📚 Year 8 Edexcel Maths Case Study: Fundraising Event | 八年级 Edexcel 数学：筹款活动案例分析实战演练
Imagine your school is planning a fundraising event to raise money for new sports equipment. You have been asked to help organise the sale of pizzas and drinks. Throughout this case study, you will apply your Year 8 maths skills — from ratios and percentages to area and data analysis — to make sure the event is a success. Let’s work through the real-world problems step by step.
想象一下，你所在的学校正在策划一场筹款活动，为购置新的体育器材募集资金。你受邀协助组织披萨和饮料的售卖。在这个案例研究中，你将运用八年级的数学技能——从比和比例、百分数到面积与数据分析——确保活动圆满成功。让我们一步步解决这些现实世界的问题。
1. Setting the Scene | 场景设定
The event will run for four hours on a Saturday afternoon. You plan to sell medium pizzas cut into 8 slices each, along with canned drinks. Your class has decided to charge £2.50 per slice and £1.20 per drink. Before buying supplies, you need to forecast how many customers to expect and how much to purchase.
活动定在周六下午，持续四个小时。你们计划售卖中型披萨（每张切成 8 块）以及罐装饮料。班级决定每块披萨售价 2.50 英镑，每罐饮料 1.20 英镑。在采购之前，你需要预测预计的顾客数量以及采购量。
- English: Based on a survey of the school, you estimate 250 people will attend. On average, each person might buy 1.5 slices of pizza and 1 drink. Calculate the expected number of pizza slices needed.
- 中文：根据对学校的调查，你们预计 250 人参加。平均而言，每人可能购买 1.5 块披萨和 1 罐饮料。计算所需的披萨块数。
Expected slices = 250 × 1.5 = 375 slices. Since each pizza has 8 slices, number of pizzas = 375 ÷ 8 = 46.875, so you round up to 47 pizzas.
预计块数 = 250 × 1.5 = 375 块。由于每张披萨有 8 块，披萨数量 = 375 ÷ 8 = 46.875，所以向上取整为 47 张披萨。
2. Budgeting: Estimating Costs and Revenue | 预算：估算成本与收入
You need to create a budget to see if the event will make a profit. The cost of each pizza from the supplier is £12.50, and a pack of 24 drinks costs £18.00. You will sell individual drinks so you can unpack the packs.
你需要制定预算，看看活动是否能盈利。供应商提供的每张披萨成本为 12.50 英镑，一箱 24 罐装饮料成本为 18.00 英镑。你们会拆分销售，所以可以打开包装单卖。
Total pizza cost = 47 × £12.50 = £587.50. For drinks, estimated demand = 250 × 1 = 250 drinks. Packs needed = 250 ÷ 24 = 10.416…, so buy 11 packs. Drink cost = 11 × £18.00 = £198.00.
披萨总成本 = 47 × 12.50 = 587.50 英镑。饮料预计需求量 = 250 × 1 = 250 罐。需要箱数 = 250 ÷ 24 = 10.416…，所以购买 11 箱。饮料成本 = 11 × 18.00 = 198.00 英镑。
Total cost = £587.50 + £198.00 = £785.50. Expected revenue from pizza = 375 slices × £2.50 = £937.50. Revenue from drinks = 250 × £1.20 = £300. Total revenue = £937.50 + £300 = £1237.50.
总成本 = 587.50 + 198.00 = 785.50 英镑。披萨预期收入 = 375 块 × 2.50 = 937.50 英镑。饮料收入 = 250 × 1.20 = 300 英镑。总收入 = 937.50 + 300 = 1237.50 英镑。
Forecasted profit = £1237.50 – £785.50 = £452.00. This is promising, but remember to consider leftover stock.
预期利润 = 1237.50 – 785.50 = 452.00 英镑。这看起来不错，但记得要考虑剩余库存。
3. Recipe Ratios: Pizza Dough Ingredients | 食谱配比：披萨面团配料
Suppose you were making the pizza dough yourself instead of buying ready-made ones. The basic dough recipe for 2 medium pizzas uses 500 g of flour, 300 ml of water, 7 g of yeast, and 2 tablespoons of olive oil. You need to scale this up for 47 pizzas.
