📚 Year 7 AQA Maths: Case Study Practice | 案例分析实战演练
Maths isn't just about numbers on a page. It helps us make decisions every day. In this article, we'll work through real-life case studies that link directly to the Year 7 AQA Mathematics curriculum. Each case shows how topics like number, ratio, measurement, and statistics can be applied. By solving these problems step by step, you'll build confidence and see why maths matters in the world around you.
数学不只是纸面上的数字。它帮助我们在日常生活中做决定。在本文中，我们将通过真实的生活案例来练习，这些案例与七年级 AQA 数学课程紧密相连。每个案例都展示了数字、比例、测量和统计等知识如何被实际应用。通过逐步解决这些问题，你将建立自信，并明白数学在周围世界中的重要性。
1. Introduction to Case Studies in Maths | 数学案例学习入门
A case study in maths is a scenario that requires you to use different skills to find a solution. You might need to add, subtract, multiply, divide, work with fractions or percentages, interpret data, or calculate measurements. The key is to read carefully, identify the maths involved, and show all your working out.
数学中的案例学习是一种需要你运用多种不同技能来寻找答案的情境。你可能需要做加减乘除、处理分数或百分比、解读数据，或计算测量值。关键在于仔细阅读、识别涉及的数学知识，并展示出你所有的解题步骤。
2. Case Study 1: Budgeting for a School Trip | 案例1：学校旅行预算
A school plans a trip for 60 students. The coach costs £360. Entry to the museum is £4.50 per student. The school wants to include a packed lunch costing £2.80 per student. How much money must each student bring if costs are shared equally? Round your answer to the nearest penny.
一所学校计划组织 60 名学生出游。大巴花费 360 英镑。博物馆门票为每名学生 4.50 英镑。学校希望包含一份每人 2.80 英镑的打包午餐。如果费用均摊，每名学生需要带多少钱？将答案四舍五入到最接近的便士。
Step 1: Find the total cost. Coach: £360. Entry: 60 × £4.50 = £270. Lunch: 60 × £2.80 = £168. Total = 360 + 270 + 168 = £798.
第一步：求总花费。大巴：360 英镑。门票：60 × 4.50 = 270 英镑。午餐：60 × 2.80 = 168 英镑。总计 = 360 + 270 + 168 = 798 英镑。
Step 2: Divide total by 60: 798 ÷ 60 = 13.3. So each student must bring £13.30.
第二步：用总金额除以 60：798 ÷ 60 = 13.3。因此每名学生须带 13.30 英镑。
Check: 13.30 × 60 = £798. This is a good example of using whole number and decimal operations together.
检验：13.30 × 60 = 798 英镑。这是在同一个问题中使用整数和小数运算的一个好例子。
3. Case Study 2: Scaling a Recipe | 案例2：食谱的比例缩放
A recipe for 8 pancakes uses 200 g of flour, 2 eggs, 400 ml of milk, and 50 g of butter. You need to make 20 pancakes for a party. How much of each ingredient do you need?
一份制作 8 个煎饼的食谱需要 200 克面粉、2 个鸡蛋、400 毫升牛奶和 50 克黄油。你需要为一次聚会制作 20 个煎饼。每种原料需要多少？
multiplier: new amount ÷ original amount = 20 ÷ 8 = 2.5. Multiply each ingredient by 2.5.
首先，求倍数：新数量 ÷ 原数量 = 20 ÷ 8 = 2.5。将每种原料的用量乘以 2.5。
- Flour: 200 g × 2.5 = 500 g
- Eggs: 2 × 2.5 = 5 eggs
- Milk: 400 ml × 2.5 = 1000 ml (1 litre)
- Butter: 50 g × 2.5 = 125 g
- 面粉：200 克 × 2.5 = 500 克
- 鸡蛋：2 × 2.5 = 5 个
- 牛奶：400 毫升 × 2.5 = 1000 毫升（1 升）
- 黄油：50 克 × 2.5 = 125 克
Scaling recipes directly uses ratio and proportion. You can also use a unitary method: find for 1 pancake, then multiply by 20.
