📚 Year 11 Eduqas Maths: Case Study Problem-Solving Practice | Year 11 Eduqas 数学：案例分析实战演练
Eduqas GCSE Mathematics frequently challenges students with real-world scenarios that require more than just mechanical calculation. These case study questions test your ability to extract relevant information, choose the right mathematical tools, and interpret your results in context. This article provides a step-by-step walkthrough of strategies, thought processes, and worked mini case studies to help you master the skills needed to tackle any contextual problem with confidence.
Eduqas GCSE 数学经常通过与现实世界相关的情景题来考查学生，而不仅仅是机械计算。这类案例分析问题考验你提取相关信息、选择合适数学工具并在情境中解读结果的能力。本文将逐步讲解策略、思维过程，并通过若干小型案例演练，帮助你掌握自信应对任何情境问题所需的技能。
1. Understanding Case Study Questions in Eduqas Maths | 理解 Eduqas 数学中的案例分析问题
A case study question in the Eduqas exam is a multi-part problem embedded in a real-life situation, such as planning a trip, comparing mobile phone tariffs, designing a garden, or interpreting health data. The scenario provides numerical information, often displayed in tables, graphs, or written descriptions. Your task is to identify what needs to be calculated and decide which area of maths to apply.
Eduqas 考试中的案例分析题往往是一个嵌在真实生活场景里的多步骤问题，比如规划旅行、对比手机话费套餐、设计花园或解读健康数据。题目会给出数字信息，常以表格、图表或文字描述呈现。你的任务是确定需要计算什么，并决定应用哪部分数学知识。
Unlike standard procedural questions, case studies rarely tell you which formula to use. You must recognise that a situation involves percentages, compound measures, algebra, or probability. The mark scheme rewards correct interpretation and clear working just as much as the final numerical answer.
与标准的程序化题目不同，案例分析很少直接告诉你使用哪个公式。你必须识别出场景涉及百分数、复合单位、代数或概率。评分方案对正确的解读和清晰的步骤给分不亚于对最终数值答案的给分。
2. How to Read and Analyse the Problem Scenario | 如何阅读与分析问题情境
Begin by reading the whole question carefully, noticing any units, currencies, or time periods. Underline key pieces of data: lengths, costs, rates, and conditions. Then, re-read the specific instructions for each part to see what is actually being asked.
首先仔细通读全题，注意单位、货币或时间段。画出关键数据：长度、费用、比率和条件。然后再次阅读每个小问的具体要求，弄清到底需要你求什么。
Convert all measurements into consistent units early on. If a garden plan mixes metres and centimetres, change everything to the same unit. For money problems, ensure all amounts are in the same currency, usually pounds (£) or pence, before doing calculations.
尽早将所有测量值转换为一致的单位。如果花园设计图上米和厘米混用，就把所有数据统一成同一单位。涉及金钱问题时，确保计算前全部金额为同一种货币单位，通常是英镑 (£) 或便士。
Draw a simple diagram or annotate the given graph to visualise the scenario. A quick sketch often reveals relationships – such as similar triangles or symmetry – that are not immediately obvious from the text.
画一个简图或在给出的图上做标注，把场景形象化。一个速写草图常能揭示文字中没有直接显现的关系，例如相似三角形或对称性。
3. Identifying Key Mathematical Concepts and Variables | 识别关键数学概念与变量
Once you have the data, ask yourself: ‘What mathematical topic lies behind this scenario?’ Questions about growth over time may involve sequences, compound interest, or linear graphs. Optimising area could require quadratic expressions, while comparing two linear cost plans suggests solving simultaneous equations.
掌握数据后问自己：“这个场景背后是什么数学主题？”关于随时间增长的问题可能涉及序列、复利或线性图。优化面积可能需要二次表达式，而比较两个线性费用方案则意味着解联立方程。
Define variables for the unknowns. For example, let n be the number of people attending an event, let t be the time in hours, or let d be the distance travelled. Writing these down clarifies your approach and helps you formulate equations.
为未知量定义变量。例如，设 n 为参加活动的人数，设 t 为时间（小时），或设 d 为行驶距离。写下这些变量可以使你的思路更清晰，并有助于构建方程。
- “The total cost includes a fixed booking fee plus a charge per person” → a linear model: y = mx + c.
