📚 Case Study Practice: Real-World Maths Problems | 案例分析实战演练：生活中的数学问题
In Edexcel GCSE Mathematics, case study questions test your ability to apply mathematical concepts to everyday situations. These problems often combine topics like percentages, ratio, algebra, geometry and statistics. This revision guide will walk you through realistic scenarios, showing step-by-step how to deconstruct a problem, choose the right method and present your solution clearly. Working through these examples will boost your confidence and sharpen your problem-solving skills for the exam.
在 Edexcel GCSE 数学中，案例分析题考查你将数学概念应用到实际情境的能力。这类题目常常结合百分比、比例、代数、几何和统计等多个知识点。本复习指南将带你走进真实场景，逐步展示如何拆解问题、选择正确方法以及清晰地呈现解答。通过演练这些例题，你将增强信心，提升应考解题能力。
1. Understanding Case Study Questions | 理解案例分析题
Case study questions usually provide a block of text describing a real-life scenario, along with data in tables, charts or diagrams. Your first task is to read carefully and identify the mathematical topics involved. Underline key numbers and what you are asked to find. Then plan a strategy: will you need to change units, set up an equation, compare ratios or draw a graph? Organise your working logically and always check if your final answer makes sense in context.
案例分析题通常给出一段描述真实情境的文字，并配有表格、图表或示意图中的数据。你的首要任务是仔细阅读并识别涉及的数学主题。标出关键数字以及题目要求你求出的量。然后制定策略：你是否需要转换单位、建立方程、比较比例或者绘制图表？逻辑清晰地组织解题过程，并始终检查最终答案在实际情况中是否合理。
2. Case Study 1: Savings Account Interest | 案例1：储蓄账户利息
Olivia deposits £1200 in a savings account offering 2.5% compound interest per annum. She plans to keep her money untouched for 3 years. How much will she have at the end of the period? This is a compound interest scenario, where the interest earned each year is added to the principal for the following year.
奥利维亚将1200英镑存入年利率为2.5%的复利储蓄账户。她打算3年内不动用这笔钱。期末她将拥有多少钱？这是一个复利场景，每年所获利息会加入本金用于下一年的计息。
3. Calculating Compound Interest | 计算复利
100) = 1 + 0.025 = 1.025. So the amount after 3 years is 1200 × 1.025³. Step-by-step: 1.025² = 1.050625, then multiply by 1.025 again to get 1.076890625. Multiply by 1200 to get £1292.27 (to the nearest penny). Always round money to two decimal places.
我们可以通过将初始金额乘以利息因子的年数次幂来求解。利息因子 = 1 + (利率/100) = 1 + 0.025 = 1.025。因此3年后的金额为1200 × 1.025³。逐步计算：1.025² = 1.050625，再乘以1.025得到1.076890625。乘以1200得到1292.27英镑（精确到便士）。货币结果务必保留两位小数。
Final amount = £1200 × 1.025³ = £1292.27
An alternative method is to find the interest year by year:
- Year 1: £1200 × 0.025 = £30 interest; new balance £1230.
- Year 2: £1230 × 0.025 = £30.75; balance £1260.75.
- Year 3: £1260.75 × 0.025 = £31.52 (approx); balance £1292.27.
另一种方法是逐年计算利息：
- 第1年：1200 × 0.025 = 30英镑利息；新余额1230英镑。
- 第2年：1230 × 0.025 = 30.75英镑；余额1260.75英镑。
- 第3年：1260.75 × 0.025 ≈ 31.52英镑；余额1292.27英镑。
4. Case Study 2: Best Value Shopping | 案例2：最佳购物方案
A supermarket sells rice in two sizes: a 500 g bag for £1.80 and a 2 kg bag for £6.50. Which bag offers better value for money? To compare, we find the cost per 100 g or per kilogram. Here we'll calculate price per kilogram. For the 500 g bag: 500 g is 0.5 kg, so price per kg = £1.80 ÷ 0.5 = £3.60 per kg. For the 2 kg bag: price per kg = £6.50 ÷ 2 = £3.25 per kg. The larger bag is better value.
