📚 Year 9 CCEA Maths: Case Study Practical Drills | 九年级 CCEA 数学：案例分析实战演练
Real-life problems in CCEA Mathematics are not just about numbers – they require you to understand the situation, identify the maths needed, and apply the right methods step by step. This article presents a series of case study drills designed to strengthen your problem-solving skills for the Year 9 CCEA curriculum. Each case mimics an everyday scenario, from shopping discounts to energy consumption, so you can see how the topics you learn in class appear in the world around you.
CCEA 数学中的实际问题不仅仅是数字运算——它要求你理解情境、识别所需的数学知识，并一步步运用正确的方法。本文提供了一系列案例分析训练，旨在强化你应对九年级 CCEA 课程的解题能力。每个案例都模拟了日常生活中的场景，从购物折扣到能源消耗，让你看到课堂上学到的知识是如何体现在周围世界中的。
1. Shopping Discounts | 购物折扣
Anna wants to buy a hoodie originally priced at £48. The shop is offering a 15% discount. What is the sale price?
安娜想买一件原价为 48 英镑的连帽衫。商店提供 15% 的折扣。售价是多少？
First, calculate the discount amount: 15% of £48.
首先，计算折扣金额：48 英镑的 15%。
Discount = 0.15 × 48 = £7.20
Sale price = Original price – Discount = £48 – £7.20 = £40.80.
售价 = 原价 – 折扣 = 48 英镑 – 7.20 英镑 = 40.80 英镑。
Always check if the percentage is already written as a decimal or fraction. To find a percentage of an amount, multiply by the percentage and divide by 100, or simply multiply by the decimal form.
始终检查百分比是否已经写成小数或分数。计算一个数的百分之几，可以乘以百分比再除以 100，或直接乘以小数形式。
2. Designing a Garden | 设计花园
A rectangular garden measures 6 m by 4 m. A triangular flower bed with base 3 m and height 2 m is to be dug in one corner. How much lawn area remains?
一个矩形花园长 6 米、宽 4 米。在一个角落打算挖一个底为 3 米、高为 2 米的三角形花坛。还剩下多少草坪面积？
Area of the rectangle = length × width = 6 × 4 = 24 m².
矩形面积 = 长 × 宽 = 6 × 4 = 24 平方米。
Area of the triangle = ½ × base × height = ½ × 3 × 2 = 3 m².
三角形面积 = ½ × 底 × 高 = ½ × 3 × 2 = 3 平方米。
Remaining lawn area = 24 – 3 = 21 m².
剩余草坪面积 = 24 – 3 = 21 平方米。
You may be asked to find the cost of turfing the remaining area. If turf costs £5 per square metre, the cost would be 21 × £5 = £105.
你可能会被问到铺设剩余草坪的费用。如果草皮每平方米 5 英镑，那么费用是 21 × 5 英镑 = 105 英镑。
3. Planning a Trip | 规划旅行
A family plans to travel 210 miles. Their car uses fuel at a rate of 42 miles per gallon. How many gallons of fuel are needed for the journey?
一家人计划行驶 210 英里。他们的汽车耗油量为每加仑 42 英里。这段旅程需要多少加仑燃料？
Number of gallons = Total distance ÷ Miles per gallon = 210 ÷ 42 = 5 gallons.
加仑数 = 总距离 ÷ 每加仑英里数 = 210 ÷ 42 = 5 加仑。
If the fuel cost is £1.45 per litre and 1 gallon ≈ 4.55 litres, what is the total fuel cost?
如果燃料价格为每升 1.45 英镑，且 1 加仑约等于 4.55 升，总燃料费是多少？
Total litres = 5 × 4.55 = 22.75 litres. Cost = 22.75 × £1.45 = £32.99 (rounded to nearest penny).
总升数 = 5 × 4.55 = 22.75 升。费用 = 22.75 × 1.45 英镑 ≈ 32.99 英镑（四舍五入到分）。
Remember to convert units correctly before multiplying. Use the conversion factor carefully.
记得在乘法前正确转换单位。谨慎使用换算系数。
4. Interpreting Survey Data | 解读调查数据
A survey of 200 students asked their favourite fruit. Results: Apples 60, Bananas 50, Oranges 40, Grapes 30, Others 20. Draw a pie chart to represent this data and find the percentage for apples.
