Case Study Practical Exercises | 案例分析实战演练

📚 Case Study Practical Exercises | 案例分析实战演练

In SQA Year 8 Mathematics, case studies bridge the gap between textbook exercises and real-world applications. These practical problems require you to interpret information, select appropriate operations, and justify your reasoning. This article guides you through several case study scenarios to build your problem-solving confidence.

在SQA八年级数学中，案例分析是连接课本练习与现实应用之间的桥梁。这些实际问题要求你解读信息、选择合适的运算方法，并证明你的推理过程。本文将引导你通过多个案例场景，建立解决问题的信心。

1. Case Study 1: Organising a School Fair | 案例1：组织学校义卖会

The student council is planning a fair. Tickets cost £4.50 each. Costs include £220 for a bouncy castle, £95 for craft supplies, and £40 for prizes. They want to raise at least £300 for charity. How many tickets must they sell?

学生会正在筹备一场义卖会。门票每张4.5英镑。支出包括蹦床220英镑、手工材料95英镑和奖品40英镑。他们希望为慈善机构至少筹集300英镑。他们需要卖出多少张门票？

Step 1: Total expenses = £220 + £95 + £40 = £355.

步骤1：总支出 = 220 + 95 + 40 = 355 英镑。

Step 2: Money needed = expenses + charity target = £355 + £300 = £655.

步骤2：所需收入 = 支出 + 慈善目标 = 355 + 300 = 655 英镑。

Step 3: Ticket number = £655 / £4.50. Since 4.50 x 100 = 450, 655 / 4.5 = 6550 / 45 = 145.555…, so they must sell 146 tickets (rounding up because selling 145 only gives £652.50).

步骤3：门票张数 = 655 ÷ 4.5。将除数化为整数，6550 ÷ 45 ≈ 145.56，因此需向上取整至146张（因为145张仅收入652.5英镑，不满足目标）。

Step 4: Check: 146 x £4.50 = £657. Profit = £657 – £355 = £302, which is at least £300.

步骤4：验证：146 × 4.5 = 657 英镑。利润 = 657 – 355 = 302 英镑，满足至少300英镑的要求。

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2. Case Study 2: Interpreting a Weather Chart | 案例2：解读天气图表

A weather station recorded these midday temperatures for a week: Monday 12°C, Tuesday 14°C, Wednesday 11°C, Thursday 15°C, Friday 13°C, Saturday 18°C, Sunday 16°C. Find the mean temperature and the range. On how many days was the temperature above the mean?

气象站记录了一周正午温度：周一12°C，周二14°C，周三11°C，周四15°C，周五13°C，周六18°C，周日16°C。计算平均气温和极差。有多少天的气温高于平均值？

Sum of temperatures = 12 + 14 + 11 + 15 + 13 + 18 + 16 = 99°C. Number of days = 7. Mean = 99 / 7 = 14.14°C (approx. 14.1°C).

温度总和 = 12+14+11+15+13+18+16 = 99°C。天数 = 7。平均值 = 99 ÷ 7 ≈ 14.1°C。

The range is the difference between the highest and lowest: 18 – 11 = 7°C.

极差为最高与最低之间的差：18 − 11 = 7°C。

Check each day: Tuesday (14°C) is just below the mean? 14 is less than 14.14, so only Thursday (15°C), Saturday (18°C) and Sunday (16°C) are above the mean. That is 3 days.

逐日比较：周二14°C略低于平均值14.1°C，因此只有周四15°C、周六18°C和周日16°C高于平均，共3天。

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3. Case Study 3: Measuring for New Curtains | 案例3：测量新窗帘

A window is 2.4 m wide and 1.8 m high. You need two panels, each half the width. Fabric costs £12 per metre length and is 1.2 m wide. Allow 15 cm extra per panel for hems and 10% extra total length for waste. How many whole metres should you buy and what is the cost?

一扇窗户宽2.4米，高1.8米。需要两块窗帘，各覆盖一半宽度。布料每米12英镑，幅宽1.2米。每块需加长15厘米做卷边，且总长度再加10%考虑浪费。应购买多少整米布料，费用是多少？

Each panel covers 1.2 m width, which matches the fabric width, so no joining is needed. Height per panel before waste = 1.8 m + 0.15 m = 1.95 m. Two panels: total length without waste = 1.95 m x 2 = 3.9 m.

每块窗帘覆盖宽度1.2米，正好与幅宽一致，无需拼接。每块使用长度（不含浪费）= 1.8 + 0.15 = 1.95米。两块总需长度（不含浪费）= 1.95 × 2 = 3.9米。

Add 10% waste: total with waste = 3.9 m x 1.10 = 4.29 m. Since fabric is sold by the metre, round up to 5 metres. Cost = 5 x £12 = £60.

加上10%浪费：含浪费的总长度 = 3.9 × 1.10 = 4.29米。因布料按整米出售，需向上取整至5米。费用 = 5 × 12 = 60英镑。

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4. Case Study 4: Planning a Bus Journey | 案例4：规划公交出行

Buses from school to the city centre depart at 07:45, 08:15 and 08:50. The journey takes exactly 40 minutes. You must arrive by 09:00. What is the latest bus you can catch? If you miss the 08:15 bus, what time do you arrive and by how many minutes are you late?

从学校到市中心的公交车发车时间为07:45、08:15和08:50。行程恰好需要40分钟。你必须在09:00前到达。你最晚可以乘坐哪一班车？如果错过了08:15的车，你将何时到达，迟到多少分钟？

07:45 + 40 min = 08:25 (arrives on time). 08:15 + 40 min = 08:55 (arrives on time). 08:50 + 40 min = 09:30 (arrives late). Therefore, the latest bus is the 08:15 departure.

