📚 Case Study Practical Exercises: School Carnival Planning | 案例分析实战演练：学校嘉年华规划
Welcome to this practical case study designed to sharpen your advanced mathematics skills. You will take on the role of a student committee member responsible for planning a school carnival. Along the way, you will apply number operations, fractions, decimals, percentages, geometry, probability, and data handling to solve real-world problems. Work through each section carefully, showing your reasoning step by step.
欢迎参加本次旨在提升你进阶数学技能的实战案例分析。你将扮演一名负责策划学校嘉年华的学生委员会成员。在此过程中，你将运用整数运算、分数、小数、百分数、几何、概率和数据处理来解决真实问题。请仔细完成每个部分，逐步展示你的推理过程。
1. Introduction to the Case Study | 案例介绍
Your school has decided to hold a carnival to raise money for new sports equipment. A budget of £500 has been allocated for initial expenses. The committee must plan the layout of stalls, set ticket prices, order supplies, and predict profits. You will also analyse data from last year's event to make informed decisions.
你所在学校决定举办一场嘉年华，为新的体育器材筹集资金。已拨款 £500 作为启动经费。委员会需要规划摊位布局、设定门票价格、订购物资并预测利润。你还将分析去年活动的数据以做出合理的决策。
The goal of this case study is to integrate various mathematical topics into a single, cohesive project. By the end, you will have calculated costs, designed a floor plan, used probability to win games, and presented your findings in a clear report. Ready? Let's begin.
本案例分析的目标是将多个数学主题整合到一个连贯的项目中。最终，你将计算成本、设计平面图、利用概率赢得游戏，并以清晰的报告呈现你的发现。准备好了吗？我们开始吧。
2. Understanding the Budget | 理解预算
The initial £500 must be divided among three categories: stall decorations (25%), food ingredients (45%), and prizes for games (30%). First, calculate the exact amount allocated to each category. Show your working in pounds and pence.
初始的 £500 必须在三个类别间分配：摊位装饰 (25%)、食品原料 (45%) 和游戏奖品 (30%)。首先计算每个类别的具体拨款金额。请用英镑和便士展示你的计算过程。
Decorations: 25% of £500 = 0.25 × 500 = £125.00. Food ingredients: 45% of £500 = 0.45 × 500 = £225.00. Prizes: 30% of £500 = 0.30 × 500 = £150.00.
装饰：£500 的 25% = 0.25 × 500 = £125.00。食品原料：£500 的 45% = 0.45 × 500 = £225.00。奖品：£500 的 30% = 0.30 × 500 = £150.00。
spending by one-fifth and move that saving into prizes. What is the new amount for prizes? The saving is 1/5 of £125 = £25. New prizes amount = £150 + £25 = £175.00.
现在假设委员会决定将装饰支出减少五分之一，并将节省的金额转入奖品类别。奖品的新金额是多少？节省金额为 £125 的 1/5 = £25。新奖品金额 = £150 + £25 = £175.00。
3. Area and Perimeter of the Venue | 场地面积与周长
The school field is rectangular, measuring 60 metres in length and 40 metres in width. Calculate the perimeter and area of the field. These measurements will help you plan the stall arrangement.
学校操场为矩形，长 60 米，宽 40 米。计算操场的周长和面积。这些测量数据将帮助你规划摊位布置。
Perimeter = 2 × (length + width) = 2 × (60 m + 40 m) = 2 × 100 m = 200 m
Area = length × width = 60 m × 40 m = 2400 m²
central performance stage that is a square of side 10 m. What is the remaining usable area? Stage area = 10 m × 10 m = 100 m². Remaining area = 2400 m² − 100 m² = 2300 m².
你决定设置一个边长 10 米的正方形中央表演舞台。剩余可用面积是多少？舞台面积 = 10 m × 10 m = 100 m²。剩余面积 = 2400 m² − 100 m² = 2300 m²。
ires a rectangular space of 3 m by 4 m. Including walking paths, you estimate that only 70% of the remaining area can be used for stalls. How many stalls can fit? Usable stall area = 70% 2 m². Number of stalls = 1610 ÷ 12 = 134.166..., so a maximum of 134 stalls can be placed.