“总费用包括固定预约费加每人收费标准” → 线性模型：y = mx + c。 - “The price is reduced by 15% in a sale” → percentage decrease, multiplicative factor 0.85.
“打折降价15%” → 百分比减少，乘数因子0.85。 - “The volume of the container is doubled” → scale factor on lengths, area, or volume.
“容器容积翻倍” → 长度、面积或体积的比例因子。
4. Building Algebraic Models to Represent Relationships | 建立代数模型表达关系
Real-world case studies often require you to write an expression or formula. A fitness app charging a monthly subscription of £4.99 plus £0.30 per workout session can be written as:
现实案例分析常要求你写出表达式或公式。一款健身应用每月收费4.99英镑，外加每次锻炼0.30英镑，可以写成：
C = 4.99 + 0.30w
where C is the total monthly cost and w is the number of workouts. This model lets you answer ‘how many workouts for a budget of £10?’ by solving an equation.
其中 C 为每月总费用，w 为锻炼次数。这个模型可以让你通过解方程回答“预算10英镑能锻炼多少次？”
For non-linear relationships, such as a rectangular field with a wall on one side, expressing the area A in terms of width x might give a quadratic like A = x(50 − 2x). You can then find the maximum area by completing the square or using the symmetry of the parabola.
对于非线性关系，比如一边靠墙的矩形场地，将面积 A 用宽度 x 表示，可能会得到二次式 A = x(50 − 2x)。然后你可以通过配方法或利用抛物线的对称性求出最大面积。
Always state the meaning of each variable and the units when you present your model. The examiner expects clarity, not just a string of symbols.
在展示你的模型时，务必说明每个变量的含义及单位。考官期望表述清晰，而不仅仅是一串符号。
5. Tackling Graphs and Geometry in Context | 处理图形与几何情境
Graphs in context are more than lines and curves; they represent real changes. A distance–time graph shows speed, a conversion graph links currencies or units, and a cumulative frequency graph reveals median and quartiles. Always check the axes labels and scales before interpreting.
情境中的图形不仅仅是线条和曲线，它们代表真实的变化。距离–时间图显示速度，换算图链接货币或单位，累计频率图揭示中位数和四分位数。解读前一定要先查看坐标轴标签和刻度。
When a question provides a scale drawing or a plan, use the given scale to convert lengths. Pythagoras’ theorem and trigonometry are common tools here. For instance, a wheelchair ramp scenario may ask for the slope length given a rise of 0.8 m and a horizontal run of 4.2 m. Apply c² = a² + b² to find the ramp length.
当题目提供比例图或平面图时，用所给的比例尺转换长度。勾股定理和三角函数是这里的常用工具。例如，一个轮椅坡道场景可能会给出升高0.8米，水平距离4.2米，求坡道长度。应用 c² = a² + b² 求出坡道长。
| Shape / Context 图形/情境 | Key Maths 关键数学 |
|---|---|
| Garden with circular pond 带有圆形池塘的花园 | Area of a circle, subtraction 圆面积，减法 |
| Warehouse stacking boxes 仓库堆叠箱子 | Volume, packing problems 体积，装箱问题 |
| Map with bearings 带有方位角的地图 | Bearings, sine/cosine rules 方向角，正弦/余弦定理 |
6. Data Interpretation and Statistical Reasoning | 数据解读与统计推理
Case studies often include tables, bar charts, or pie charts showing survey results or sales figures. You might need to calculate the mean, median, mode, or range, and then make a comparative statement. Always quote the figures you used before drawing a conclusion.
案例分析常包含展现调查结果或销售数据的表格、条形图或饼图。你可能需要计算平均数、中位数、众数或极差，然后做比较陈述。得出结论前一定要引用你所用到的数据。
Interpreting grouped frequency tables requires care: use the midpoints of intervals to estimate the mean. A statement such as ‘On average, customers spend more on weekends than on weekdays’ must be supported by calculated means of £24.50 and £18.70, for example.