假设你们决定自制披萨面团，而不是购买现成的。制作 2 张中型披萨的基础食谱需要：500 克面粉、300 毫升水、7 克酵母以及 2 汤匙橄榄油。你需要将这个食谱按比例扩大到 47 张披萨的量。
First, find the amount per pizza: flour per pizza = 500 ÷ 2 = 250 g. Water per pizza = 300 ÷ 2 = 150 ml. Yeast per pizza = 7 ÷ 2 = 3.5 g. Oil per pizza = 2 ÷ 2 = 1 tablespoon.
首先，计算每张披萨的用量：每张披萨面粉 = 500 ÷ 2 = 250 克。每张披萨水 = 300 ÷ 2 = 150 毫升。每张披萨酵母 = 7 ÷ 2 = 3.5 克。每张披萨油 = 2 ÷ 2 = 1 汤匙。
Then for 47 pizzas: flour = 47 × 250 = 11,750 g = 11.75 kg. Water = 47 × 150 = 7,050 ml = 7.05 litres. Yeast = 47 × 3.5 = 164.5 g. Oil = 47 × 1 = 47 tablespoons (about 47 × 15 ml = 705 ml).
然后计算 47 张披萨：面粉 = 47 × 250 = 11,750 克 = 11.75 千克。水 = 47 × 150 = 7,050 毫升 = 7.05 升。酵母 = 47 × 3.5 = 164.5 克。油 = 47 × 1 = 47 汤匙（约 47 × 15 毫升 = 705 毫升）。
This ratio work uses multiplication and division, essential for scaling recipes.
这种配比计算用到乘法和除法，是调整食谱份量的基本技能。
4. Percentage Discounts: Buying Drinks | 百分数折扣：购买饮料
The supplier offers a 15% discount on drink packs if you order more than 10 packs. Since you are buying 11 packs, you can claim the discount. What is the new cost for drinks?
供应商规定，如果订购超过 10 箱饮料，可享受 15% 的折扣。既然你们购买 11 箱，就可以享受这个折扣。饮料的新成本是多少？
Original cost for 11 packs = £198.00. Discount amount = 15% of £198.00 = 0.15 × 198 = £29.70. New cost = £198.00 – £29.70 = £168.30.
11 箱的原价 = 198.00 英镑。折扣金额 = 198.00 的 15% = 0.15 × 198 = 29.70 英镑。新成本 = 198.00 – 29.70 = 168.30 英镑。
You can also find the multiplier method: new cost = £198.00 × 0.85 = £168.30. Now the total cost for the event becomes £587.50 + £168.30 = £755.80, increasing profit to £1237.50 – £755.80 = £481.70.
你也可以用乘数法：新成本 = 198.00 × 0.85 = 168.30 英镑。现在活动的总成本变为 587.50 + 168.30 = 755.80 英镑，利润增加到 1237.50 – 755.80 = 481.70 英镑。
Percentages are very useful when comparing deals and maximising profit.
在比较优惠和最大化利润时，百分数非常有用。
5. Area and Perimeter: Comparing Pizza Sizes | 面积与周长：比较披萨大小
You are curious whether a large pizza cut into 8 slices would give more pizza per slice than a medium. A medium pizza has a diameter of 12 inches, a large has a 14-inch diameter. Calculate the area of each using the formula A = πr². Use π ≈ 3.14.
你好奇如果将大披萨切成 8 块，每块是否比中型披萨块更大。中型披萨直径为 12 英寸，大型为 14 英寸。用公式 A = πr² 计算每种披萨的面积。π 取 3.14。
Medium radius r = 12 ÷ 2 = 6 inches. Area = 3.14 × 6² = 3.14 × 36 = 113.04 square inches.
Large radius r = 14 ÷ 2 = 7 inches. Area = 3.14 × 7² = 3.14 × 49 = 153.86 square inches.
