{
  "source_prefix": "MGS 2023 P6 Prelim",
  "models": [
    {
      "n": 9,
      "answer": "\\(\\dfrac{1}{3}\\)",
      "answer_matches_key": true,
      "working": "Split the 36 pens into 6 equal units (each \\(=6\\) pens). Red \\(=\\dfrac{1}{3}=2\\) units \\(=12\\). Blue \\(=50\\%=3\\) units \\(=18\\). Green is the last \\(1\\) unit \\(=6\\). Green as a fraction of blue \\(=\\dfrac{6}{18}=\\dfrac{1}{3}\\).",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "36 pens", "segments": [
            {"u": 1, "text": "Red", "fill": "#ffd6d6"},
            {"u": 1, "text": "Red", "fill": "#ffd6d6"},
            {"u": 1, "text": "Blue", "fill": "#dbe9ff"},
            {"u": 1, "text": "Blue", "fill": "#dbe9ff"},
            {"u": 1, "text": "Blue", "fill": "#dbe9ff"},
            {"u": 1, "text": "Green", "fill": "#d6f5d6"}
          ], "brace": {"text": "36"}}
        ]
      }
    },
    {
      "n": 12,
      "answer": "$4.20",
      "answer_matches_key": true,
      "working": "Lisa: \\(3\\) files \\(+\\,3\\) pencils \\(=\\$18.90\\), so \\(1\\) file \\(+\\,1\\) pencil \\(=\\$6.30\\). Howard: \\(2\\) files \\(+\\,1\\) pencil \\(=\\$10.50\\). The difference is \\(1\\) extra file: \\(\\$10.50-\\$6.30=\\$4.20\\).",
      "spec": {
        "unit_px": 90,
        "bars": [
          {"label": "Howard", "segments": [
            {"u": 1, "text": "file", "fill": "#dbe9ff"},
            {"u": 1, "text": "file", "fill": "#dbe9ff"},
            {"u": 1, "text": "pencil", "fill": "#ffe7c2"}
          ], "brace": {"text": "$10.50"}},
          {"label": "1 set", "segments": [
            {"u": 1, "text": "file", "fill": "#dbe9ff"},
            {"u": 1, "text": "pencil", "fill": "#ffe7c2"}
          ], "brace": {"text": "$6.30"}}
        ]
      }
    },
    {
      "n": 13,
      "answer": "\\(12 : 11\\)",
      "answer_matches_key": true,
      "working": "Bala : Chelsea \\(=10:11\\). Tianwei : Bala \\(=6:5=12:10\\). Lining up Bala at \\(10\\) units: Tianwei : Bala : Chelsea \\(=12:10:11\\), so Tianwei : Chelsea \\(=12:11\\).",
      "spec": {
        "unit_px": 22,
        "bars": [
          {"label": "Tianwei", "segments": [{"u": 12, "fill": "#dbe9ff"}], "brace": {"text": "12 units"}},
          {"label": "Bala", "segments": [{"u": 10, "fill": "#ffe7c2"}], "brace": {"text": "10 units"}},
          {"label": "Chelsea", "segments": [{"u": 11, "fill": "#d6f5d6"}], "brace": {"text": "11 units"}}
        ]
      }
    },
    {
      "n": 15,
      "answer": "80",
      "answer_matches_key": true,
      "working": "Take Dalia's start as \\(1\\) unit, so Huda \\(=1\\) unit \\(-30\\). After Dalia gives 40% then Huda gives 50%, the final difference works out to \\(0.6\\) of Dalia's start \\(=48\\). So Dalia's start \\(=48\\div0.6=80\\) stickers.",
      "spec": {
        "unit_px": 28,
        "bars": [
          {"label": "Before: Dalia", "segments": [{"u": 5, "fill": "#dbe9ff"}, {"u": 3, "text": "30 more", "fill": "#cdddff"}], "brace": {"text": "?"}},
          {"label": "Before: Huda", "segments": [{"u": 5, "fill": "#ffe7c2"}]},
          {"label": "After: Dalia", "segments": [{"u": 6, "fill": "#dbe9ff"}, {"u": 4, "text": "48 more", "fill": "#cdddff"}]},
          {"label": "After: Huda", "segments": [{"u": 6, "fill": "#ffe7c2"}]}
        ]
      }
    },
    {
      "n": 33,
      "answer": "70 min",
      "answer_matches_key": true,