对 200 名学生进行调查，询问他们最喜欢的水果。结果：苹果 60、香蕉 50、橙子 40、葡萄 30、其他 20。请画出饼图表示这些数据，并求出苹果所占的百分比。
Percentage for apples = (60 ÷ 200) × 100 = 30%.
苹果的百分比 = (60 ÷ 200) × 100 = 30%。
To draw a pie chart, calculate the angle for each sector: Angle = (Frequency ÷ Total) × 360°.
要画饼图，计算每个扇区的角度：角度 = (频数 ÷ 总数) × 360°。
Apple angle = (60 ÷ 200) × 360° = 0.3 × 360° = 108°. Bananas: (50 ÷ 200) × 360° = 90°, and so on.
苹果的角度 = (60 ÷ 200) × 360° = 0.3 × 360° = 108°。香蕉：(50 ÷ 200) × 360° = 90°，以此类推。
When interpreting the chart, you can compare sector sizes to understand proportions.
解读图表时，你可以比较扇区大小来理解比例关系。
5. Profit and Loss in a School Tuck Shop | 学校小卖部的盈利与亏损
A school tuck shop buys bottled water for £0.40 each and sells them for £0.75. They sold 320 bottles in a week. Calculate the total profit and the percentage profit on the cost price.
学校小卖部以每瓶 0.40 英镑的价格购入瓶装水，并以 0.75 英镑出售。一周内卖出了 320 瓶。计算总利润以及成本价的百分比利润。
Profit per bottle = Selling price – Cost price = £0.75 – £0.40 = £0.35.
每瓶利润 = 售价 – 成本价 = 0.75 英镑 – 0.40 英镑 = 0.35 英镑。
Total profit = 320 × £0.35 = £112.
总利润 = 320 × 0.35 英镑 = 112 英镑。
Percentage profit = (Profit ÷ Cost price) × 100 = (0.35 ÷ 0.40) × 100 = 87.5%.
百分比利润 = (利润 ÷ 成本价) × 100 = (0.35 ÷ 0.40) × 100 = 87.5%。
This shows how a small profit per item can add up to a significant amount when selling many units.
这表明，当销售大量商品时，每件商品的小额利润会累积成可观的金额。
6. Mixing Paint Colours | 混合油漆颜色
A painter needs to mix blue and yellow paint in the ratio 3:2 to make 25 litres of green paint. How many litres of each colour does she need?
一位油漆工需要将蓝色和黄色油漆按 3:2 的比例混合，配制出 25 升绿色油漆。她需要每种颜色多少升？
Total number of parts = 3 + 2 = 5 parts.
总份数 = 3 + 2 = 5 份。
Volume of one part = 25 litres ÷ 5 = 5 litres.
一份的体积 = 25 升 ÷ 5 = 5 升。
Blue paint = 3 parts × 5 litres = 15 litres. Yellow paint = 2 parts × 5 litres = 10 litres.
蓝色油漆 = 3 份 × 5 升 = 15 升。黄色油漆 = 2 份 × 5 升 = 10 升。
If the ratio had been given as a comparison of volumes directly, such as ‘for every 3 tins of blue use 2 tins of yellow’, you could use proportions: blue = (3/5) of total, yellow = (2/5).
如果比例是直接以体积对比给出的，比如“每 3 罐蓝色配 2 罐黄色”，你可以用占比计算：蓝色 = 总量的 3/5，黄色 = 总量的 2/5。
7. Comparing Mobile Phone Plans | 比较手机套餐
Two contract plans are available. Plan A: £10 per month + 3p per text. Plan B: £15 per month + 1.5p per text. If you send 400 texts a month on average, which plan is cheaper, and by how much?
现有两种合约套餐。套餐 A：月租 10 英镑 + 每条短信 3 便士。套餐 B：月租 15 英镑 + 每条短信 1.5 便士。如果平均每月发送 400 条短信，哪种套餐更便宜？便宜多少？
Convert pence to pounds for easy calculation: 3p = £0.03, 1.5p = £0.015.
为了方便计算，将便士转换为英镑：3 便士 = 0.03 英镑，1.5 便士 = 0.015 英镑。
Plan A cost = £10 + (400 × £0.03) = £10 + £12 = £22.