中型半径 r = 12 ÷ 2 = 6 英寸。面积 = 3.14 × 6² = 3.14 × 36 = 113.04 平方英寸。
大型半径 r = 14 ÷ 2 = 7 英寸。面积 = 3.14 × 7² = 3.14 × 49 = 153.86 平方英寸。
Area per slice: medium = 113.04 ÷ 8 = 14.13 in²; large = 153.86 ÷ 8 = 19.2325 in². So a large slice gives about 36% more pizza! However, you need to consider cost and selling price.
每块面积：中型 = 113.04 ÷ 8 = 14.13 平方英寸；大型 = 153.86 ÷ 8 = 19.2325 平方英寸。因此大块披萨比中块多约 36%！但你需要考虑成本和售价。
If a large pizza costs £16.00 and you still sell 8 slices at £2.50 each, revenue per pizza is £20.00, making a profit of £4.00 per pizza compared to medium’s £2.50 × 8 – £12.50 = £7.50 profit. So medium gives higher profit margin per pizza. Maths helps decision-making.
如果大披萨成本为 16.00 英镑，你仍然按每块 2.50 英镑卖 8 块，每张披萨收入 20.00 英镑，利润为 4.00 英镑，而中型披萨利润为 2.50 × 8 – 12.50 = 7.50 英镑。因此中型披萨的单张利润率更高。数学帮助做出决策。
6. Scaling Up: Adjusting the Recipe for More People | 比例缩放：为更多人调整食谱
On the day, unexpectedly 320 people attend instead of 250. You need to make more pizza quickly using the same dough recipe. Recalculate the ingredients needed for an extra 70 people, assuming they consume pizza at the same rate (1.5 slices each).
活动当天，出乎意料有 320 人参加而不是 250 人。你需要快速按同样的面团食谱制作更多披萨。重新计算为额外 70 人所需的配料，假设他们以同样速度消费（每人 1.5 块）。
Extra slices needed = 70 × 1.5 = 105 slices. Extra pizzas needed = 105 ÷ 8 = 13.125, so 14 more pizzas. Now total pizzas = 47 + 14 = 61.
额外所需块数 = 70 × 1.5 = 105 块。额外所需披萨数量 = 105 ÷ 8 = 13.125，所以需要 14 张更多披萨。现在披萨总量 = 47 + 14 = 61。
Using the per-pizza recipe amounts: flour = 61 × 250 = 15,250 g = 15.25 kg; water = 61 × 150 = 9,150 ml = 9.15 L; yeast = 61 × 3.5 = 213.5 g; oil = 61 tbsp.
使用每张披萨的配料用量：面粉 = 61 × 250 = 15,250 克 = 15.25 千克；水 = 61 × 150 = 9,150 毫升 = 9.15 升；酵母 = 61 × 3.5 = 213.5 克；油 = 61 汤匙。
Scaling up uses direct proportion. If the original recipe for 2 pizzas is doubled for 4, tripled for 6, etc., you can use the unitary method to find values for any number.
比例缩放使用的是正比例关系。如果 2 张披萨的食谱翻倍就是 4 张，三倍就是 6 张，你可以用归一法求出任意数量的用量。
7. Data and Statistics: Surveying Favourite Flavours | 数据与统计：调查最受欢迎的口味
To plan the toppings, you surveyed 120 students. The results: 45 chose Margherita, 30 Pepperoni, 25 Hawaiian, and 20 Vegetarian. You want to present this in a pie chart and calculate the proportion of each.
为了计划配料，你们调查了 120 名学生。结果如下：45 人选玛格丽特，30 人选辣香肠，25 人选夏威夷，20 人选素食。你想用饼图展示并计算每种所占比例。
Calculate angle for each sector: total = 120. Margherita = (45/120) × 360° = 135°. Pepperoni = (30/120) × 360° = 90°. Hawaiian = (25/120) × 360° = 75°. Vegetarian = (20/120) × 360° = 60°.
计算每个扇区的角度：总数 = 120。玛格丽特 = (45/120) × 360° = 135°。辣香肠 = (30/120) × 360° = 90°。夏威夷 = (25/120) × 360° = 75°。素食 = (20/120) × 360° = 60°。
You could also express these as fractions: Margherita 45/120 = 3/8, Pepperoni 30/120 = 1/4, Hawaiian 25/120 = 5/24, Vegetarian 20/120 = 1/6. Use these to plan how many of each type to order.