07:45 + 40 分钟 = 08:25（准时到达）；08:15 + 40 分钟 = 08:55（准时到达）；08:50 + 40 分钟 = 09:30（迟到）。因此最晚可乘08:15一班。

If you miss the 08:15 bus, you must take the 08:50. Arrival time = 09:30. Scheduled arrival time was 09:00, so you are 30 minutes late.

如果错过08:15的车，必须乘08:50的班次。到达时间为09:30。原定到达时间为09:00，因此迟到30分钟。

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5. Case Study 5: Adjusting a Smoothie Recipe | 案例5：调整奶昔配方

This recipe serves 2 people: 150 g yoghurt, 300 ml milk, 2 bananas, and 60 g honey. How much of each ingredient is needed for 5 people?

这份食谱供2人食用：酸奶150克，牛奶300毫升，香蕉2根，蜂蜜60克。5人份各需要多少？

The scaling factor is 5 / 2 = 2.5. Yoghurt: 150 g x 2.5 = 375 g. Milk: 300 ml x 2.5 = 750 ml. Bananas: 2 x 2.5 = 5 bananas. Honey: 60 g x 2.5 = 150 g.

比例因子 = 5 ÷ 2 = 2.5。酸奶：150 g × 2.5 = 375 g；牛奶：300 ml × 2.5 = 750 ml；香蕉：2 × 2.5 = 5根；蜂蜜：60 g × 2.5 = 150 g。

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6. Case Study 6: Designing a Survey and Creating a Pie Chart | 案例6：设计调查并绘制饼图

30 pupils were asked their favourite subject: 12 chose Maths, 8 chose Science, 6 chose English, and the rest chose ‘Other’. Calculate the angle for each sector on a pie chart and verify the total is 360°.

30名学生被询问最喜欢的科目：12人选数学，8人选科学，6人选英语，其余选择“其他”。计算饼图中每个扇形的角度，并验证总和为360°。

Maths angle = (12 / 30) x 360° = 144°. Science = (8 / 30) x 360° = 96°. English = (6 / 30) x 360° = 72°. Other = 30 – 12 – 8 – 6 = 4, so angle = (4 / 30) x 360° = 48°. Check: 144 + 96 + 72 + 48 = 360°.

数学扇形角度 = (12/30)×360° = 144°；科学 = (8/30)×360° = 96°；英语 = (6/30)×360° = 72°；其他 = 30−12−8−6 = 4人，角度 = (4/30)×360° = 48°。验证：144+96+72+48 = 360°。

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7. Case Study 7: Comparing Discounts | 案例7：比较折扣

Item A costs £80. Shop X gives 30% off; Shop Y gives 1/3 off. Which is cheaper, and by how much? Item B costs £45. Shop X offers 20% off then an additional 10% off the reduced price; Shop Y offers a straight 30% off. Which shop is cheaper for Item B?

商品A原价80英镑。X店打七折（减30%），Y店减1/3。哪家更便宜，便宜多少？商品B原价45英镑。X店先打八折再打九折，Y店直接打七折。对于商品B，哪家更便宜？

Item A: Shop X price = £80 x 0.7 = £56. Shop Y price = £80 x (2/3) = £160/3 ≈ £53.33. Shop Y is cheaper by £56 – £53.33 = £2.67.

商品A：X店价格 = 80 × 0.7 = 56英镑；Y店价格 = 80 × 2/3 ≈ 53.33英镑。Y店便宜约56 − 53.33 = 2.67英镑。

Item B: Shop X: first discount price = £45 x 0.8 = £36; second discount = £36 x 0.9 = £32.40. Shop Y: £45 x 0.7 = £31.50. Shop Y is cheaper by £0.90.

商品B：X店第一次折后 = 45 × 0.8 = 36英镑，第二次折后 = 36 × 0.9 = 32.40英镑；Y店 = 45 × 0.7 = 31.50英镑。Y店便宜0.90英镑。

Note: successive percentage discounts do not simply add (20% + 10% ≠ 30%).

注意：连续的百分比折扣不能简单相加（20%+10%≠30%）。

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8. Case Study 8: Mobile Data Usage | 案例8：移动数据使用

A mobile plan includes 5 GB of data per month. In 15 days, 1.2 GB have been used. Convert the remaining data into MB (1 GB = 1024 MB) and find the average MB used per day so far. If usage continues at the same rate, will the user exceed the allowance? How much data would remain after 30 days?

手机套餐每月包含5 GB流量。15天内已用1.2 GB。将剩余流量转换为MB（1 GB = 1024 MB），并计算截至目前平均每天用了多少MB。如果保持相同用量，是否会超出限额？30天后还剩余多少？

Total data allowance = 5 GB = 5 x 1024 = 5120 MB. Used = 1.2 GB = 1.2 x 1024 = 1228.8 MB. Remaining = 5120 – 1228.8 = 3891.2 MB.

总流量限额 = 5 GB = 5 × 1024 = 5120 MB。已用 = 1.2 × 1024 = 1228.8 MB。剩余 = 5120 − 1228.8 = 3891.2 MB。

Average per day for the first 15 days = 1228.8 MB / 15 = 81.92 MB per day. Projected total for 30 days = 81.92 x 30 = 2457.6 MB = 2.4 GB, which is well within the 5 GB cap. Remaining after 30 days = 5120 – 2457.6 = 2662.4 MB.

前15天日均用量 = 1228.8 ÷ 15 = 81.92 MB/天。按此速率30天预计总用量 = 81.92 × 30 = 2457.6 MB = 2.4 GB，远低于5 GB限额。30天后剩余 = 5120 − 2457.6 = 2662.4 MB。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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