📚 Year 10 AQA Maths: Case Study Practical Workout | 十年级 AQA 数学：案例分析实战演练
In this article, you will work through a realistic scenario that brings together many Year 10 AQA Maths topics – from ratios and percentages to geometry, statistics and probability. The case study follows a group of students organising a charity bake sale stall, allowing you to practise applying mathematical skills to make decisions, solve problems and interpret data just as you would in an exam and in the real world.
在本文中，你将通过一个真实的情境，把十年级 AQA 数学的许多主题——从比率、百分比到几何、统计和概率——串联起来。该案例分析追踪了一群学生组织慈善烘焙义卖摊位的过程，让你有机会练习运用数学技能来做出决策、解决问题和解读数据，就像你在考试和现实世界中需要做的那样。
1. Understanding the Scenario | 理解情境
The school is hosting a charity fair, and your team has decided to run a bake sale stall selling three products: cupcakes, cookies and brownies. The goal is to raise at least £200 for charity after covering all costs. You must plan the quantities to bake, work out the budget, set prices, design the stall layout, and then analyse the sales data to improve future events. Every decision will be supported by mathematical reasoning.
学校正在举办一场慈善游园会，你的小组决定经营一个烘焙义卖摊位，出售三种产品：纸杯蛋糕、曲奇饼干和布朗尼。目标是扣除所有成本后至少为慈善机构筹集 £200。你必须规划烘焙数量、计算预算、设定价格、设计摊位布局，然后分析销售数据以改进未来的活动。每一个决定都将由数学推理来支持。
2. Scaling Recipes: Ratios and Proportions | 缩放食谱：比率与比例
, 4 large eggs and 120 ml milk. You expect to sell 120 cupcakes, so you need to scale the recipe by a factor of 5 (since 120 ÷ 24 = 5). Using direct proportion, multiply every ingredient quantity by 5.
你的基础食谱可以制作 24 个香草纸杯蛋糕，需要以下材料：400 克中筋面粉、200 克细砂糖、150 克无盐黄油、4 个大鸡蛋和 120 毫升牛奶。你预计能卖出 120 个纸杯蛋糕，因此需要用比例因子 5 来缩放食谱（因为 120 ÷ 24 = 5）。利用正比例关系，把所有材料数量乘以 5。
The scaled amounts become: flour 2000 g (2 kg), sugar 1000 g (1 kg), butter 750 g, 20 eggs and 600 ml milk. The table below summarises the conversion; this is a classic ratio application where you keep the ingredient ratios constant.
放大后的用量变为：面粉 2000 克（2 千克）、砂糖 1000 克（1 千克）、黄油 750 克、20 个鸡蛋和 600 毫升牛奶。下表总结了换算过程；这是比率保持不变的一个典型比例应用。
| Ingredient | For 24 | ×5 | For 120 |
|---|---|---|---|
| Plain flour | 400 g | ×5 | 2000 g |
| Caster sugar | 200 g | ×5 | 1000 g |
| Butter | 150 g | ×5 | 750 g |
| Eggs | 4 | ×5 | 20 |
| Milk | 120 ml | ×5 | 600 ml |
Always check that your scaled quantities are realistic and that you have the equipment to mix larger batches. For the cookies and brownies you will perform similar calculations, reinforcing your skills with the unitary method and equivalent ratios.
始终要检查放大后的用量是否切合实际，以及你是否有设备来制作更大批量的产品。对于曲奇和布朗尼，你将进行类似的计算，从而强化单位法和等价比率的运用。
3. Cost Analysis and Percentages | 成本分析与百分比
gredient list finalised, you need to price the shopping. Supermarket prices show: flour is £0.75 per 1.5 kg bag, sugar £0.90 per 1 kg, butter £1.20 per 250 g block, eggs £2.00 per dozen, and milk £0.90 per litre. Since you need 2 kg of flour, you must buy two bags; the flour cost is 2 × £0.75 = £1.50. Similar logic gives the total ingredient cost for 120 cupcakes as £10.90. Fixed costs (tablecloths, disposable plates) add £8.00, so total cost is £18.90.
随着材料清单确定下来，你需要为采购标价。超市价格为：面粉每袋 1.5 千克 £0.75，砂糖每千克 £0.90，黄油每 250 克块 £1.20，鸡蛋每打 £2.00，牛奶每升 £0.90。因为你需 2 千克面粉，就必须买两袋；面粉成本为 2 × £0.75 = £1.50。类似的逻辑可得出 120 个纸杯蛋糕的总原料成本为 £10.90。固定成本（桌布、一次性盘子）再加 £8.00，因此总成本为 £18.90。
Now calculate the percentage of the fundraising target that this cost represents: £18.90 ÷ £200 × 100 = 9.45%. This tells you that about 9.5% e if ingredient prices rise by a given rate, practising percentage change calculations that appear regularly in AQA exams.