你也可以用分数表示：玛格丽特 45/120 = 3/8，辣香肠 30/120 = 1/4，夏威夷 25/120 = 5/24，素食 20/120 = 1/6。用这些数据来计划每种口味订购多少张。
If you make 61 pizzas, numbers: Margherita ≈ 61 × 3/8 = 22.875 → 23 pizzas, Pepperoni = 61 × 1/4 = 15.25 → 15, Hawaiian = 61 × 5/24 ≈ 12.7 → 13, Vegetarian = 61 × 1/6 ≈ 10.17 → 10. Sum = 23+15+13+10=61. Nice.
如果制作 61 张披萨，数量为：玛格丽特 ≈ 61 × 3/8 = 22.875 → 23 张，辣香肠 = 61 × 1/4 = 15.25 → 15，夏威夷 = 61 × 5/24 ≈ 12.7 → 13，素食 = 61 × 1/6 ≈ 10.17 → 10。总和 = 23+15+13+10=61。正好。
8. Profit Calculations and Break-Even | 利润计算与盈亏平衡
You need to know the break-even point—the number of slices you must sell to cover costs. With the updated total cost of £755.80 (using discount) and selling price per slice £2.50, let’s calculate.
你需要知道盈亏平衡点——即必须卖出多少块披萨才能收回成本。更新后的总成本为 755.80 英镑（使用折扣），每块披萨售价 2.50 英镑，我们来计算。
Break-even point in slices = Total cost ÷ Price per slice = £755.80 ÷ £2.50 = 302.32 slices. So you need to sell at least 303 slices to avoid a loss. Remember, drink sales also contribute but we are focusing on pizza only here for simplicity. In reality, combined break-even requires considering both item margins.
按块数计算的盈亏平衡点 = 总成本 ÷ 每块售价 = 755.80 ÷ 2.50 = 302.32 块。所以至少需要卖出 303 块披萨才能避免亏损。请记住，饮料销售也有贡献，但这里我们简化只考虑披萨。现实中，综合盈亏平衡需考虑两种商品的利润。
If you sell all 61 pizzas as slices: 61 × 8 = 488 slices. Profit from pizza = 488 × 2.50 – 755.80 = 1220 – 755.80 = £464.20. Plus drink profit (assuming you buy the right amount). This is a healthy profit.
如果你将所有 61 张披萨按块售出：61 × 8 = 488 块。披萨利润 = 488 × 2.50 – 755.80 = 1220 – 755.80 = 464.20 英镑。加上饮料利润（假设采购量适当），这是一笔可观的利润。
9. Time Management and Scheduling | 时间管理与日程安排
The event is 4 hours (2:00 PM – 6:00 PM). You need to bake pizzas in batches. Each pizza takes 12 minutes to prepare and cook, and you have two ovens that can each hold 2 pizzas at a time. How long to cook all 61 pizzas?
活动时长 4 小时（下午 2:00 至 6:00）。你需要分批烤制披萨。每张披萨准备和烤制需 12 分钟，你们有两台烤箱，每台一次可烤 2 张披萨。烤完所有 61 张披萨需要多长时间？
Capacity per batch = 2 ovens × 2 pizzas = 4 pizzas. Number of batches = 61 ÷ 4 = 15.25, so 16 batches. Total time = 16 × 12 minutes = 192 minutes = 3 hours 12 minutes. This fits within the 4-hour window if you start on time, but it’s tight. Perhaps you need to pre-cook some pizza bases.
每批产能 = 2 台烤箱 × 2 张披萨 = 4 张。批次数 = 61 ÷ 4 = 15.25，即 16 批。总时间 = 16 × 12 分钟 = 192 分钟 = 3 小时 12 分钟。如果准时开始，这能在 4 小时窗口内完成，但时间很紧。或许你需要提前预制一些披萨饼底。
You can also analyse with a timeline chart. For instance, if you start baking at 1:00 PM, you can have plenty ready by opening. Mathematical scheduling helps avoid delays.