每个摊位需要一个 3 米 × 4 米的矩形空间。考虑到过道，你估计只有剩余面积的 70% 可用于摆放摊位。最多能容纳多少个摊位？摊位可用面积 = 2300 m² 的 70% = 0.70 × 2300 = 1610 m²。每个摊位面积 = 12 m²。摊位数量 = 1610 ÷ 12 = 134.166...，因此最多可放置 134 个摊位。
4. Ticket Pricing Strategy | 门票定价策略
You plan to sell two types of tickets: adult tickets at £3.50 each and child tickets at £1.80 each. Last year, the ratio of adult to child tickets sold was 2:5. If you expect to sell 1,400 tickets in total, how many adult and child tickets do you predict?
你计划出售两种门票：成人票每张 £3.50，儿童票每张 £1.80。去年售出的成人票与儿童票比例为 2:5。如果预计共售出 1400 张门票，你预测成人票和儿童票各售出多少？
t = 1400 ÷ 7 = 200 tickets. Adult tickets = 2 × 200 = 400. Child tickets = 5 × 200 = 1000.
比例的总份数 = 2 + 5 = 7 份。一份 = 1400 ÷ 7 = 200 张。成人票 = 2 × 200 = 400 张。儿童票 = 5 × 200 = 1000 张。
Calculate the total revenue from ticket sales. Revenue from adults = 400 × £3.50 = £1400. Revenue from children = 1000 × £1.80 = £1800. Total = £1400 + £1800 = £3200.
计算门票销售总收入。成人票收入 = 400 × £3.50 = £1400。儿童票收入 = 1000 × £1.80 = £1800。总收入 = £1400 + £1800 = £3200。
If you decide to offer a 15% discount on all tickets during a pre-sale week and predict 30% s will be sold at this discount, calculate the new total revenue. Discounted adult price = £3.50 × 0.85 = £2.975 ≈ £2.98. Discounted child price = £1.80 × 0.85 = £1.53. Pre-sale adult tickets = 30% of 400 = 120. Pre-sale child tickets = 30% of 1000 = 300. Revenue from pre-sale = (120 × £2.98) + (300 × £1.53) = £357.60 + £459.00 = £60. Remaining tickets at full price: adults 280, children 700. Full price revenue = (280 × £3.50) + (700 × £1.80) = £980 + £1260 = £2240. Total new revenue = £816.60 + £2240 = £3056.60.
如果你决定在预售周对所有门票给予 15% 的折扣，并预测总票量的 30% 将以此折扣售出，计算新的总收入。折扣成人票价 = £3.50 × 0.85 = £2.975 ≈ £2.98。折扣儿童票价 = £1.80 × 0.85 = £1.53。预售成人票 = 400 的 30% = 120 张。预售儿童票 = 1000 的 30% = 300 张。预售收入 = (120 × £2.98) + (300 × £1.53) = £357.60 + £459.00 = £816.60。余下全价票：成人 280 张，儿童 700 张。全价收入 = (280 × £3.50) + (700 × £1.80) = £980 + £1260 = £2240。新总收入 = £816.60 + £2240 = £3056.60。
5. Food Stall Costs and Profits | 食品摊位成本与利润
You plan to run a hot dog stall. The ingredients for one hot dog cost £0.65. You will sell each hot dog for £2.20. What is the profit made on each hot dog sold?
你计划经营一个热狗摊。一个热狗的原料成本为 £0.65。每个热狗的售价为 £2.20。每卖出一个热狗的利润是多少？
Profit per hot dog = Selling price − Cost price = £2.20 − £0.65 = £1.55
If 340 hot dogs are sold during the carnival, what is the total profit from this stall? Total profit = 340 × £1.55 = £527.00.
如果在嘉年华期间售出 340 个热狗，该摊位的总利润是多少？总利润 = 340 × £1.55 = £527.00。
However, there is a fixed stall rental fee of £30. The stall also spent £12 on sauces and napkins. Deduct these to find the net profit. Total expenses = £30 + £12 = £42. Net profit = £527.00 − £42.00 = £485.00.
但是，还有 £30 的固定摊位租赁费，以及 £12 的酱料和纸巾支出。扣除这些费用后计算净利润。总费用 = £30 + £12 = £42。净利润 = £527.00 − £42.00 = £485.00。
Suppose you want the net profit to be at least £ 2 = 600 => 1.55n = 642 => n = 642 ÷ 1.55 ≈ 414.19. So you must sell at least 415 hot dogs.