现在计算这一成本占筹款目标的百分比：£18.90 ÷ £200 × 100 = 9.45%。这告诉你大约 9.5% 的收入会在盈利前被开支吞噬，从而凸显成本控制的重要性。你还可以计算如果原料价格上涨一定比例所带来的成本增加，练习在 AQA 考试中经常出现的百分比变化计算。
4. Pricing Strategy and Profit Maximisation | 定价策略与利润最大化
You must decide between two pricing options for cupcakes. Option A: £1.50 each, selling an estimated 150 units. Option B: £2.00 each, selling an estimated 100 units. Revenue for A = 150 × 1.50 = £225, profit = £225 − £18.90 = £206.10. Revenue for B = 100 × 2.00 = £200, profit = £200 − £18.90 = £ ant to know the break-even price and how changes in sales projections affect the decision.
你必须在两种纸杯蛋糕定价方案中做出选择。方案 A：每个 £1.50，预计销售 150 个。方案 B：每个 £2.00，预计销售 100 个。A 的收入 = 150 × 1.50 = £225，利润 = £225 − £18.90 = £206.10。B 的收入 = 100 × 2.00 = £200，利润 = £200 − £18.90 = £181.10。显然方案 A 带来的利润更高，但你还想知道盈亏平衡价格以及销售预测的变化如何影响决策。
elling price p that makes profit zero when 120 cupcakes are sold: 120p − 18.90 = 0, so p = £ o set up an equation to discover how many cakes must be sold at a given price to reach the £200 target. For instance, at £ you need at least 122 cakes. Such equation-solving is essential for AQA algebra questions.
为了练习代数，假设销售 120 个纸杯蛋糕，求使利润为零的售价 p：120p − 18.90 = 0，因此 p = £0.1575（约 16 便士）。这虽不现实，但显示了覆盖成本的最低价格。你还可以建立方程来发现在给定价格下需要卖出多少个饼才能达到 £200 的目标。例如，每个 £1.80 时：1.80x − 18.90 = 200 → 1.80x = 218.90 → x ≈ 121.6，因此至少需要 122 个蛋糕。此类方程求解对于 AQA 代数题目至关重要。
5. Geometric Design: Planning the Stall Layout | 几何设计：规划摊位布局
de, side by side along their length. The total occupied rectangle is 1.80 m by 1.50 m. Is there enough space to walk around? 0 + 2 × 0.60 = 2.70 m. The given width is only 2.50 m, so the layout is too wide by 0.20 m.
游园会组织者给了你一块 3.0 米长、2.5 米宽的矩形场地。你打算把两张长 1.80 米、宽 0.75 米的桌子长边并排放置。两张桌子占据的矩形区域为 1.80 米 × 1.50 米。是否有足够的空间让人们在周围走动？你决定在四周至少留出 0.60 米的通道。场地所需长度为 1.80 + 2 × 0.60 = 3.00 米，所需宽度为 1.50 + 2 × 0.60 = 2.70 米。给定宽度只有 2.50 米，因此该布局宽度超标了 0.20 米。
You explore turning one table perpendicularly, creating an L-shape. Using accurate scale drawings and applying Pythagoras' theorem helps check clearances: the diagonal of a single table is √1.80² + 0.75² = √3.24 + 0.5625 = √3 s, along with calculating perimeter for decorative ribbon, give you practical experience with compound shapes and area calculations, which appear frequently in AQA geometry problems.
你尝试把一张桌子转 90° 垂直放置，形成 L 形布局。借助精确的比例图并应用勾股定理可以检查间距：单张桌子的对角线长度为 √1.80² + 0.75² = √3.24 + 0.5625 = √3.8025 ≈ 1.95 米。这些几何检查，再加上为装饰丝带计算周长，为你提供了应对复合图形和面积计算的实际经验，这些内容在 AQA 几何题中频繁出现。
6. Data Collection and Representation | 数据收集与表示
During the bake sale, you record the number of each item sold every hour. The raw data for cupcakes: 12, 18, 25, 20, 15 (hours 1–5). For cookies: 8, 14, 22, 19, 11. For brownies: 5, 9, 13, 12, 7. A well-drawn bar chart or line graph helps visualise the sales pattern. Make sure to label the axes clearly, choose an appropriate scale, and title your chart – all of which earn marks in AQA statistics questions.
在义卖过程中，你每小时记录了每种产品的销售数量。纸杯蛋糕的原始数据为：12, 18, 25, 20, 15（第 1–5 小时）。曲奇：8, 14, 22, 19, 11。布朗尼：5, 9, 13, 12, 7。绘制良好的条形图或折线图有助于直观呈现销售模式。务必清晰地标注坐标轴、选择合适的刻度和图表标题——这些在 AQA 统计题中都能得分。
You could also draw a composite bar chart comparing the three products side by side or a component bar chart showing the proportion of total sales each hour that each product accounts for. These activities strengthen your handling of grouped and ungrouped data, a key skill for the exam.
你还可以绘制并列复合条形图比较三种产品，或绘制百分比堆积条形图显示每种产品每小时在总销售量中所占的份额。这些活动加强了你处理分组数据和未分组数据的能力，这是考试的关键技能。
7. Statistical Analysis: Interpreting Sales Data | 统计分析：解读销售数据
red: 12, 15, 18, 20, 25 → median = 18. The mode is no single value repeated, but you notice that sales peaked in the third hour. The range is 25 − 12 = 13, indicating variability.