你也可以用时间线图进行分析。例如，如果下午 1:00 开始烤制，你可以在开业时备好大量披萨。数学日程安排有助于避免延误。
10. Handling Leftovers and Waste | 处理剩余与损耗
At the end of the event, you have 3 whole pizzas unsold and 15 drinks remaining. Calculate the loss of potential profit and decide what to do. The unsold pizzas cost £12.50 each (or maybe they were homemade so cheaper). For simplicity, assume the store-bought cost.
活动结束时，你有 3 整张披萨未售出，还剩 15 罐饮料。计算潜在利润的损失并决定如何处理。未售出的披萨每张成本为 12.50 英镑（如果是自制则更便宜）。为简单起见，假设为购买的成本。
Cost of unsold pizzas = 3 × £12.50 = £37.50. If you could have sold them as slices: 3 × 8 × £2.50 = £60.00 revenue, so lost profit = £60.00 – £37.50 = £22.50. Drinks: cost of 15 drinks = £18.00/24 = £0.75 per can; 15 × £0.75 = £11.25 cost. Lost revenue if sold = 15 × £1.20 = £18.00, lost profit = £18.00 – £11.25 = £6.75. Total lost potential profit = £29.25.
未售出披萨的成本 = 3 × 12.50 = 37.50 英镑。如果将它们按块售出：3 × 8 × 2.50 = 60.00 英镑收入，因此利润损失 = 60.00 – 37.50 = 22.50 英镑。饮料：15 罐的成本 = 18.00/24 = 每罐 0.75 英镑；15 × 0.75 = 11.25 英镑成本。若售出则收入损失 = 15 × 1.20 = 18.00 英镑，利润损失 = 18.00 – 11.25 = 6.75 英镑。总计潜在利润损失 = 29.25 英镑。
To reduce waste, you could sell remaining slices at half price near closing, or donate them. This teaches inventory management using percentages and proportion.
为减少浪费，你可以在快结束时半价出售剩余披萨，或者捐赠。这教会你用百分数和比例进行库存管理。
11. Reflection: The Power of Maths in Action | 反思：数学在行动中的力量
This case study shows that Year 8 maths is not just about solving exercises in a book. Ratio, proportion, percentages, area, data handling, and simple algebra were used to plan, budget, scale up, and forecast. You analysed costs, maximised profit, and scheduled tasks — all by applying mathematical thinking.
这个案例研究表明，八年级数学不仅仅是解决课本上的习题。比和比例、百分数、面积、数据处理以及简单的代数都被用来规划、预算、扩大规模以及预测。你通过应用数学思维分析了成本、最大化了利润并安排了任务。
The skills you practised here are transferrable to real-life situations: organising events, running a small business, or even managing personal finance. Always ask yourself: “How can maths help me make better decisions?”
你在这里练习的技能可以迁移到现实生活情境中：组织活动、经营小生意，甚至管理个人财务。始终问自己：“数学如何能帮助我做出更好的决策？”
Keep revisiting these concepts and try creating your own case studies with different scenarios — a bake sale, a sports tournament, or a school trip. The more you practice, the more confident you’ll become.
不断复习这些概念，并尝试用不同的场景创建自己的案例研究——义卖蛋糕、体育比赛或者学校旅行。练习得越多，你就会越自信。
12. Extra Challenge: What If? | 额外挑战：如果……呢？
What if the attendance was 25% higher than expected, and each person bought 2 slices on average? Recalculate the number of pizzas needed and the new profit. (Attendance = 320 × 1.25 = 400 people; slices = 400 × 2 = 800 slices; pizzas = 800 ÷ 8 = 100 pizzas. Then redo the budget with discounts and costs. This can be your homework!)
如果参加人数比预期高出 25%，并且平均每人购买 2 块披萨呢？重新计算所需的披萨数量以及新利润。（人数 = 320 × 1.25 = 400 人；块数 = 400 × 2 = 800 块；披萨数 = 800 ÷ 8 = 100 张。然后重新计算包含折扣和成本的预算。这可以作为你的家庭作业！）
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