一家超市出售两种规格的大米：一袋500克售价1.80英镑，另一袋2千克售价6.50英镑。哪种包装更实惠？为了比较，我们计算每100克或每千克的价格。这里我们计算每千克价格。500克袋：500克=0.5千克，因此每千克价格=1.80 ÷ 0.5 = 每千克3.60英镑。2千克袋：每千克价格=6.50 ÷ 2 = 每千克3.25英镑。大包装更划算。
5. Using Unit Cost to Compare | 使用单位成本比较
Always convert to a common unit to avoid mistakes. In this example, we used kilograms. Another approach is to compare the cost per gram: 500 g for £1.80 gives £0.0036 per gram; 2000 g for £6.50 gives £0.00325 per gram. Whichever method, the conclusion is the same. When dealing with multipack offers, remember to check if the items are identical in weight or volume. Edexcel exams often test your ability to spot misleading packaging or discounts.
始终转换为同一单位以避免错误。本例中我们使用了千克。另一种方法是比较每克成本：500克1.80英镑，每克0.0036英镑；2000克6.50英镑，每克0.00325英镑。无论采用哪种方法，结论都相同。处理多件装优惠时，记得检查商品净重或容量是否相同。Edexcel考试经常测试你识别具有误导性的包装或折扣的能力。
6. Case Study 3: Journey Planning | 案例3：旅行计划
iles. Calculate the total journey time. This problem requires splitting the journey into two legs and adding the break. For the first leg: distance 120 miles, speed 60 mph, time = 120 ÷ 60 = 2 hours. Then a 0.5-hour break. Second leg: remaining distance = 210 - 120 = 90 miles, speed still 60 mph, time = 90 ÷ 60 = 1.5 hours. Total time = 2 + 0.5 + 1.5 = 4 hours.
一辆汽车从伦敦开往曼彻斯特，全程210英里，平均车速60英里/小时。司机在行驶120英里后休息30分钟。计算整个行程耗时。此题需要将行程分成两段并加上休息时间。第一段：距离120英里，速度60英里/小时，时间 = 120 ÷ 60 = 2小时。然后休息0.5小时。第二段：剩余距离 = 210 - 120 = 90英里，速度仍为60英里/小时，时间 = 90 ÷ 60 = 1.5小时。总时间 = 2 + 0.5 + 1.5 = 4小时。
7. Speed, Distance and Time Relationship | 速度、距离与时间关系
The key formula linking speed, distance and time is:
Speed = Distance ÷ Time
s and time in hours. When time includes minutes, convert to decimal hours (e.g., 15 minutes = 0.25 hours). In Edexcel questions, you may need to convert between metric and imperial units, so familiarise yourself with common conversions like 1 mile ≈ 1.6 km.
连接速度、距离和时间的关键公式为：
速度 = 距离 ÷ 时间
变形得：时间 = 距离 ÷ 速度，距离 = 速度 × 时间。务必确保单位一致：如果速度单位是英里/小时，距离就必须用英里，时间用小时。当时间包含分钟时，应转换为小时的小数形式（例如15分钟=0.25小时）。在Edexcel考题中，你可能需要在公制和英制单位间转换，因此要熟悉常用换算，如1英里≈1.6千米。
8. Case Study 4: Designing a Garden | 案例4：花园设计
A rectangular garden measures 8 m by 6 m. The owner wants to lay a square patio of side 3 m in one corner and cover the rest with turf. She also plans to erect a fence around the whole garden. Find the area of the patio, the area to be turfed, and the length of fencing required. This blends area, perimeter and real-world layout.
