📚 Year 11 Cambridge Mathematics: Case Study Practice & Mastery | Year 11 Cambridge 数学：案例分析实战演练
Case study problems in Year 11 Cambridge IGCSE Mathematics require you to apply mathematical concepts to real-world situations. These questions often combine algebra, geometry, statistics, and probability, testing not just your calculation skills but also your ability to interpret information and communicate reasoning. Mastering case studies gives you a huge advantage in the exam, as they appear in both core and extended papers, often carrying high marks. This article will guide you through a structured approach, provide worked examples across key topics, and help you avoid common pitfalls.
在 Year 11 剑桥 IGCSE 数学中，案例分析题要求你将数学概念应用到现实情境里。这类问题常常结合代数、几何、统计与概率，不仅考验计算能力，也考查你对信息的解读与推理表达能力。掌握案例分析技巧在考试中优势巨大，因为它们在核心卷和拓展卷中都会出现，且分值较高。本文将带你梳理结构化解题方法，通过重点主题的典型例题演练，并帮助你避开常见陷阱。
1. Understanding Case Study Questions | 理解案例分析题
Case study questions are word problems presented in a narrative style, often describing a scenario such as a journey, a business situation, or a measurement task. The information is rarely given in a ready-to-use mathematical form. You need to identify key numbers, relationships, and the ultimate question being asked.
案例分析题是以叙述方式呈现的应用题，通常会描述一段旅程、一个商业情境或一个测量任务。题目给出的信息很少直接可用，你需要从中识别关键数字、数量关系和最终所问的问题。
Typical command words include 'Calculate', 'Find', 'Show that', or 'Hence'. You might also need to justify your answer or explain why a certain model is valid. Always read the question twice – once for the big picture, and once to underline numerical data and units.
常见的指令词有“计算”、“求出”、“证明”或“由此”。你可能还需要证明答案的正确性或解释为什么某个模型是合理的。务必把题目读两遍：第一遍把握整体情景，第二遍划出数字和单位。
2. The Three-Step Approach | 三步解题法
A reliable way to tackle any case study is to follow a three-step framework: (1) Interpret and organise the given information, (2) Translate it into mathematical language, and (3) Solve and verify the result. This method keeps your working logical and ensures you don't miss vital details.
解决案例分析的一种可靠方法是遵循三步框架：(1) 解读并整理所给信息，(2) 将其翻译为数学语言，(3) 求解并验证结果。这个方法能让你的解答过程更有条理，避免遗漏关键细节。
Many students lose marks by jumping straight into equations without clearly defining variables. Taking a minute to draw a diagram or write a summary table can make a significant difference. The three steps apply equally to algebra, geometry, and statistics problems.
许多学生因为不先明确定义变量就匆忙列方程而失分。花一分钟画出简图或写出汇总表格，效果会大不相同。这三步同样适用于代数、几何和统计问题。
3. Step One: Interpret and Organise | 第一步：解读并整理信息
Begin by highlighting all numerical values and noting their units. Identify what is unknown and assign a variable, for example 'Let the speed be v km/h' or 'Let the width be w metres'. Be precise with the variable definition, including its unit if possible.
首先标出所有数值并注意单位。找出未知量并设好变量，例如“设速度为 v 公里/小时”或“设宽为 w 米”。变量定义要精确，尽量注明单位。
If the scenario involves motion, sketch a simple diagram with distances and directions. For business problems, list fixed costs, variable costs, and selling prices. Organising data into two columns – known and unknown – often clarifies the path forward.
如果场景涉及运动，画一个标有距离和方向的简图。对于商业问题，列出固定成本、变动成本和售价。把数据整理成“已知”和“未知”两栏，往往能让解题思路一目了然。
4. Step Two: Translate into Mathematics | 第二步：转化为数学语言
t each relationship described in words into an algebraic expression or equation. For speed–distance– member that Profit = Revenue − Total Cost. For geometry, apply area or perimeter formulas.
把题目中用文字描述的每个关系转化为代数式或方程。对于速度-距离-时间问题，使用公式：距离 = 速度 × 时间。对于利润问题，记住：利润 = 收入 − 总成本。对于几何问题，套用面积或周长公式。
If the problem contains inequalities or constraints (e.g., 'at least', 'no more than'), express them mathematically. Also, recognise when a quadratic might appear – for example, when two quantities are multiplied and one is expressed in terms of the other.
