📚 Case Study Practice: Year 10 CCEA Maths in Real Life | 案例分析实战演练：Year 10 CCEA 数学实际应用
In Year 10 CCEA Mathematics, case studies bring together number, algebra, geometry, statistics and probability to solve real-world problems. This article walks you through a complete case study—organising a charity bake sale—showing how to apply your maths skills step by step.
在 Year 10 CCEA 数学中，案例分析将数字、代数、几何、统计和概率融合起来，解决现实生活问题。本文将带你完成一个完整的案例研究——组织一场慈善烘焙义卖，逐步展示如何应用你的数学技能。
1. The Scenario: Planning a Charity Bake Sale | 案例情境：策划慈善烘焙义卖
Your school has decided to hold a charity bake sale to raise money for a local animal shelter. As a member of the organising committee, you must handle budgeting, survey favourite snacks, design the stall layout, forecast sales and maximise profit. Every decision relies on Year 10 mathematical methods.
你的学校决定举办一场慈善烘焙义卖，为当地动物收容所筹款。作为组委会成员，你需要处理预算、调查最受欢迎的零食、设计摊位布局、预测销售量并最大化利润。每一个决策都依赖 Year 10 的数学方法。
2. Budgeting with Numbers and Decimals | 运用数字和小数进行预算
You start by listing costs: hiring a table and gazebo costs £25.50, baking ingredients for 60 cupcakes total £18.90, and packaging (boxes, napkins) adds £5.30. Use addition and subtraction to find the total fixed cost.
你首先列出各项成本：租用一张桌子和一个遮阳棚花费 25.50 英镑，60 个纸杯蛋糕的烘焙原料总计 18.90 英镑，包装材料（盒子、纸巾）再加 5.30 英镑。运用加减法求出总固定成本。
Total fixed cost = £25.50 + £18.90 + £5.30 = £49.70
Always align decimal points when adding money. The total fixed cost of £49.70 must be covered before any profit is made.
计算金额时务必对齐小数点。总固定成本为 49.70 英镑，必须在产生任何利润之前收回。
3. Algebra: Break-Even Analysis | 代数：盈亏平衡分析
You decide to sell each cupcake for £1.50. Each cupcake costs £0.32 in ingredients and packaging (variable cost). Let n be the number of cupcakes sold. The profit equation can be set up using algebra.
你决定每个纸杯蛋糕售价 1.50 英镑。每个蛋糕的原料和包装可变成本为 0.32 英镑。设 n 为售出的蛋糕数量。可以用代数建立利润方程。
Revenue R = 1.50 × n
Variable cost V = 0.32 × n
Profit P = R − V − Fixed cost = 1.50n − 0.32n − 49.70 = 1.18n − 49.70
Break-even occurs when profit is zero. Solve 1.18n − 49.70 = 0. This gives n = 49.70 ÷ 1.18 ≈ 42.12. Since you cannot sell a fraction of a cupcake, you need to sell at least 43 cupcakes to start making a profit.
盈亏平衡点在利润为零时出现。解方程 1.18n − 49.70 = 0，得到 n = 49.70 ÷ 1.18 ≈ 42.12。由于不能卖出部分蛋糕，你需要至少售出 43 个蛋糕才能开始盈利。
4. Geometry and Measures: Stall Layout | 几何与测量：摊位布局
The stall area is a rectangular space of 3 metres by 2.5 metres. You need to fit a table that is 1.8 m long and 0.8 m wide, leaving a serving gap. Calculate the perimeter to estimate how much bunting you need, and the remaining area for standing space.
摊位区域是一个 3 米 × 2.5 米的矩形空间。你需要摆放一张长 1.8 米、宽 0.8 米的桌子，并留出服务间隙。计算周长以估算需要多少彩旗，以及剩余的站立空间面积。
Stall perimeter = 2 × (3.0 + 2.5) = 11.0 m
Table area = 1.8 × 0.8 = 1.44 m²
Total stall area = 3.0 × 2.5 = 7.5 m²
Free area = 7.5 − 1.44 = 6.06 m²
The large free area allows at least two volunteers to move comfortably. The 11 m perimeter will require exactly one pack of 12 m bunting from the store.
