📚 Year 10 OCR Maths: Case Study Workout | 十年级 OCR 数学：案例分析实战演练
In the OCR GCSE Mathematics specification, case study questions are designed to test your ability to apply mathematical concepts to real-world situations. These multi‑step problems often combine several topic areas – such as number, algebra, geometry and statistics – within a single context. They reward clear working, logical reasoning and effective communication of methods. This article will walk you through typical case study scenarios you may encounter in Year 10, offering practical strategies and worked examples to build your confidence and accuracy.
在 OCR GCSE 数学大纲中，案例分析题旨在考查你将数学概念应用于实际情境的能力。这类多步骤问题常常将一个情境下的数、代数、几何和统计等多个领域糅合在一起。清晰的步骤、合理的逻辑和有效的表达方式是得分的关键。本文将带你走过十年级常见的典型案例场景，提供实用策略和详细示例，帮助你增强信心、提高准确率。
1. Understanding the OCR Case Study Approach | 理解 OCR 案例分析题型
An OCR case study typically begins with a paragraph of contextual information, such as a budget plan, a travel itinerary, a design brief or survey data. Following this, several sub‑questions appear, each focusing on different mathematical skills. You are expected to extract relevant figures, decide which operations and formulas to use, and present your working clearly.
一道 OCR 案例分析题通常以一段情境信息开头，比如预算方案、旅行行程、设计概要或调查数据。随后会提出若干子问题，每道子问题侧重不同的数学技能。你需要从中提取相关数字，决定选用哪种运算或公式，并清晰地展示解题过程。
Key tip: do not be overwhelmed by the volume of text. Underline numbers, units and key conditions as you read. Treat the scenario as a puzzle where the maths is your toolkit.
关键提示：不要被大段文字吓倒。阅读时划出数字、单位和关键条件。把情境看作一个谜题，而数学就是你的工具包。
2. Reading and Extracting Key Data | 阅读与提取关键数据
Begin by scanning the whole case study to understand its overall theme, then read each sub‑question carefully. Highlight any given measurements, costs, percentages, time intervals or constraints. Write these down in a structured list before attempting any calculations. This prevents mixing up figures and makes it easier to spot the relevant data for each part.
首先，快速浏览整个案例，把握主题大意，然后仔细阅读每一个子问题。圈出已知的度量、成本、百分比、时间区间或限制条件。在动手计算之前，先把它们列成清单。这样做可以避免数字混淆，也能让你更快地找到每个部分所需的数据。
Also pay attention to implicit information. For example, 'a 20% discount' implies you will pay 80% of the original price; 'return journey' means the distance is doubled. Learning to decode these phrases saves time and reduces errors.
还要留意隐含信息。比如“打八折”意味着支付原价的 80%；“往返行程”则表示总路程要翻倍。学会破译这些用语能节省时间并减少错误。
3. Financial Mathematics: Budgeting a School Trip | 金融数学：学校旅行预算
A common case study involves planning a school trip. You might be given the cost per student, booking fees, meal allowances and an exchange rate if the trip is abroad. Questions often ask you to calculate total cost, work out how much each student must pay after a grant, or determine whether a given budget is sufficient.
一个常见的案例是规划学校旅行。你可能会拿到每名学生的费用、预订手续费、餐费补贴，以及如果是国外旅行还会给出汇率。题目通常要求计算总费用、减去补助后每名学生应付的金额，或判断给定预算是否足够。
Example: A trip to Paris costs £245 per pupil for 28 pupils. There is a 15% early‑bird discount if booked 8 weeks in advance. Additionally, the school contributes £360 from its fund. How much does each pupil need to pay if they book early?
示例：去巴黎旅行每名学生 245 英镑，共有 28 名学生。提前8周预订可享受 15% 早鸟折扣。此外学校从基金中拨出 360 英镑。如果提前预订，每名学生需要支付多少？
Step‑by‑step: Total without discount = 28 × 245 = £6860. Discount = 0.15 × 6860 = £1029. Cost after discount = 6860 − 1029 = £5831. Subtract school contribution: £5831 − 360 = £5471. Amount per pupil = 5471 ÷ 28 = £195.39 (to the nearest penny).