一个矩形花园长8米、宽6米。主人想在花园一角铺设一个边长3米的正方形露台，其余区域铺上草皮。她还计划沿花园整个边界建造围栏。求露台面积、铺草皮区域面积以及所需围栏长度。这道题融合了面积、周长及实际布局的应用。
9. Area and Perimeter Applications | 面积与周长应用
erimeter of rectangle = 2 × (length + width) = 2 × (8 + 6) = 2 × 14 = 28 m. Note that the patio placement does not reduce the perimeter because the fence goes around the garden boundary; the patio is inside. Always sketch a diagram to avoid confusion.
露台面积 = 边长 × 边长 = 3 × 3 = 9平方米。花园总面积 = 8 × 6 = 48平方米。草皮面积 = 总面积 - 露台面积 = 48 - 9 = 39平方米。围栏长度即花园周长。矩形周长 = 2 × (长 + 宽) = 2 × (8 + 6) = 2 × 14 = 28米。注意，露台的位置并不会减少围栏长度，因为围栏是沿着花园边界修建的；露台在内部。最好画出示意图以避免混淆。
If the fence was to enclose only the turf area (an L-shaped region), the perimeter would be different, but the question stated 'whole garden'. Be careful to read the wording precisely. Edexcel often tests whether you distinguish between area and perimeter.
如果围栏只围起草皮区域（L形区域），周长会不同，但题目明确说明“整个花园”。仔细审题很关键。Edexcel经常考查你是否能区分面积与周长。
10. Case Study 5: School Survey Analysis | 案例5：学校调查分析
A school surveyed 360 students about their favourite after-school activity. Results: 120 chose sport, 90 chose music, 60 chose art, and the rest chose reading. Draw a pie chart and find the percentage of students who chose reading. This problem combines data representation and percentage calculations.
一所学校对360名学生进行了关于最喜爱的课后活动调查。结果如下：120人选体育，90人选音乐，60人选美术，其余人选阅读。绘制饼图并求出选择阅读的学生所占百分比。此题结合了数据表示和百分比计算。
11. Interpreting Pie Charts and Percentages | 解读饼图与百分比
(120 + 90 + 60) = 360 - 270 = 90 students. Percentage for reading = (90 ÷ 360) × 100 = 25% ngles are: Sport = 120°, Music = 90°, Art = 60°, Reading = 90°. You would draw these sectors with a protractor, label them and give the chart a title.
首先，求出选择阅读的学生人数：360 - (120 + 90 + 60) = 360 - 270 = 90人。阅读的百分比 = (90 ÷ 360) × 100 = 25%。绘制饼图时，由于360名学生对应圆的360°，每个学生代表1度（360 ÷ 360 = 1°/人）。因此各扇区角度为：体育120°、音乐90°、美术60°、阅读90°。你需要用量角器画出这些扇区，标注清楚并给图表加上标题。
Numeracy skills like these frequently appear in Edexcel exams, often asking you to interpret an existing pie chart or complete a table and then construct one. Always double-check that the angles sum to 360° and that percentages add up to 100%.
类似的计算能力在Edexcel考试中很常见，常要求你解读已有饼图，或先完成表格再绘制饼图。务必检查扇区角度总和是否为360°，百分比总和是否为100%。
12. Key Takeaways for Edexcel | 关键考点总结
Case study questions reward methodical working and clear presentation. Always show your steps, use correct units and round appropriately. Practise converting wordy descriptions into mathematical expressions, and never hesitate to draw diagrams or tables. To prepare effectively, work through past Edexcel practice papers and review mark schemes to understand where marks are awarded. With regular practice, you will become confident in tackling any real-world problem the exam throws at you.
案例分析题倾向于奖励有条理的解题过程和清晰的呈现方式。务必展示步骤、使用正确的单位并进行恰当的舍入。练习将冗长的文字描述转换为数学表达式，并毫不犹豫地绘制示意图或表格。为了高效备考，应多做Edexcel往年试题并研读评分标准，了解每一步的得分点。通过反复练习，你将充满信心地应对考试中出现的任何实际问题。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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