假设你希望净利润至少为 £600。确定你需要售出的最少热狗数量。方程：(1.55 × n) − 42 = 600，1.55n = 642，n = 642 ÷ 1.55 ≈ 414.19，因此需售出至少 415 个热狗。
6. Probability at Game Stalls | 游戏摊位的概率
One game involves spinning a fair spinner divided into 8 equal sections: 3 red, 2 blue, 2 yellow, and 1 gold. A player wins a prize if the spinner lands on gold. What is the theoretical probability of winning?
一个游戏需旋转一个均分为 8 等份的公平转盘：3 个红色，2 个蓝色，2 个黄色，1 个金色。如果指针停在金色区域则赢得奖品。获胜的理论概率是多少？
P(win) = Number of gold sections / Total sections = 1/8
If 160 players each spin once, how many players would you expect to win? Expected winners = 160 × (1/8) = 20 players.
如果有 160 名玩家各旋转一次，你预期有多少名玩家获胜？预期获胜者 = 160 × (1/8) = 20 人。
A prize costs £2.50. If you charge £1.00 per spin from 160 players, what is the expected profit from this game? Total income = 160 × £1.00 = £160. Cost of prizes = 20 × £2.50 = £50. Expected profit = £160 − £50 = £110.
每个奖品成本 £2.50。如果向 160 名玩家收取每人 £1.00 的旋转费，该游戏的预期利润是多少？总收入 = 160 × £1.00 = £160。奖品成本 = 20 × £2.50 = £50。预期利润 = £160 − £50 = £110。
1 white ball. A player draws one ball. What is the probability of not drawing a white ball? P(not white) = 9/10 = 0.9.
另一个游戏使用一个装有 5 个红球、4 个绿球和 1 个白球的袋子。玩家摸出一个球。未摸出白球的概率是多少？P（非白）= 9/10 = 0.9。
7. Scheduling and Timetables | 时间安排与时刻表
The carnival runs from 10:30 am to 4:45 pm. Calculate the total duration in hours and minutes. From 10:30 to 16:45 is 6 hours and 15 minutes.
嘉年华从上午 10:30 开始，到下午 4:45 结束。计算总共持续的时长（以小时和分钟表示）。从 10:30 到 16:45 为 6 小时 15 分钟。
You plan to run the main performance 3 times, each lasting 35 minutes, with 15-minute breaks between them. Will the performances fit within the carnival duration if the first starts at 11:00 am? First show: 11:00–11:35. Break: 11:35–11:50. Second show: 11:50–12:25. Break: 12:25–12:40. Third show: 12:40–13:15. Yes, they fit comfortably before 13:15.
你计划进行 3 场主表演，每场持续 35 分钟，表演之间有 15 分钟休息。如果第一场于上午 11:00 开始，这些表演能否在嘉年华时间内完成？第一场：11:00–11:35。休息：11:35–11:50。第二场：11:50–12:25。休息：12:25–12:40。第三场：12:40–13:15。是的，它们在 13:15 之前轻松完成。
arrives at 8:45 am to set up and needs 1 hour 40 minutes. What time will the setup finish? 8:45 + 1 hour 40 min = 10:25 am.
一名摊主于上午 8:45 到达准备，需要 1 小时 40 分钟。准备工作什么时候完成？8:45 + 1 小时 40 分钟 = 上午 10:25。
8. Data Collection and Graphs | 数据收集与图表
Last year, the number of visitors each hour was recorded: 10–11 am: 120; 11–12: 250; 12–1 pm: 310; 1–2 pm: 300; 2–3 pm: 270; 3–4 pm: 195; 4–5 pm: 85. Construct a bar chart of these data, labelling axes clearly.
去年每小时游客人数记录如下：10–11 am: 120; 11–12: 250; 12–1 pm: 310; 1–2 pm: 300; 2–3 pm: 270; 3–4 pm: 195; 4–5 pm: 85。绘制这些数据的条形图，清晰标注坐标轴。
310+300+270+195+85 = 1530. Number of hours = 7. Mean = 1530 ÷ 7 ≈ 218.6 visitors per hour.