剩余面积较大，至少可以让两名志愿者舒适地走动。11 米的周长正好需要商店里一包 12 米长的彩旗。
5. Statistics: Surveying Favourite Flavours | 统计：调查最受欢迎的口味
To decide what to bake, you survey 50 students. They choose their favourite cupcake flavour from chocolate, vanilla and red velvet. The results: chocolate 28, vanilla 12, red velvet 10. Work out the relative frequencies as percentages.
为了决定烘焙什么，你调查了 50 名学生。他们在巧克力、香草和红丝绒三种纸杯蛋糕口味中选出最喜欢的。结果：巧克力 28 人，香草 12 人，红丝绒 10 人。计算出每种口味的相对频率百分比。
Chocolate: 28/50 × 100% = 56%
Vanilla: 12/50 × 100% = 24%
Red velvet: 10/50 × 100% = 20%
These percentages tell you that over half of the potential customers prefer chocolate. You plan to bake cupcakes in the ratio 56 : 24 : 20, which simplifies to 14 : 6 : 5 by dividing each number by 4.
这些百分比告诉你，超过一半的潜在顾客更喜欢巧克力口味。你计划按照 56 : 24 : 20 的比例烘焙，每个数除以 4 可简化为 14 : 6 : 5。
6. Ratio and Proportion: Scaling the Recipe | 比和比例：调整食谱用量
A basic chocolate cupcake recipe for 12 cupcakes requires 200 g flour, 100 g sugar, 100 g butter and 2 eggs. You need to scale it up to produce 56 cupcakes (14 parts in the ratio). Find the multiplier and calculate new quantities.
一份基础的巧克力纸杯蛋糕食谱可供制作 12 个，需要 200 克面粉、100 克糖、100 克黄油和 2 个鸡蛋。你需要将其扩大至制作 56 个蛋糕（比例中的 14 份）。求出倍数并计算新的用量。
Multiplier = 56 ÷ 12 = 14/3 ≈ 4.67
However, it is easier to think proportionally: 12 × 4 = 48, plus 12 × (2/3) = 8 adds up to 56. Multiply each ingredient by 56/12.
不过，按比例思考更简单：12 × 4 = 48，再加上 12 × (2/3) = 8 得到 56。将每样原料乘以 56/12。
Flour: 200 g × (56/12) = 200 × 14/3 = 933.33 g (round to 933 g)
Sugar: 100 g × 14/3 = 466.67 g (≈ 467 g)
Butter: 100 g × 14/3 = 467 g
Eggs: 2 × 14/3 = 28/3 = 9.33, so you need 10 eggs (practical rounding)
Similarly, scale vanilla and red velvet recipes to make 24 and 20 cupcakes respectively using their base recipes.
同样地，根据基础食谱分别调整香草和红丝绒的配方，制作 24 个和 20 个蛋糕。
7. Probability: Estimating Attendance | 概率：估算参与人数
Based on past school events, the probability that a student buys at least one cupcake is 0.45 when the weather is sunny, and 0.25 if it rains. The weather forecast gives a 60% chance of sunshine on the day. Use a probability tree to find the overall probability that a randomly selected student will buy a cupcake.
根据过往学校活动记录，晴天时一名学生至少购买一个蛋糕的概率为 0.45，雨天则为 0.25。天气预报显示活动当天晴天概率为 60%。使用概率树形图，求随机选取的一名学生购买蛋糕的总概率。
P(buy) = P(sun) × P(buy|sun) + P(rain) × P(buy|rain)
= 0.6 × 0.45 + 0.4 × 0.25 = 0.27 + 0.10 = 0.37
If 300 students are expected to pass by the stall, the estimated number of buyers is 300 × 0.37 = 111. This helps set production targets.
如果预计有 300 名学生经过摊位，估计购买人数为 300 × 0.37 = 111 人。这有助于设定生产目标。
8. Data Handling: Graphs and Interpretation | 数据处理：图表与解读Published by TutorHao | Year 10 Mathematics Revision Series | aleveler.com
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