套餐 A 费用 = 10 英镑 + (400 × 0.03 英镑) = 10 英镑 + 12 英镑 = 22 英镑。
Plan B cost = £15 + (400 × £0.015) = £15 + £6 = £21.
套餐 B 费用 = 15 英镑 + (400 × 0.015 英镑) = 15 英镑 + 6 英镑 = 21 英镑。
Plan B is cheaper by £1 per month.
套餐 B 每月便宜 1 英镑。
| Plan | 套餐 | Monthly Fee | 月租 | Cost per Text | 每条短信费 | Total for 400 Texts | 400 条短信总费用 |
|---|---|---|---|
| A | £10 | £0.03 | £22 |
| B | £15 | £0.015 | £21 |
Using a table helps compare the costs side by side, which is a useful strategy in case study questions.
使用表格有助于并排比较成本，这在案例分析题中是一种有用的策略。
8. Population Density | 人口密度
A town has an area of 12 km² and a population of 36,000. Calculate its population density in people per km². If another town has the same population but an area of 15 km², which is more densely populated?
某镇面积为 12 平方公里，人口 36,000。计算其人口密度（人/平方公里）。如果另一个镇人口相同但面积为 15 平方公里，哪个镇人口更稠密？
Population density = Population ÷ Area = 36,000 ÷ 12 = 3,000 people per km².
人口密度 = 人口 ÷ 面积 = 36,000 ÷ 12 = 3,000 人/平方公里。
Second town density = 36,000 ÷ 15 = 2,400 people per km².
第二个镇的人口密度 = 36,000 ÷ 15 = 2,400 人/平方公里。
The first town is more densely populated because it packs more people into each square kilometre.
第一个镇人口更稠密，因为每平方公里容纳了更多人。
Real-world applications include urban planning and resource allocation. Always check units: if area is given in hectares, convert to km² (1 km² = 100 ha).
实际应用包括城市规划和资源分配。始终检查单位：如果面积以公顷给出，要转换为平方公里（1 平方公里 = 100 公顷）。
9. Energy Consumption | 能源消耗
A household’s electricity meter readings were 34,560 kWh at the start of January and 35,280 kWh at the end of March. Find the total units used in these 3 months and the average monthly consumption. If the cost is 12p per unit, calculate the quarterly bill.
一个家庭在 1 月初的电表读数是 34,560 千瓦时，3 月底的读数是 35,280 千瓦时。求这 3 个月内使用的总用电量以及月平均消耗量。如果电费是每千瓦时 12 便士，计算季度账单。
Units used = 35,280 – 34,560 = 720 kWh.
用电量 = 35,280 – 34,560 = 720 千瓦时。
Average monthly consumption = 720 ÷ 3 = 240 kWh.
月平均消耗量 = 720 ÷ 3 = 240 千瓦时。
Cost = 720 × £0.12 = £86.40.
费用 = 720 × 0.12 英镑 = 86.40 英镑。
Being able to interpret meter readings is a practical skill. Check whether a standing charge is included; if so, add it to the total.
能够解读电表读数是一项实用技能。注意是否包含固定月租费；如果有，要加在总额上。
10. Water Usage | 用水量
A school’s water tank holds 2,500 litres. On Monday, it was full. By Friday, the water level had dropped to 1,200 litres. If the school day has 6 hours and water is used steadily, how much water was used per hour on average?
一所学校的水箱容量为 2,500 升。周一水箱是满的。到周五，水位降至 1,200 升。如果每天上课 6 小时且用水量稳定，平均每小时用了多少升水？
Total water used = 2,500 – 1,200 = 1,300 litres over the week.
本周总共用水 = 2,500 – 1,200 = 1,300 升。
Number of school days from Monday to Friday = 5 days.
从周一到周五的上学天数 = 5 天。
Total hours of use = 5 × 6 = 30 hours.
总用水小时数 = 5 × 6 = 30 小时。
Average use per hour = 1,300 ÷ 30 ≈ 43.33 litres per hour (or 43.3 litres to one decimal place).
平均每小时用水 = 1,300 ÷ 30 ≈ 43.33 升/小时（或四舍五入到一位小数为 43.3 升/小时）。
Such problems combine subtraction, multiplication, and division. Always identify the total change and the total time period.
这类问题综合了减法、乘法和除法。始终先确定总变化量和总时长。
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