      "working": "\\(3\\) hours \\(=180\\) min. Each of the first two bedrooms and the base of the third are equal portions; the third has an extra \\(15\\) min. Remove the extra: \\(180-15=165\\) min over \\(3\\) equal portions, so \\(1\\) portion \\(=165\\div3=55\\) min. Third bedroom \\(=55+15=70\\) min.",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Bedroom 1", "segments": [{"u": 1, "fill": "#dbe9ff"}]},
          {"label": "Bedroom 2", "segments": [{"u": 1, "fill": "#dbe9ff"}]},
          {"label": "Bedroom 3", "segments": [{"u": 1, "fill": "#dbe9ff"}, {"u": 1, "text": "15", "fill": "#ffe7c2"}], "brace": {"text": "?"}}
        ],
        "total": {"text": "180 min", "bars": [0, 1, 2]}
      }
    },
    {
      "n": 38,
      "answer": "14",
      "answer_matches_key": true,
      "working": "Each square is cut into \\(3\\) rectangles, adding \\(2\\) extra pieces per square. The total grew from \\(40\\) to \\(92\\), an increase of \\(52\\). Number of squares \\(=52\\div2=26\\). Rectangular papers bought \\(=40-26=14\\).",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "Bought", "segments": [
            {"u": 1, "text": "squares", "fill": "#dbe9ff"},
            {"u": 1, "text": "rectangles", "fill": "#ffe7c2"}
          ], "brace": {"text": "40 pieces"}},
          {"label": "After cutting", "segments": [
            {"u": 1, "text": "squares ×3", "fill": "#cdddff"},
            {"u": 1, "text": "rectangles", "fill": "#ffe7c2"}
          ], "brace": {"text": "92 pieces"}}
        ]
      }
    },
    {
      "n": 43,
      "answer": "(a) 720 m (b) 1530 m",
      "answer_matches_key": true,
      "working": "The post office is the midpoint of Ali's and Nelson's houses, and the library is \\(360\\) m past it on Nelson's side. (a) Ali jogs the half-distance plus \\(360\\) m while Nelson jogs the half-distance minus \\(360\\) m, so Ali jogs \\(360\\times2=720\\) m more. (b) Ali's speed \\(=85+40=125\\) m/min; they jog the same time. Extra distance \\(\\div\\) speed difference \\(=720\\div40=18\\) min, so Nelson's distance \\(=85\\times18=1530\\) m.",
      "spec": {
        "unit_px": 70,
        "bars": [
          {"label": "Ali → Library", "segments": [
            {"u": 2, "text": "half", "fill": "#dbe9ff"},
            {"u": 1, "text": "360", "fill": "#cdddff"}
          ], "brace": {"text": "?"}},
          {"label": "Nelson → Library", "segments": [
            {"u": 1, "text": "half − 360", "fill": "#ffe7c2"}
          ]}
        ]
      }
    },
    {
      "n": 44,
      "answer": "(a) 34 cm (b) 1.17 m",
      "answer_matches_key": true,
      "working": "Two box lengths span the \\(150\\) cm height, so \\(1\\) length \\(=75\\) cm. The side gaps differ by \\(30-13=17\\) cm, which equals \\(1\\) length \\(-2\\) breadths. (a) The top gap \\(=17\\times2=34\\) cm. (b) \\(1\\) breadth \\(=(75-17)\\div2=29\\) cm; cupboard width \\(=75+29+13=117\\) cm \\(=1.17\\) m.",
      "spec": {
        "unit_px": 80,
        "bars": [
          {"label": "A: width", "segments": [
            {"u": 1, "text": "length 75", "fill": "#dbe9ff"},
            {"u": 1, "text": "breadth", "fill": "#ffe7c2"},
            {"u": 1, "text": "30 gap", "fill": "#eeeeee"}
          ]},
          {"label": "B: width", "segments": [
            {"u": 1, "text": "breadth", "fill": "#ffe7c2"},
            {"u": 1, "text": "breadth", "fill": "#ffe7c2"},