解读分组频率表需要谨慎：用组中值来估算平均数。像“总体而言，顾客周末消费比平日多”这样的陈述，必须用计算出的平均值，例如24.50英镑和18.70英镑作为支撑。
When the scenario involves comparing two sets of data, consider both a measure of central tendency and a measure of spread. A higher mean but a smaller interquartile range suggests more consistent high performance.
当场景涉及比较两组数据时，要同时考虑集中量数和离散量数。较高的平均数但较小的四分位距意味着表现更稳定且水平较高。
7. Ratio, Proportion and Real-Life Applications | 比率、比例与实际应用
Recipes, scale models, and currency exchanges are all about ratio and proportion. To adapt a recipe for 6 people to one for 10, multiply each quantity by the scale factor 10/6 or 5/3. For best value problems, find the unit cost: pence per gram or pound per litre, then compare.
食谱、比例模型和货币兑换都离不开比和比例。要将一份6人份的食谱调整为10人份，需将每种量乘以比例因子10/6或5/3。遇到最佳性价比问题时，计算单位成本：每克便士或每升英镑，然后进行对比。
Direct and inverse proportion appear in many practical contexts. The time to complete a task is inversely proportional to the number of workers, assuming equal efficiency. The cost of fuel is directly proportional to the distance travelled. Set up an equation using the proportional constant k.
正比例和反比例出现在许多实际场景中。假设效率相等，完成一项任务的时间与工人数量成反比。燃料费用与行驶距离成正比。用比例常数 k 建立方程。
T = k/n (for inverse proportion 反比例)
Always convert to the same units before forming a ratio. For example, if a map scale is 1 : 50 000, a 2 cm line represents 2 × 50 000 = 100 000 cm = 1 km. Careful unit conversion avoids scale errors.
在构建比之前始终换算成相同单位。例如，地图比例尺为 1 : 50 000，一条2厘米的线代表 2 × 50 000 = 100 000 厘米 = 1 公里。细致地进行单位转换能避免比例错误。
8. Probability and Risk Assessment | 概率与风险分析
Case studies might ask you to evaluate how likely a particular event is, such as the chance of rain on two consecutive days or the probability of a product being faulty. Use probability tree diagrams to organise the outcomes.
案例分析可能会要求你评估某一特定事件发生的可能性，比如连续两天下雨的概率或产品有缺陷的概率。用概率树状图来整理所有可能结果。
If the question provides experimental data (relative frequencies), you can use these to estimate probabilities. For risk assessment, compare the expected number of occurrences: expected value = probability × number of trials. For example, if the probability of a battery being defective is 0.03, then in a batch of 2000 batteries you would expect 2000 × 0.03 = 60 defective ones.
如果题目提供实验数据（相对频率），你可以用这些来估算概率。进行风险评估时，比较期望发生次数：期望值 = 概率 × 试验次数。例如，一节电池有缺陷的概率为0.03，那么在一批2000节电池中，可以预期有2000 × 0.03 = 60 节有缺陷。
Always state whether events are independent (e.g., rolling a dice twice) or conditional (e.g., picking marbles without replacement). Wrongly assuming independence is a common pitfall in case study probability questions.
始终要说明事件是独立的（如掷两次骰子）还是条件相关的（如不放回地摸弹珠）。错误地假设独立性是案例分析概率题中常见的陷阱。
9. Checking Answers and Validating Reasonableness | 检查答案与合理性验证
After obtaining a numerical answer, always return to the context and ask: ‘Does this make sense?’ If you find that a car’s fuel consumption is 0.2 litres per 100 km, you have probably mistaken the unit rate. Reread the units given in the problem.
得到数值答案后，务必要回顾情境问自己：“这合理吗？”如果你算出一辆汽车的油耗是每100公里0.2升，很可能是弄错了单位比率。重读题目给出的单位。
Perform a rough estimate before or after calculation to serve as a sanity check. For multi-part questions, carry forward your earlier answers accurately but check for any rounding instructions stated in the question.
在计算前或计算后做一个粗略估算，充当“头脑清醒校验”。对于多小问的题目，要准确沿用前面得出的答案，同时检查题目中是否提过任何有关四舍五入的说明。
If time permits, substitute your solution back into the original model. For example, if you found that a rectangular field has width 8 m, verify that the corresponding length and area match the constraints. This habit catches algebra or arithmetic slips.