食谱的缩放直接运用了比和比例的知识。你也可以使用归一法：先求出一个煎饼的用量，再乘以 20。
4. Case Study 3: Measuring and Tiling a Floor | 案例3：测量与铺地砖
A rectangular kitchen floor is 3.6 m long and 2.4 m wide. Tiles are squares with side 30 cm. How many tiles are needed to cover the floor? The tiles are sold in boxes of 10. How many boxes should be bought?
一个长方形厨房地面的长为 3.6 米，宽为 2.4 米。地砖是边长为 30 厘米的正方形。需要多少块地砖才能铺满地面？地砖按每盒 10 块出售。应购买多少盒？
nits. Option: change metres to centimetres. Length: 3.6 m = 360 cm. Width: 2.4 m = 240 cm.
将所有单位统一。可选择将米转换为厘米。长度：3.6 米 = 360 厘米。宽度：2.4 米 = 240 厘米。
0 = 86,400 cm². Area of one tile = 30 × 30 = 900 cm². Number of tiles = 86,400 ÷ 900 = 96.
地面面积 = 360 × 240 = 86,400 平方厘米。一块地砖的面积 = 30 × 30 = 900 平方厘米。所需地砖数量 = 86,400 ÷ 900 = 96 块。
ust buy 10 boxes. This shows rounding up in context, because you cannot buy part of a box.
所需盒数：96 ÷ 10 = 9.6，因此你必须购买 10 盒。这表明在真实情境下需要向上取整，因为你无法购买零散的盒数。
5. Case Study 4: Interpreting Timetables | 案例4：解读时间表
A bus timetable shows departures from Town A at 07:35, 08:15, 09:00, and every 30 minutes until 18:00. The journey to Town C takes 1 hour 45 minutes. What time does the bus that departs at 08:15 arrive at Town C? If you need to reach Town C by 12:00, what is the latest bus you can take from Town A?
一份公共汽车时刻表显示，从 A 镇发车的时间为 07:35、08:15、09:00，之后直到 18:00 每 30 分钟一班。前往 C 镇的行程需要 1 小时 45 分钟。08:15 出发的那班车什么时候到达 C 镇？如果你需要在 12:00 之前到达 C 镇，你最晚可以乘坐从 A 镇出发的哪一班车？
:15 departure: 08:15 + 1 hour 45 min = 08:15 + 1:45 = 10:00. So that bus arrives at 10:00.
08:15 出发的到达时间：08:15 + 1 小时 45 分钟 = 10:00。所以那班车在 10:00 到达。
2:00. 12:00 minus 1 hour 45 min = 10:15. So the bus must depart Town A at or before 10:15.
要在 12:00 前到达的最晚班次：从 12:00 倒推。12:00 减去 1 小时 45 分钟 = 10:15。所以班次必须在 10:15 或之前从 A 镇出发。
Departures after 09:00 are at 09:30, 10:00, 10:30… 10:15, so the 10:00 bus is the latest. Check: 10:00 + 1:45 = 11:45, which is before 12:00.
09:00 之后的发车时间是 09:30、10:00、10:30…… 10:30 的那班车在 10:15 之后出发，所以 10:00 那班是最晚的。检查：10:00 + 1:45 = 11:45，在 12:00 之前。
6. Case Study 5: Data from a Survey | 案例5：调查数据分析
A class survey asks: "How many hours of sport do you do per week?" The results from 20 students are: 3, 5, 2, 4, 6, 3, 1, 5, 3, 4, 2, 6, 2, 3, 5, 1, 4, 3, 2, 4. Construct a frequency table and find the mean number of hours.
