📚 Case Study Practice in Year 7 CCEA Maths | 七年级 CCEA 数学案例分析实战演练
In this article, we work through a realistic case study – organising a charity bake sale – to show how Year 7 maths skills can be used to solve everyday problems. Each section focuses on a different topic, such as money, percentages, measurement, ratios, time, data and fractions, giving you plenty of practice with multi-step reasoning.
在本文中，我们通过一个真实的案例研究——组织一次慈善烘焙义卖——来展示如何运用七年级数学技能解决日常问题。每一节重点介绍不同的数学主题，例如货币、百分比、测量、比例、时间、数据和分数，为你提供大量的多步推理练习。
1. Setting the Budget – Addition and Subtraction | 设定预算 – 加法与减法
We plan to sell cupcakes and biscuits at the bake sale. The selling prices are £1.50 per cupcake and £0.80 per biscuit. We expect to sell 40 cupcakes and 35 biscuits.
我们计划在义卖会上出售纸杯蛋糕和饼干。售价为每个纸杯蛋糕 1.50 英镑，每块饼干 0.80 英镑。我们预计可以卖出 40 个纸杯蛋糕和 35 块饼干。
To find the total income, we multiply the number of each item by its price and then add the results: Income from cupcakes = 40 × £1.50 = £60.00; income from biscuits = 35 × £0.80 = £28.00.
为了计算总收入，我们用每种商品的数量乘以单价，然后将结果相加：纸杯蛋糕收入 = 40 × 1.50 英镑 = 60.00 英镑；饼干收入 = 35 × 0.80 英镑 = 28.00 英镑。
Total income = £60.00 + £28.00 = £88.00
Next, we list the costs of ingredients and packaging. The table below shows our expected spending.
接下来，我们列出原料和包装的成本。下表显示我们的预期支出。
| Item | Cost (£) |
| Flour | 2.60 |
| Sugar | 1.85 |
| Butter | 3.20 |
| Eggs | 2.10 |
| Packaging | 1.50 |
| Total cost | 11.25 |
We subtract the total cost of £11.25 from the income of £88.00 to find the initial profit: £88.00 – £11.25 = £76.75.
我们用 88.00 英镑的收入减去 11.25 英镑的总成本，得出初始利润：88.00 英镑 – 11.25 英镑 = 76.75 英镑。
2. Working with Discounts – Percentages | 计算折扣 – 百分比
The local supermarket offers a 10% discount on all baking ingredients if we buy in bulk. We need to calculate the new cost after the discount and work out the updated profit.
当地超市对批量采购的所有烘焙原料提供 10% 的折扣。我们需要计算折扣后的新成本，并计算更新后的利润。
A 10% discount means we pay 90% of the original price. As a decimal, 90% = 0.90, so we multiply the original cost by 0.90.
10% 的折扣意味着我们支付原价的 90%。作为小数，90% = 0.90，因此我们将原成本乘以 0.90。
Discounted cost = £11.25 × 0.90 = £10.125
In money problems, we usually round to two decimal places. £10.125 rounds to £10.13. The new profit becomes:
在货币问题中，我们通常四舍五入到小数点后两位。10.125 英镑四舍五入为 10.13 英镑。新的利润变为：
New profit = £88.00 – £10.13 = £77.87
We can also find the saving from the discount: £11.25 – £10.13 = £1.12, which is exactly 10% of the original cost.
我们也可以计算节省的金额：11.25 英镑 – 10.13 英镑 = 1.12 英镑，这正好是原成本的 10%。
3. Measuring Tables – Perimeter and Area | 测量桌子 – 周长与面积
We need to cover a rectangular stall table with a decorative cloth. The table measures 1.8 m in length and 0.75 m in width. We must find the perimeter to attach a trim and the area to buy enough fabric.
我们需要用装饰布覆盖一个长方形的摊位桌子。桌子的长度为 1.8 米，宽度为 0.75 米。我们需要计算周长以便缝制装饰边，并计算面积以购买足够的布料。
The perimeter of a rectangle is calculated by adding the lengths of all four sides: P = l + w + l + w, or P = 2(l + w).
长方形的周长通过将所有四条边的长度相加来计算：P = 长 + 宽 + 长 + 宽，或 P = 2(长 + 宽)。
Perimeter = 2 × (1.8 m + 0.75 m) = 2 × 2.55 m = 5.1 m
The area of a rectangle is length multiplied by width.
长方形的面积等于长乘以宽。
Area = 1.8 m × 0.75 m = 1.35 m²
Now we know we need 5.1 metres of trim and 1.35 square metres of fabric. We should buy slightly extra to allow for mistakes.
