{
  "source_prefix": "Catholic High 2023 P6 Prelim",
  "models": [
    {
      "n": 13,
      "answer": "84",
      "answer_matches_key": true,
      "working": "End amount \\(=\\dfrac{1}{5}\\) of the total \\(=\\dfrac{3}{7}\\) of the remainder after the morning. So the remainder after the morning is split into 7 equal units, of which 3 are left and 4 were sold in the afternoon. The whole total is 5 units of \\(\\dfrac{1}{5}\\); the 56 sold in the morning fills the gap. Solving, total \\(=105\\), left \\(=21\\), so sold altogether \\(=105-21=84\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Total", "segments": [
            {"u": 1, "text": "56", "fill": "#ffe7c2"},
            {"u": 4, "text": "afternoon", "fill": "#ffd0d0"},
            {"u": 1, "text": "left 21", "fill": "#dbe9ff"}],
           "brace": {"text": "105"}}
        ],
        "total": {"text": "sold = 84", "bars": [0]}
      }
    },
    {
      "n": 33,
      "answer": "45",
      "answer_matches_key": true,
      "working": "\\(30-3=27\\) pupils still fold. Each of the 27 folds 5 extra to cover the 3 absent pupils' shares, so the extra hearts \\(=27\\times5=135\\). These 135 equal the 3 absentees' original share \\(=3\\) units. So 1 unit (each pupil's first amount) \\(=135\\div3=45\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "3 absent", "segments": [
            {"u": 1, "fill": "#ffd0d0"}, {"u": 1, "fill": "#ffd0d0"}, {"u": 1, "fill": "#ffd0d0"}],
           "brace": {"text": "shared out = 135"}},
          {"label": "27 share", "segments": [{"u": 3, "text": "27 \\times 5 = 135 extra"}]}
        ]
      }
    },
    {
      "n": 34,
      "answer": "168",
      "answer_matches_key": true,
      "working": "The total stays the same. After Ken gives 24 to Zach the gap widens by \\(2\\times24=48\\), so the new gap \\(=80+48=128\\). Now Zach \\(=3\\times\\)Ken, so the gap is 2 equal units \\(=128\\); 1 unit \\(=64\\). Ken after \\(=64\\), Zach after \\(=192\\). Zach at first \\(=192-24=168\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Zach after", "segments": [
            {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}]},
          {"label": "Ken after", "segments": [{"u": 1, "text": "64", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "gap = 128 (2 units)", "bars": [0, 1]}
      }
    },
    {
      "n": 37,
      "answer": "$22",
      "answer_matches_key": true,
      "working": "Both spend the same price, so the difference in their money stays \\(\\$92-\\$50=\\$42\\). The left-over ratio is \\(5:2\\), whose difference is \\(5-2=3\\) units \\(=\\$42\\), so 1 unit \\(=\\$14\\). Mabel's 2 units left \\(=\\$28\\), so the book \\(=\\$50-\\$28=\\$22\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Nathan left", "segments": [
            {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}, {"u": 1, "fill": "#dbe9ff"}]},
          {"label": "Mabel left", "segments": [
            {"u": 1, "fill": "#ffe7c2"}, {"u": 1, "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "diff = $42 (3 units)", "bars": [0, 1]}
      }
    },
    {
      "n": 47,
      "answer": "\\(\\dfrac{16}{35}\\)",
      "answer_matches_key": true,
      "working": "Take Abel's tarts as 35 equal units. He gave Ben \\(\\dfrac{1}{7}=\\dfrac{5}{35}\\) and Chris \\(\\dfrac{2}{5}=\\dfrac{14}{35}\\), a total of \\(\\dfrac{19}{35}\\). Fraction left \\(=1-\\dfrac{19}{35}=\\dfrac{16}{35}\\).",
      "spec": {
        "unit_px": 60,
        "bars": [
          {"label": "Abel", "segments": [