根据五小时的纸杯蛋糕销售数据，计算平均数：(12+18+25+20+15) ÷ 5 = 90 ÷ 5 = 18 个/小时。按序排列后的中位数为第三个数：12, 15, 18, 20, 25 → 中位数 = 18。没有重复的众数，但你注意到销售在第三小时达到峰值。全距为 25 − 12 = 13，表明存在波动。
, maximum 25. The interquartile range (IQR) is 20 − 15 = 5, which shows that the middle 50% d. Comparing these statistics across products helps you decide which items to bake more of – ability to compare data sets using averages and spread is routinely tested in AQA papers.
为了进行更深度的分析，你使用五数概括法构建箱线图：最小值 12、下四分位数 15、中位数 18、上四分位数 20、最大值 25。四分位距（IQR）为 20 − 15 = 5，这表明中间 50% 的小时销售量集中度较高。将这些统计量在三种产品间进行比较，可以帮助你决定多烘烤哪种产品——纸杯蛋糕显然最受欢迎。利用平均数和离散度比较数据集的能力在 AQA 试卷中经常考查。
8. Applying Probability to Stock Decisions | 概率在备货决策中的应用
s, 40 cookies, 40 brownies), what is the probability that the next customer buys a cupcake? dence, the probability both buy cookies is (40/150) × (40/150) = (4/15)² = 16/225 ≈ 0.071.
在卖出的总共 150 个商品中（70 个纸杯蛋糕、40 个曲奇、40 个布朗尼），下一个顾客购买纸杯蛋糕的概率是多少？P(纸杯蛋糕) = 70 ÷ 150 = 7/15。该值可写成分数、小数或百分比。如果两位顾客接连到来，在假定独立的前提下，两人都买曲奇的概率为 (40/150) × (40/150) = (4/15)² = 16/225 ≈ 0.071。
s. This application of relative frequency and theoretical probability links directly to the AQA specification and gives you a solid reason to choose your stock quantities based on evidence rather than guesswork.
你还使用期望频数来决定在未来预计卖出 200 个商品的活动中准备多少个布朗尼：40/150 × 200 ≈ 53 个。这种相对频率和理论概率的应用与 AQA 大纲直接关联，为你基于证据而非凭空猜测来选择备货量提供了可靠的理由。
9. Checking Solutions and Assumptions | 检查解答与假设
good practice to test the sensitivity of your profit. Suppose flour costs increased by 10%: new flour cost = £1.50 × 1.10 = £1.65. The total cost becomes £18.90 + (£1.65 − £1.50) = £19.05. Profit under Option A falls to £225 − £19.05 = £205.95 – so the plan is robust. Understanding percentage increase and decrease in context is vital.
数学建模总是依赖于假设。你假定了每小时需求稳定、无损耗和售价不变。测试利润的敏感性是一种良好的实践。假设面粉成本上涨 10%：新面粉成本 = £1.50 × 1.10 = £1.65。总成本变为 £18.90 + (£1.65 − £1.50) = £19.05。方案 A 下的利润降至 £225 − £19.05 = £205.95——变化很小，因此计划是稳健的。在情境中理解百分比增减至关重要。
You also re-check all calculations: recipe scaling, cost tallies, graph scales and statistical summaries. A simple estimating technique, like rounding to the nearest pound for a quick mental total, can flag big errors. These checking habits are exactly what AQA examiners want to see in a well-structured solution.
你还重新检查了所有计算：食谱缩放、成本合计、图表刻度和统计摘要。简单的估算技巧——比如把金额取整到最接近的英镑快速心算总数——能够发现大的错误。这些检查习惯正是 AQA 阅卷人希望在条理清晰的解答中看到的。
10. Presenting Findings and Reflecting | 呈现结果与反思
Finally, you summarise your mathematics in a short report. The profit projection of £206.10 exceeds the £200 target, the stall layout fits the pitch after redesign, and data analysis shows cupcakes should be the main product. You highlight that ratios simplified costing, equations drove pricing, geometric reasoning solved the space problem, and statistics guided stocking decisions.
最后，你在简短的报告中总结了你的数学运用。利润预测 £206.10 超出了 £200 的目标，经过重新设计后摊位布局适合场地，数据分析表明纸杯蛋糕应为主要产品。你强调比率简化了成本计算，方程驱动了定价，几何推理解决了空间问题，而统计为备货决策提供了指导。
Reflecting on the process, you recognise how each AQA topic – number, algebra, ratio, geometry, probability and statistics – played a part in one coherent project. This holistic approach mirrors real-life problem solving and strengthens your ability to tackle multi-step exam questions with confidence.
反思整个过程，你认识到每一个 AQA 主题——数、代数、比率、几何、概率与统计——如何在一个连贯的项目中发挥作用。这种整体性的方法反映了现实生活的问题解决模式，并增强了你自信应对多步骤考试题目的能力。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图