如果时间允许，将你的解代入原始模型。例如，你求出矩形场地宽为8米，验证相应的长和面积是否符合给定条件。这种习惯能发现代数或算术疏忽。
10. Comprehensive Case Study Practice: From Shopping Budget to Event Planning | 实战案例综合演练：从购物预算到赛事规划
Let’s work through a mini case study that combines several skills. Scenario: A school is organising a charity fun run. The track is a rectangular field 120 m long and 80 m wide. Participants run 5 laps along the outside boundary. The school hires a marquee for £250 plus £3 per participant. Each participant pays an entry fee of £8 and is expected to raise £15 in sponsorship on average. The event aims to raise a total profit of at least £2000 after all costs.
我们来完整演练一个综合了多种技能的小型案例。情景：一所学校正在组织慈善趣味跑。场地是120米长、80米宽的矩形区域，参赛者沿外边跑5圈。学校租用大帐篷，费用为250英镑加每位参赛者3英镑。每位参赛者缴纳8英镑报名费，并预计平均募集15英镑赞助。活动目标是在扣除所有成本后总利润至少达到2000英镑。
Step 1: Identify the given quantities and unknown n = number of participants. Step 2: The distance one lap is the perimeter of the field: P = 2(120 + 80) = 2 × 200 = 400 m. Total running distance = 5 × 400 = 2000 m = 2 km. This might be useful for health and safety but is not directly needed for the financial model.
第1步：识别已知量，设未知数 n = 参赛人数。第2步：一圈长度为场地周长：P = 2(120 + 80) = 2 × 200 = 400米。总跑步距离 = 5 × 400 = 2000米 = 2公里。这可能对健康安全有用，但与财务模型无直接关系。
Step 3: Model the total income and costs. Income from entry fees = 8n. Sponsorship income = 15n. Total income = 8n + 15n = 23n. Total costs = fixed marquee £250 + variable cost £3n. Profit = total income − total costs → P = 23n − (250 + 3n) = 20n − 250.
第3步：建立总收入和成本模型。报名费收入 = 8n。赞助收入 = 15n。总收入 = 8n + 15n = 23n。总成本 = 固定帐篷费250 + 变动成本3n。利润 = 总收入 − 总成本 → P = 23n − (250 + 3n) = 20n − 250。
Step 4: Inequality for the target profit. We need P ≥ 2000, so 20n − 250 ≥ 2000 → 20n ≥ 2250 → n ≥ 112.5. Since n must be a whole number, at least 113 participants are required.
第4步：目标利润的不等式。我们需要 P ≥ 2000，所以 20n − 250 ≥ 2000 → 20n ≥ 2250 → n ≥ 112.5。因为n必须为整数，所以至少需要113人参加。
Step 5: Sense-check. With 113 participants, income = 113 × 23 = £2599, cost = 250 + 3×113 = £589, profit = £2010, just over £2000. For 112 participants, profit = £1990, which is below target. The model works.
第5步：合理性检查。113人参加时，收入 = 113 × 23 = £2599，成本 = 250 + 3×113 = £589，利润 = £2010，刚好超过£2000。112人时利润为£1990，低于目标。模型成立。
Step 6: If the school also plans to sell refreshments and expects another fixed cost of £80 plus £1.50 per participant, the cost function becomes C = 250 + 80 + (3 + 1.50)n = 330 + 4.5n. Then profit P = 23n − (330 + 4.5n) = 18.5n − 330. To reach £2000 profit: 18.5n − 330 ≥ 2000 → 18.5n ≥ 2330 → n ≥ 125.95, so 126 participants needed. This shows how additional layers can be added in a case study.
第6步：如果学校还计划售卖茶点，追加固定费用£80和每人£1.50，那么成本函数变为 C = 250 + 80 + (3 + 1.50)n = 330 + 4.5n。此时利润 P = 23n − (330 + 4.5n) = 18.5n − 330。要达到£2000利润：18.5n − 330 ≥ 2000 → 18.5n ≥ 2330 → n ≥ 125.95，需126人参加。这演示了案例分析中如何逐步叠加新的层次。
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