一项班级调查询问：“你每周进行几小时的体育运动？” 20 名学生的回答结果如下：3, 5, 2, 4, 6, 3, 1, 5, 3, 4, 2, 6, 2, 3, 5, 1, 4, 3, 2, 4。制作一张频数表并求平均小时数。
Frequency table:
| Hours | Frequency |
| 1 | 2 |
| 2 | 4 |
| 3 | 5 |
| 4 | 4 |
| 5 | 3 |
| 6 | 2 |
4+5+4+3+2 = 20. Sum of values: (1×2)+(2×4)+(3×5)+(4×4)+(5×3)+(6×2) = 2+8+15+16+15+12 = 68.
学生总数 = 2+4+5+4+3+2 = 20。数值总和：(1×2)+(2×4)+(3×5)+(4×4)+(5×3)+(6×2) = 2+8+15+16+15+12 = 68。
8 ÷ 20 = 3.4 hours. The mean gives a typical amount of sport, but note there is variation.
平均值 = 68 ÷ 20 = 3.4 小时。平均值展示了一个典型的运动量，但注意存在差异。
7. Case Study 6: Comparing Mobile Phone Plans | 案例6：比较手机套餐
Plan A: £15 per month with 500 free minutes, then 10p per extra minute. Plan B: £20 per month with unlimited minutes. In a month you use 650 minutes. Which plan is cheaper, and by how much?
套餐 A：每月 15 英镑，包含 500 分钟免费通话，之后每分钟 10 便士。套餐 B：每月 20 英镑，通话不限分钟。某个月你使用了 650 分钟。哪种套餐更便宜，便宜多少钱？
Cost for Plan A: First 500 minutes included in £15. Extra minutes = 650 – 500 = 150. Extra cost = 150 × £0.10 = £15.00. Total = £15 + £15 = £30.
套餐 A 的费用：前 500 分钟包含在 15 英镑内。额外分钟数 = 650 – 500 = 150。额外收费 = 150 × 0.10 英镑 = 15.00 英镑。总计 = 30 英镑。
Plan B total = £20. So Plan B is £10 cheaper.
套餐 B 总计 = 20 英镑。因此套餐 B 便宜了 10 英镑。
This case uses subtraction, multiplication, and money calculations. It reminds us to read the conditions carefully.
这个案例使用了减法、乘法和货币计算。它提醒我们要仔细阅读条件。
8. Case Study 7: Designing a Garden | 案例7：设计花园
A rectangular garden is 15 m long and 10 m wide. There is a square patio of side 4 m in one corner. The rest will be lawn. Grass seed costs £2.50 per square metre to buy and spread. How much will the grass seed cost in total?
一个长方形花园长 15 米，宽 10 米。在一个角落有一个边长为 4 米的正方形露台。其余部分为草坪。草籽的购买和播撒成本为每平方米 2.50 英镑。草籽的总费用是多少？
Area of entire garden = 15 × 10 = 150 m². Area of patio = 4 × 4 = 16 m². Lawn area = 150 – 16 = 134 m². Cost = 134 × £2.50.
整个花园的面积 = 15 × 10 = 150 平方米。露台面积 = 4 × 4 = 16 平方米。草坪面积 = 150 – 16 = 134 平方米。费用 = 134 × 2.50 英镑。
134 × 2.5 = 335. So total cost = £335.
134 × 2.5 = 335。所以总费用是 335 英镑。
This combines area of rectangles and squares, subtraction, and money multiplication.
这个问题综合了长方形与正方形的面积、减法以及货币乘法。
9. Case Study 8: Converting Currency | 案例8：货币兑换
You are travelling to the USA. The exchange rate is £1 = $1.30. You exchange £400 into dollars. How many dollars do you receive? On return, you have $52 left. How many pounds is that, if the rate hasn't changed?
你要去美国旅行。汇率为 1 英镑 = 1.30 美元。你将 400 英镑兑换成美元。你能获得多少美元？返回时，你还剩 52 美元。如果汇率不变，这能换回多少英镑？
Dollars from £400: 400 × 1.30 = 520 dollars.
400 英镑可兑换的美元：400 × 1.30 = 520 美元。
Convert $52 back to pounds: divide by 1.30. 52 ÷ 1.30 = 40. So you get £40.