计算每小时游客数的平均值。总和 = 120+250+310+300+270+195+85 = 1530。小时数 = 7。平均值 = 1530 ÷ 7 ≈ 218.6 人/小时。
What was the percentage of total visitors that came between 12 pm and 2 pm? Visitors in 12–2 pm = 310 + 300 = 610. Percentage = (610 / 1530) × 100% ≈ 39.9%.
中午 12 点到下午 2 点之间的游客数占总数的百分比是多少？12–2 pm 游客数 = 310 + 300 = 610。百分比 = (610 / 1530) × 100% ≈ 39.9%。
If you want to schedule extra activities during the busiest 2-hour period, which slot would you choose and why? The slot 12–2 pm with 610 visitors is the busiest, so that is the best choice.
如果你想在最繁忙的 2 小时时段安排额外活动，你会选择哪个时段，为什么？12–2 pm 时段有 610 名游客，是最繁忙的，因此是最佳选择。
9. Break-Even Analysis | 盈亏平衡分析
The carnival's total fixed costs (rent, licences, advertising) amount to £640. The variable cost per visitor is £1.20 (for free samples, security, etc.), and the average revenue per visitor from tickets and stalls is £4.80. How many visitors are needed to break even?
嘉年华的总固定成本（租金、许可证、广告）为 £640。每位游客的变动成本为 £1.20（用于免费样品、安保等），每位游客的平均收入（来自门票和摊位）为 £4.80。需要多少游客才能达到盈亏平衡？
Variable cost per visitor) = 640 ÷ (4.80 − 1.20) = 640 ÷ 3.60 = 177.77... ≈ 178 visitors.
If 350 visitors attend, what is the total profit? e cost per visitor) × Number of visitors − Fixed costs = (3.60 × 350) − 640 = 1260 − 640 = £620.
如果有 350 名游客参加，总利润是多少？利润 = (每位游客收入 − 每位游客变动成本) × 游客数量 − 固定成本 = (3.60 × 350) − 640 = 1260 − 640 = £620。
ing the ticket price to attract more visitors. The new average revenue per visitor becomes £4.00. How many visitors are now needed for the profit to reach £800? 20)n − 640 = 800 => 2.80n = 1440 => n = 1440 ÷ 2.80 ≈ 514.3, so 515 visitors are required.
委员会考虑降低门票价格以吸引更多游客。新的每位游客平均收入变为 £4.00。现在需要多少游客才能使利润达到 £800？设游客数为 n。(4.00 − 1.20)n − 640 = 800，2.80n = 1440，n = 1440 ÷ 2.80 ≈ 514.3，因此需要 515 名游客。
10. Final Report and Reflection | 最终报告与反思
Summarise your financial plan. From earlier calculations, if 350 visitors attend with full pricing, total revenue from tickets is £3,056.60 (from pre-sale and full price). Food stall net profit is £485.00, game profit is £110.00, and overall carnival profit is £620 (from break-even analysis). Note that ticket revenue and stall profits are combined in the per-visitor average, so the £620 is the overall carnival profit.
总结你的财务计划。根据之前的计算，如果有 350 名游客并以全价票参与，门票总收入为 £3056.60（含预售和全价）。食品摊净利润为 £485.00，游戏利润为 £110.00，嘉年华总利润为 £620（来自盈亏平衡分析）。注意门票收入和摊位利润已合并到每位游客平均值中，因此 £620 为嘉年华整体利润。
Reflect on which assumptions might change the outcome. For example, if the ratio of adult to child visitors differs, or if the spinner probabilities are not exactly as planned due to real-world imperfections. Mathematics provides a model, but real events often vary.
反思哪些假设可能会改变结果。例如，如果成人票与儿童票的比例不同，或者转盘概率因现实不完美而未完全符合计划。数学提供了一个模型，但实际活动往往存在差异。
Present your findings in a short report using charts, tables, and clear calculations. This case study shows how advanced mathematics can be applied to plan, predict, and improve a school event.
使用图表、表格和清晰的计算，将你的发现写成一份简短报告。本案例分析展示了如何应用进阶数学来规划、预测和改进学校活动。
Published by TutorHao | Advanced Mathematics Revision Series | aleveler.com
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