如果问题包含不等式或约束条件（如“至少”“不超过”），用数学形式写出来。另外，当两个量相乘且其中一个可以用另一个表示时，很可能会产生二次方程，要能识别出来。
5. Step Three: Solve and Verify | 第三步：求解与验证
Once the equation or inequality is set up, solve it using appropriate algebraic techniques. For linear equations, isolate the variable. For quadratics, factorise or use the quadratic formula. Remember to reject any negative solutions that do not make sense in the context (e.g., negative lengths or times).
方程或不等式建立后，运用合适的代数方法求解。线性方程要移项整理；二次方程可因式分解或使用求根公式。注意舍去在情境中不合理的负根（如负的长度或时间）。
Always substitute your answer back into the original scenario to check if it satisfies all conditions. For a business case, does the profit become zero at the break-even point? For a speed problem, do the times add up? Verification is a quick step that can catch calculation errors.
一定要把答案代回原题情境，检验是否满足所有条件。商业案例中，盈亏平衡点利润是否为零？速度问题中，时间之和是否正确？验证这一步很简短，却能帮你发现计算错误。
6. Type 1: Speed, Distance and Time | 类型一：速度、距离与时间
Problem: Two cars start at the same time towards each other from towns 150 km apart. Car A travels at v km/h, Car B travels at (v+5) km/h. Car B arrives 1 hour earlier than Car A. Find the speed of each car.
问题：两辆车同时从相距 150 公里的两镇相向而行。车 A 速度为 v km/h，车 B 速度为 (v+5) km/h。车 B 比车 A 早到 1 小时。求两车的速度。
Let time taken by A be t₁ = 150/v hours. Time taken by B is t₂ = 150/(v+5). The condition 'B arrives 1 hour earlier' means t₁ − t₂ = 1.
设车 A 所用时间为 t₁ = 150/v 小时，车 B 为 t₂ = 150/(v+5)。条件“车 B 早到 1 小时”即 t₁ − t₂ = 1。
150/v − 150/(v+5) = 1
50(v+5) − 150v = v(v+5) → 150v + 750 − 150v = v² + 5v → 750 = v² + 5v → v² + 5v − 750 = 0.
两边同乘 v(v+5)：150(v+5) − 150v = v(v+5) → 750 = v² + 5v → v² + 5v − 750 = 0。
25 km/h, Car B = 30 km/h. Check: 150/25 = 6 hours, 150/30 = 5 hours, difference = 1 hour.
因式分解：(v+30)(v−25)=0，正根 v=25。车速：车 A 25 km/h，车 B 30 km/h。检验：150/25=6 h，150/30=5 h，差 1 h。
7. Type 2: Cost, Revenue and Profit | 类型二：成本、收入与利润
A school fundraiser produces custom T-shirts. Fixed setup cost is £200, and each shirt costs £5 to print. They sell each shirt for £12. How many shirts must be sold to break even?
某学校筹款活动制作定制 T 恤。固定准备成本为 200 英镑，每件印刷成本 5 英镑。每件售价 12 英镑。需要卖出多少件才能达到盈亏平衡？
shirts be x. Total cost C = 5x + 200. Total revenue R = 12x. Break-even occurs when R = C.
设 T 恤数量为 x。总成本 C = 5x + 200，总收入 R = 12x。盈亏平衡时 R = C。
12x = 5x + 200 → 7x = 200 → x ≈ 28.57
29 shirts must be sold to start making a profit. For x=29, profit = 12(29) − (5(29)+200) = £3.
由于不能卖出零头，因此必须卖出 29 件才能开始盈利。当 x=29 时，利润 = 12×29 − (5×29+200)= 3 英镑。
8. Type 3: Geometry and Measurement | 类型三：几何与测量
A rectangular garden has length 3 m longer than its width. The area is 54 m². Find the dimensions of the garden.
一个矩形花园的长比宽多 3 米，面积为 54 平方米。求花园的尺寸。
Let width be w metres, so length = w + 3. Area = w(w + 3) = 54.