            {"u": 1, "text": "13 gap", "fill": "#eeeeee"}
          ]}
        ],
        "total": {"text": "same width", "bars": [0, 1]}
      }
    },
    {
      "n": 45,
      "answer": "\\(9 : 35\\)",
      "answer_matches_key": true,
      "working": "Total A : Total B \\(=2:3\\). Scale to \\(30:45\\) so each hall splits cleanly. Hall A girls : boys \\(=7:3\\) of \\(30\\) units \\(=21:9\\). Hall B girls : boys \\(=2:7\\) of \\(45\\) units \\(=10:35\\). So boys A : boys B \\(=9:35\\).",
      "spec": {
        "unit_px": 18,
        "bars": [
          {"label": "Hall A (30)", "segments": [
            {"u": 21, "text": "girls 21", "fill": "#ffd6d6"},
            {"u": 9, "text": "boys 9", "fill": "#dbe9ff"}
          ]},
          {"label": "Hall B (45)", "segments": [
            {"u": 10, "text": "girls 10", "fill": "#ffd6d6"},
            {"u": 35, "text": "boys 35", "fill": "#dbe9ff"}
          ]}
        ]
      }
    },
    {
      "n": 46,
      "answer": "285",
      "answer_matches_key": true,
      "working": "Combining both halls, total girls : total boys \\(=(21+10):(9+35)=31:44\\) (75 units). After \\(375\\) boys leave, girls are 62% and boys 38%. Girls stay \\(31\\) units \\(=62\\%\\), so \\(1\\%=\\dfrac{1}{2}\\) unit and boys \\(=38\\%=19\\) units. Boys who left \\(=44-19=25\\) units \\(=375\\), so \\(1\\) unit \\(=15\\). Boys remaining \\(=19\\times15=285\\).",
      "spec": {
        "unit_px": 9,
        "bars": [
          {"label": "Girls", "segments": [{"u": 31, "text": "31 units (62%)", "fill": "#ffd6d6"}]},
          {"label": "Boys at first", "segments": [
            {"u": 19, "text": "19 stay", "fill": "#dbe9ff"},
            {"u": 25, "text": "25 left = 375", "fill": "#cdddff"}
          ]}
        ]
      }
    },
    {
      "n": 47,
      "answer": "30",
      "answer_matches_key": true,
      "working": "Money adult : child \\(=14:3\\), total \\(17\\) units \\(=\\$9180\\), so \\(1\\) unit \\(=\\$540\\). Adult money \\(=14\\times\\$540=\\$7560\\); child money \\(=3\\times\\$540=\\$1620\\). Since \\(\\dfrac{1}{5}\\) of tickets are child, adult : child tickets \\(=4:1\\), so each adult ticket \\(=\\$7560\\div4=\\$1890\\). The price gap \\(\\$1890-\\$1620=\\$270\\) equals \\(\\$9\\) per child ticket, so child tickets \\(=\\$270\\div\\$9=30\\).",
      "spec": {
        "unit_px": 30,
        "bars": [
          {"label": "Adult $", "segments": [{"u": 14, "text": "14 units", "fill": "#dbe9ff"}]},
          {"label": "Child $", "segments": [{"u": 3, "text": "3 units", "fill": "#ffe7c2"}]}
        ],
        "total": {"text": "$9180", "bars": [0, 1]}
      }
    },
    {
      "n": 48,
      "answer": "240",
      "answer_matches_key": true,
      "working": "Each tray uses cupcakes : tarts \\(=3:5\\). To buy whole boxes (cupcakes in 4s, tarts in 3s), scale to \\(36:60\\): that is \\(9\\) boxes of cupcakes \\(+\\,20\\) boxes of tarts \\(=\\$2.10\\times9+\\$1.60\\times20=\\$18.90+\\$32.00=\\$50.90\\) per set. \\(\\$203.60\\div\\$50.90=4\\) sets. Tarts \\(=4\\times60=240\\).",
      "spec": {
        "unit_px": 22,
        "bars": [
          {"label": "1 set: cupcakes", "segments": [{"u": 36, "text": "36 (9 boxes)", "fill": "#dbe9ff"}]},
          {"label": "1 set: tarts", "segments": [{"u": 60, "text": "60 (20 boxes)", "fill": "#ffe7c2"}]}
        ]
      }
    }
  ]
}