现在我们知道了需要 5.1 米的装饰边和 1.35 平方米的布料。我们应该略多买一点以便留有误差余地。
4. Baking Ratios – Ratio and Proportion | 烘焙比例 – 比例
A recipe for 12 cupcakes requires 250 g of flour and 150 g of sugar. We need to make 40 cupcakes. To adjust the recipe, we use ratio reasoning.
一份制作 12 个纸杯蛋糕的食谱需要 250 克面粉和 150 克糖。我们需要制作 40 个纸杯蛋糕。为了调整食谱，我们使用比例推理。
First, find the scaling factor by dividing the new amount by the original: 40 ÷ 12 = 10/3, or approximately 3.333.
首先，通过将新数量除以原始数量来求出缩放因子：40 ÷ 12 = 10/3，大约为 3.333。
Multiply each ingredient by this factor. For flour: 250 g × (40/12) = 250 g × (10/3).
将每种原料乘以这个因子。面粉：250 克 × (40/12) = 250 克 × (10/3)。
Flour needed = 250 g × 10 ÷ 3 = 2500 ÷ 3 ≈ 833.3 g
For sugar: 150 g × (10/3) = 1500 ÷ 3 = 500 g exactly.
糖：150 克 × (10/3) = 1500 ÷ 3 = 500 克（恰好）。
We can also express the recipe as a ratio of flour to sugar: 250:150, which simplifies to 5:3. The scaled amounts keep the same ratio: 833.3:500 ≈ 5:3.
我们也可以将食谱表示为面粉与糖的比例：250:150，化简为 5:3。缩放后的数量保持相同的比例：833.3:500 ≈ 5:3。
5. Time Planning – Time Calculations | 时间规划 – 时间计算
The bake sale starts at 11:00 am. Baking and decorating will take 2 hours 45 minutes, and setting up the stall takes 40 minutes. We need to find the latest time we must start baking.
义卖会上午 11:00 开始。烘焙和装饰需要 2 小时 45 分钟，布置摊位需要 40 分钟。我们需要找出必须开始烘焙的最晚时间。
First, add the two durations: 2 hours 45 min + 40 min = 2 hours 85 min. Since 85 minutes is 1 hour 25 minutes, the total preparation time is 3 hours 25 minutes.
首先，将两个时长相加：2 小时 45 分 + 40 分 = 2 小时 85 分。由于 85 分钟是 1 小时 25 分，总准备时间为 3 小时 25 分。
Now subtract this from 11:00 am. Counting back 3 hours gives 8:00 am, then counting back a further 25 minutes gives 7:35 am.
现在从上午 11:00 减去这个时长。往回数 3 小时得到上午 8:00，再往回数 25 分钟得到上午 7:35。
Latest start time = 7:35 am
If we allow a 15-minute break, we must start baking by 7:20 am. Using a timeline or number line helps check the calculation.
如果我们留出 15 分钟的休息时间，就必须在上午 7:20 之前开始烘焙。使用时间线或数轴有助于验算。
6. Survey Results – Data Handling and Averages | 调查结果 – 数据处理与平均数
Before the sale, we surveyed 30 classmates about their favourite cupcake flavour. The results were: Vanilla (8 votes), Chocolate (12 votes), Strawberry (6 votes) and Lemon (4 votes).
义卖前，我们调查了 30 位同学最喜欢的纸杯蛋糕口味。结果如下：香草味（8 票）、巧克力味（12 票）、草莓味（6 票）和柠檬味（4 票）。
We can present this data in a frequency table and a bar chart. The mode is the flavour with the highest frequency, which is Chocolate (12).
我们可以用频数表和条形图展示这些数据。众数是频数最高的口味，即巧克力味（12 票）。
Table: Flavour Preferences
表格：口味偏好
| Flavour | Tally | Frequency |
| Vanilla | IIII III | 8 |
| Chocolate | IIII IIII II | 12 |
| Strawberry | IIII I | 6 |
| Lemon | IIII | 4 |
The mean number of votes per flavour is not very useful here, but we can find the median flavour by ordering the frequencies: 4, 6, 8, 12. With four values, the median is the average of the two middle numbers: (6 + 8) ÷ 2 = 7.
这里每种口味的平均票数意义不大，但我们可以通过将频数排序来找出中位数：4, 6, 8, 12。有四个数值时，中位数是中间两个数的平均数：(6 + 8) ÷ 2 = 7。
Based on the survey, we plan to make more chocolate and vanilla cupcakes to match the demand.
基于调查结果，我们计划多制作巧克力味和香草味的纸杯蛋糕以满足需求。
7. Dividing the Cakes – Fractions in Action | 切分蛋糕 – 分数应用
We baked a large rectangular cake that will be sold as fractional slices. The cake is cut into 16 equal slices. We want to share these slices equally among 6 volunteers. How many slices does each volunteer get, and what fraction of the whole cake is left over?