设宽为 w 米，则长 = w + 3。面积 w(w + 3) = 54。
w² + 3w − 54 = 0 → (w + 9)(w − 6) = 0
Positive root w = 6. Hence width = 6 m, length = 9 m. Verification: 6 × 9 = 54 m².
取正根 w = 6。因此宽为 6 m，长为 9 m。验证：6×9 = 54 m²。
9. Type 4: Probability and Tree Diagrams | 类型四：概率与树状图
A bag contains 3 red and 2 blue balls. Two balls are drawn with replacement. Find the probability that both balls are red.
一个袋子里有 3 个红球和 2 个蓝球。每次抽取后放回，连抽两次。求两次都抽到红球的概率。
replaced, the probabilities for the second draw are the same. The events are independent.
第一次抽取，P(红) = 3/5，P(蓝) = 2/5。由于放回，第二次抽取的概率不变，事件相互独立。
P(Red and Red) = (3/5) × (3/5) = 9/25
If the question asked for 'without replacement', the tree branch for the second draw would change: P(Red second | Red first) = 2/4 = 1/2. Drawing a tree diagram helps avoid confusion.
如果问题改为不放回，树状图第二分支的概率就会变化：P(第二次红|第一次红) = 2/4 = 1/2。画出树状图有助于避免混淆。
10. Type 5: Statistics and Data Interpretation | 类型五：统计与数据解读
Five students scored the following marks in a test: 45, 60, 55, 70, 80. Calculate the mean, median, and range.
五名学生的测验成绩为：45, 60, 55, 70, 80。计算平均数、中位数和极差。
Mean = (45+55+60+70+80)/5 = 310/5 = 62. Median (middle value) = 60. Range = 80 − 45 = 35.
首先排序：45, 55, 60, 70, 80。平均数 = (45+55+60+70+80)/5 = 62。中位数 = 60。极差 = 80 − 45 = 35。
In a case study, you might be given a frequency table and asked to estimate the mean from grouped data. Always use mid-interval values and remember to divide by the total frequency.
在案例分析中，可能会给你一个频数表，让你估算分组数据的平均数。务必使用组中值，并记得除以总频数。
11. Common Pitfalls and How to Avoid Them | 常见易错点与应对策略
Mistake 1: Forgetting to convert units. If a time is given in minutes but speed in km/h, convert minutes to hours. Mistake 2: Not defining the variable clearly, leading to confused equations. Mistake 3: Using rounded intermediate values too early – keep exact forms until the final step.
错误 1：忘记换算单位。如果时间单位是分钟而速度是公里/小时，要先把分钟换成小时。错误 2：变量定义不清晰，导致方程混乱。错误 3：过早使用四舍五入后的中间值——在最后一步之前保持精确形式。
Mistake 4: Ignoring the context when selecting roots. A quadratic may give a negative solution that is mathematically valid but impossible in the real scenario. Always test against the original problem statement. Mistake 5: Not checking the final answer against all constraints given.
错误 4：选取根时忽略情境。二次方程可能解出负根，虽然数学上有效但在真实场景中不成立，务必对照原题条件。错误 5：未将最终答案与所有给定约束进行核对。
12. Final Exam Tips and Practice | 考前提点与练习
In the exam, show all steps of working – even if your final answer is wrong, you can earn method marks. Write down your variable definitions and any assumptions. Use clearly labelled diagrams where appropriate. Manage your time: if a case study is worth 6 marks, spend roughly 1 minute per mark, but also allow time for reading and checking.
考试时，要把所有解题步骤写出来——即使最终答案错了，也能拿到方法分。写下变量定义和假设。适当地使用标有注释的简图。合理分配时间：如果一道案例分析题分值 6 分，大约每分花 1 分钟，同时留出阅读和检查的时间。
velled 3 km/h faster, the journey would have taken 1 hour less. Find her original speed. (Hint: set up 60/v − 60/(v+3) = 1.) Keep practising until the translation step feels automatic.
用往年真题进行限时练习。试试这个小练习：一位骑行者以恒定速度行驶 60 km，如果速度提高 3 km/h，行程将缩短 1 小时。求她原本的速度。（提示：建立方程 60/v − 60/(v+3) = 1。）持续练习，直到把文字翻译成数学语言这一步变得得心应手。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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