将 52 美元换回英镑：除以 1.30。52 ÷ 1.30 = 40。所以你能换得 40 英镑。
Currency conversion uses multiplying and dividing by decimals, often resulting in real amounts. Always check whether you multiply or divide by the exchange rate.
货币兑换运用了小数的乘除法，通常会得出真实的金额。务必检查该用汇率乘以还是除以。
10. Case Study 9: Speed and Travel | 案例9：速度与旅行
A car travels at a constant speed of 40 miles per hour (mph). How far does it travel in 2 hours 30 minutes?
一辆汽车以每小时 40 英里的恒定速度行驶。2 小时 30 分钟它能行驶多远？
hours. 30 minutes = 0.5 hour. So total time = 2.5 hours. Distance = 40 × 2.5 = 100 miles.
公式：距离 = 速度 × 时间。时间必须以小时为单位。30 分钟 = 0.5 小时。因此总时间 = 2.5 小时。距离 = 40 × 2.5 = 100 英里。
If the same car travels 180 miles, how long does it take? Time = distance ÷ speed = 180 ÷ 40 = 4.5 hours = 4 hours 30 minutes.
如果同一辆车行驶 180 英里，需要多长时间？时间 = 距离 ÷ 速度 = 180 ÷ 40 = 4.5 小时 = 4 小时 30 分钟。
Speed problems involve converting time to decimals and back, linking closely to ratio and proportion.
速度问题涉及时间与小数之间的转换，并与比和比例紧密相连。
11. Case Study 10: Sale Discounts | 案例10：折扣促销
A shop offers 30% off all items. A jumper originally costs £45. What is the sale price? The shop also applies an extra 10% off the sale price for members. How much does a member pay for the jumper?
一家商店全场七折（即减免 30%）。一件毛衣原价 45 英镑。打折后的价格是多少？商店还对会员在折扣价基础上再打九折（即再减 10%）。一位会员购买这件毛衣要付多少钱？
First discount: 30% of £45 = 0.30 × 45 = £13.50. Sale price = 45 – 13.50 = £31.50. Or use a multiplier: 100% – 30% = 70%, so 0.70 × 45 = £31.50.
第一次折扣：45 英镑的 30% = 0.30 × 45 = 13.50 英镑。折后价 = 45 – 13.50 = 31.50 英镑。或者使用乘数因子：100% – 30% = 70%，所以 0.70 × 45 = 31.50 英镑。
Member discount: 10% off £31.50. 10% of 31.50 = 0.10 × 31.50 = £3.15. Member price = 31.50 – 3.15 = £28.35. Or 90% of £31.50 = 0.9 × 31.50 = £28.35.
会员折扣：31.50 英镑再减 10%。31.50 的 10% = 0.10 × 31.50 = 3.15 英镑。会员价 = 31.50 – 3.15 = 28.35 英镑。或者 31.50 的 90% = 0.9 × 31.50 = 28.35 英镑。
Percentage discount problems often appear in real shopping situations, and we must understand whether discounts are applied successively.
百分比折扣问题常常出现在真实的购物情境中，我们必须理解折扣是否是依次叠加的。
12. Conclusion: Skills Gained | 结语：技能的收获
Through these ten case studies, you've practiced adding, subtracting, multiplying and dividing whole numbers and decimals; converting units; working with ratios and proportions; finding percentages; interpreting tables and data; calculating area; and using real-life number sense. The next time you face a real-world problem involving money, time, measurement, or data, you can break it down using the same strategies. Keep asking: what is the question asking? What numbers do I have? Which operation should I use? Does my answer make sense in context?
通过这十个案例分析，你已经练习了整数的加减乘除；单位换算；比和比例；百分比计算；解读表格与数据；计算面积；以及在真实情境中运用数感。当你下次面对涉及金钱、时间、测量或数据的实际问题时，你可以用同样的策略将其分解。始终问自己：问题在问什么？我有哪些数字？该用什么运算？我的答案放在这个情境中合理吗？
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