            {"u": 5, "text": "to Ben 5/35", "fill": "#ffe7c2"},
            {"u": 14, "text": "to Chris 14/35", "fill": "#ffd0d0"},
            {"u": 16, "text": "left 16/35", "fill": "#dbe9ff"}],
           "brace": {"text": "35 units = all of Abel's tarts"}}
        ]
      }
    },
    {
      "n": 48,
      "answer": "105",
      "answer_matches_key": true,
      "working": "After giving, \\(A:B:C=4:3:9\\); total \\(=16\\) units \\(=448\\), so 1 unit \\(=28\\). Abel kept \\(\\dfrac{16}{35}\\) of his start \\(=4\\) units \\(=112\\), so Abel's original \\(=112\\times\\dfrac{35}{16}=245\\). Abel gave Ben \\(\\dfrac{1}{7}\\times245=35\\) and Chris \\(\\dfrac{2}{5}\\times245=98\\). Ben at first \\(=3\\times28-35=49\\); Chris at first \\(=9\\times28-98=154\\). Chris had \\(154-49=105\\) more than Ben.",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Chris first", "segments": [
            {"u": 3, "text": "154", "fill": "#dbe9ff"}]},
          {"label": "Ben first", "segments": [
            {"u": 1, "text": "49", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "Chris - Ben = 105", "bars": [0, 1]}
      }
    },
    {
      "n": 53,
      "answer": "196",
      "answer_matches_key": true,
      "working": "Let the collection be 100%. Silver \\(=30\\%\\); bronze \\(=\\)silver\\(+12\\); gold \\(=100\\). So \\(100+30\\%+(30\\%+12)=100\\%\\Rightarrow 40\\%=112\\), giving total \\(=280\\). Then silver \\(=84\\), bronze \\(=96\\). Gold \\(+\\) bronze \\(=100+96=196\\).",
      "spec": {
        "unit_px": 50,
        "bars": [
          {"label": "Collection", "segments": [
            {"u": 3, "text": "silver 30% = 84", "fill": "#ffe7c2"},
            {"u": 3, "text": "bronze 84+12 = 96", "fill": "#ffd0d0"},
            {"u": 4, "text": "gold = 100", "fill": "#dbe9ff"}],
           "brace": {"text": "total = 280"}}
        ],
        "total": {"text": "gold + bronze = 196", "bars": [0]}
      }
    },
    {
      "n": 54,
      "answer": "70",
      "answer_matches_key": true,
      "working": "Gold \\(+\\) bronze \\(=196\\) does not change. After the gift, silver is \\(44\\%\\), so gold \\(+\\) bronze \\(=56\\%\\) of the new collection. New total \\(=196\\div0.56=350\\). New silver \\(=44\\%\\times350=154\\). Silver received \\(=154-84=70\\).",
      "spec": {
        "unit_px": 18,
        "bars": [
          {"label": "Silver after", "segments": [
            {"u": 4, "text": "old 84", "fill": "#ffe7c2"},
            {"u": 3, "text": "received 70", "fill": "#c9f3c9"}],
           "brace": {"text": "new silver 154 = 44% of new total 350"}},
          {"label": "Gold+bronze", "segments": [
            {"u": 10, "text": "196 = 56% (unchanged)", "fill": "#dbe9ff"}]}
        ]
      }
    },
    {
      "n": 55,
      "answer": "8",
      "answer_matches_key": true,
      "working": "Going from a 3-plate stack (20 cm) to a 7-plate stack (44 cm) adds \\(7-3=4\\) plates and \\(44-20=24\\) cm, so each stacked plate adds \\(24\\div4=6\\) cm. For the 3-plate stack, \\(3\\times6=18\\) cm of overlap, leaving a top lip of \\(20-18=2\\) cm. The height of one plate \\(=6+2=8\\) cm.",
      "spec": {
        "unit_px": 26,
        "bars": [
          {"label": "7 plates", "segments": [{"u": 7, "text": "44 cm"}]},
          {"label": "3 plates", "segments": [
            {"u": 3, "text": "20 cm", "fill": "#dbe9ff"},
            {"u": 4, "text": "4 plates = 24 cm", "fill": "#ffd0d0"}]}
        ],
        "total": {"text": "each added plate = 6 cm; plate = 6 + 2 = 8 cm", "bars": [0, 1]}
      }
    }
  ]
}