我们烤了一个长方形大蛋糕，将按份出售。蛋糕被切成 16 等份。我们想把这些份量平均分给 6 位志愿者。每位志愿者分得多少块？整块蛋糕剩下几分之几？
Divide 16 by 6: 16 ÷ 6 = 2 remainder 4. Each volunteer receives 2 whole slices, and 4 slices remain. We can cut each remaining slice into smaller pieces to share further.
将 16 除以 6：16 ÷ 6 = 2 余 4。每位志愿者得到 2 整块，剩余 4 块。我们可以将剩余的每块再切成更小的块以继续分配。
As a fraction, each volunteer receives 16/6 = 8/3 = 2 2/3 slices. The fraction of the cake they share is 6 × (2/3) = 4 slices equivalent from the remainder, plus the 12 whole slices originally given: 12 + 4 = 16 slices, which is the whole cake. Thus, 4/16 = 1/4 of the cake was redistributed as smaller pieces.
用分数表示，每位志愿者得到 16/6 = 8/3 = 2 又 2/3 块。他们分得的蛋糕部分是 6 × (2/3) = 相当于 4 整块，加上最初给出的 12 整块：共 16 块，即整个蛋糕。因此，1/4 的蛋糕被重新分配为小块。
Leftover as a fraction of the whole cake = 0 (all shared)
If we sold the slices for 50p each, the total revenue from the whole cake would be 16 × £0.50 = £8.00.
如果我们以每块 50 便士出售切片，整个蛋糕的总收入将是 16 × 0.50 英镑 = 8.00 英镑。
8. Volume of Drinks – Volume and Capacity | 饮料体积 – 体积与容量
We plan to serve orange juice in cups of 200 ml each. We expect about 80 customers and assume each customer will drink one cup. Find the total volume of juice needed in litres.
我们计划每杯提供 200 毫升的橙汁。我们预计大约有 80 位顾客，并假设每位顾客会喝一杯。求所需橙汁的总量（单位：升）。
Total volume in millilitres = 80 × 200 ml = 16 000 ml.
总容量（毫升）= 80 × 200 毫升 = 16 000 毫升。
Since 1 litre = 1000 ml, we divide by 1000.
由于 1 升 = 1000 毫升，我们除以 1000。
Volume in litres = 16 000 ÷ 1000 = 16 L
If the juice is sold in 2-litre cartons, we need 16 ÷ 2 = 8 cartons. A 10% wastage allowance means we should buy an extra carton, making 9 cartons to be safe.
如果橙汁以 2 升的纸盒出售，我们需要 16 ÷ 2 = 8 盒。考虑到 10% 的损耗余量，我们应该多买一盒，即购买 9 盒以确保安全。
9. Profit Calculation – Multi-step Problems | 利润计算 – 多步运算
Now we combine all the expenses and income to find the final profit. Besides the ingredient cost, we also bought decorations for £4.50 and paper cups for £3.20. The discounted ingredient cost was £10.13.
现在我们综合所有支出和收入来求最终利润。除了原料成本，我们还花了 4.50 英镑购买装饰品，3.20 英镑购买纸杯。打折后的原料成本为 10.13 英镑。
Total expenses = £10.13 + £4.50 + £3.20 = £17.83. Total income from selling cupcakes, biscuits, cake slices and juice must be calculated.
总支出 = 10.13 英镑 + 4.50 英镑 + 3.20 英镑 = 17.83 英镑。我们需要计算出售纸杯蛋糕、饼干、蛋糕片和果汁的总收入。
Cupcake and biscuit income was £88.00. Cake slices (16 slices at £0.50) added £8.00. Juice: 80 cups at £0.40 per cup = £32.00. Total income = £88.00 + £8.00 + £32.00 = £128.00.
纸杯蛋糕和饼干收入为 88.00 英镑。蛋糕片（16 片，每片 0.50 英镑）增加了 8.00 英镑。果汁：80 杯，每杯 0.40 英镑 = 32.00 英镑。总收入 = 88.00 英镑 + 8.00 英镑 + 32.00 英镑 = 128.00 英镑。
Final profit = £128.00 – £17.83 = £110.17
We can express the profit as a percentage of the total income: (110.17 ÷ 128.00) × 100 ≈ 86.1%. This is an excellent outcome for the charity.
我们可以用利润占总收入的百分比来表示：(110.17 ÷ 128.00) × 100 ≈ 86.1%。这对慈善事业来说是一个极好的结果。
10. Final Presentation – Interpreting and Communicating | 最终展示 – 解释与交流
To share our results, we create a summary table and a bar chart comparing income sources. The table below shows the breakdown.
为了分享我们的成果，我们制作了一个汇总表和一个比较收入来源的条形图。下表展示了明细。
| Income source | Amount (£) |
| Cupcakes & biscuits | 更多咨询请联系16621398022（同微信）
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