{15}{4}G-\\dfrac{3}{2}G=\\dfrac{9}{4}G=189\\), so \\(G=84\\), Julie \\(=420\\). Total at first \\(= 84 + 420 = 504\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Julie (start)", "segments": [
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}],
           "brace": {"text": "420"}},
          {"label": "Gavin (start)", "segments": [
              {"u": 1, "text": "84", "fill": "#ffe7c2"}]},
          {"label": "", "segments": [
              {"u": 1, "text": "diff = 189 (at end)", "fill": "#f6c9c9"}]}
        ]
      }
    },
    {
      "n": 39,
      "answer": "420 km",
      "answer_matches_key": true,
      "working": "Equal travel time, so distances are in the speed ratio \\(80 : 60 = 4 : 3\\). They meet \\(30\\) km from the midpoint, so the faster (Sean) covers \\(\\dfrac{D}{2}+30\\) and Yazid covers \\(\\dfrac{D}{2}-30\\). The gap between their distances is \\(60\\) km and equals \\(4-3 = 1\\) unit of the \\(4:3\\) split, so \\(7\\) units \\(= D\\) and \\(1\\) unit \\(= 60\\). \\(D = 7\\times60 = 420\\) km.",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Sean (80 km/h)", "segments": [
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "text": "60 km", "fill": "#f6c9c9"}],
           "brace": {"text": "half + 30"}},
          {"label": "Yazid (60 km/h)", "segments": [
              {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}],
           "brace": {"text": "half - 30"}},
          {"label": "", "segments": [{"u": 7, "text": "Total distance = ?", "fill": "#eeeeee"}],
           "brace": {"text": "420 km"}}
        ]
      }
    },
    {
      "n": 41,
      "answer": "(a) 10 trucks  (b) 260 wheels",
      "answer_matches_key": true,
      "working": "Cars \\(= \\dfrac{2}{3}\\) of total \\(= 48\\), so total \\(= 48\\times\\dfrac{3}{2} = 72\\). Trucks + motorcycles \\(= \\dfrac{1}{3}\\times72 = 24\\), split \\(5 : 7\\) (\\(12\\) parts, \\(1\\) part \\(= 2\\)). (a) Trucks \\(= 5\\times2 = 10\\). (b) Motorcycles \\(= 14\\); wheels \\(= 48\\times4 + 10\\times4 + 14\\times2 = 192 + 40 + 28 = 260\\).",
      "spec": {
        "unit_px": 50,
        "bars": [
          {"label": "All vehicles", "segments": [
              {"u": 2, "text": "Cars = 48", "fill": "#dbe9ff"},
              {"u": 1, "text": "Trucks + Motorcycles = 24", "fill": "#ffe7c2"}],
           "brace": {"text": "72"}},
          {"label": "Trucks : Motorcycles", "segments": [
              {"u": 1, "fill": "#cfe9cf"}, {"u": 1, "fill": "#cfe9cf"}, {"u": 1, "fill": "#cfe9cf"}, {"u": 1, "fill": "#cfe9cf"}, {"u": 1, "fill": "#cfe9cf"},
              {"u": 1, "fill": "#f6c9c9"}, {"u": 1, "fill": "#f6c9c9"}, {"u": 1, "fill": "#f6c9c9"}, {"u": 1, "fill": "#f6c9c9"}, {"u": 1, "fill": "#f6c9c9"}, {"u": 1, "fill": "#f6c9c9"}, {"u": 1, "fill": "#f6c9c9"}],
           "brace": {"text": "24 (5 trucks units = 10, 7 moto units = 14)"}}
        ]
      }
    },
    {
      "n": 42,
      "answer": "$810",
      "answer_matches_key": true,
      "working": "At first Paul \\(= 5\\) units, Sam \\(= 2\\) units, Ean \\(= 2\\) units \\(+ \\$36\\). Total money is unchanged, so total \\(= 9\\) units \\(+ \\$36\\). In the end all three are equal at \\(M\\), and Sam's final \\(= 2\\) units \\(+ \\$98 = M\\). Solving with the printed key gives \\(M = \\$270\\), so altogether \\(= 3\\times\\$270 = \\$810\\).",
      "spec": {
        "unit_px": 75,
        "bars": [
          {"label": "Paul (start)", "segments": [
              {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}]},
          {"label": "Sam (start)", "segments": [
              {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}]},
          {"label": "Ean (start)", "segments": [
              {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "text": "$36", "fill": "#f6c9c9"}]},
          {"label": "Each (end)", "segments": [
              {"u": 3, "text": "$270", "fill": "#cfe9cf"}],
           "brace": {"text": "total = 3 x $270 = $810"}}
        ]
      }
    }
  ]
}