逐步解答：未打折总费用 = 28 × 245 = 6860 英镑。折扣额 = 0.15 × 6860 = 1029 英镑。折后费用 = 6860 − 1029 = 5831 英镑。减去学校拨款：5831 − 360 = 5471 英镑。每名学生的金额 = 5471 ÷ 28 = 195.39 英镑（四舍五入到分）。
Always interpret the final answer in context. Would the answer be rounded up or down? For payments, it is usually rounded up to the next penny or pound to ensure the total is covered.
始终在语境中解读最终答案。对于付款，通常会向上取整到下一分或下一英镑，以保证总金额覆盖。
4. Ratio and Proportion: Mixing Paint or Scaling Recipes | 比和比例：调配油漆或调整食谱
Case studies frequently use ratio scenarios such as mixing paint colours, scaling cooking recipes or sharing profits. The key skill is to interpret part‑part or part‑whole ratios and use multiplication or division to find unknown quantities.
案例分析中经常出现比例情境，如调配油漆颜色、调整食谱分量或分配利润。核心技能是解读部分与部分或部分与整体的比，并通过乘法或除法来求取未知量。
For instance, a decorator mixes blue and white paint in the ratio 3 : 7 to make a 40‑ of one part = 40 ÷ 10 = 4 litres. So blue = 3 × 4 = 12 litres, white = 7 × 4 = 28 litres.
例如，一位装饰师按 3 : 7 的比例混合蓝色和白色油漆，要调制 40 升一桶。问每种颜色各需多少升。总份数 = 3 + 7 = 10。一份的量 = 40 ÷ 10 = 4 升。所以蓝色 = 3 × 4 = 12 升，白色 = 7 × 4 = 28 升。
redient amounts. If 500 g of flour makes 12 scones, how much flour is needed for 30 scones? 2 = 2.5, so flour = 500 × 2.5 = 1250 g. Always check that quantities remain in proportion.
正比例也会出现在调整原料用量时。如果 500 克面粉可以做 12 个司康饼，做 30 个需要多少面粉？倍数 = 30 ÷ 12 = 2.5，所以面粉 = 500 × 2.5 = 1250 克。始终检查数量是否保持比例关系。
5. Geometry and Measures: Designing a Garden | 几何与测量：设计花园
Design‑based case studies ask you to work with area, perimeter and volume within a realistic project. A typical example: a rectangular garden measures 15 m by 8 m. The owner wants to cover two‑thirds of it with turf and the remaining third with decking. Questions may involve calculating the area of turf, the cost of materials, or the number of slabs needed for the decking.
设计类案例要求你在一个实际项目中运用面积、周长和体积的知识。典型例子：一个矩形花园长 15 米，宽 8 米。主人想用三分之二的面积铺设草坪，剩余三分之一铺木甲板。问题可能涉及计算草坪面积、材料成本或甲板所需的板材数量。
garden = 15 × 8 = 120 m². Turf area = ⅔ × 120 = 80 m². Decking area = 40 m². If turf costs £5.40 per square metre, turf cost = 80 × 5.40 = £ careful to convert if quantities are given in different units, e.g. centimetres to metres.
花园面积 = 15 × 8 = 120 平方米。草坪面积 = ⅔ × 120 = 80 平方米。甲板面积 = 40 平方米。若草坪每平方米 5.40 英镑，草坪花费 = 80 × 5.40 = 432 英镑。若每块甲板覆盖 0.2 平方米，所需数量 = 40 ÷ 0.2 = 200 块。始终带上单位，并注意单位换算，比如需将厘米转换为米。
Perimeter may also be asked if fencing is needed. Evaluate whether the answer makes sense – the length of a fence cannot be negative, and you usually round up when buying whole panels.
如果涉及围栏，还需要计算周长。验证答案是否合理——围栏长度不能为负值，而且购买整片围栏时通常向上取整。
6. Statistics and Data Handling: Interpreting Survey Results | 统计与数据处理：解读调查结果
Many case studies include tables, pie charts or bar charts showing survey data. You might be asked to calculate the mean, median, mode or range, or to compare two data sets by commenting on averages and spread. Interpreting percentages and proportions from charts is also common.
许多案例分析包含展示调查数据的表格、饼图或条形图。你可能会被要求计算平均数、中位数、众数或极差，或者通过比较平均数和离散程度来对比两组数据。从图表中解读百分比和比例也很常见。
favourite sport. The pie chart shows football 150°, netball 75°, tennis 60°, and the rest 'other'. To find the number for tennis: 60°/360° × 240 = 40 students. To find the angle for 'other', subtract the known angles from 360°: 360 − (150+75+60) = 75°.
示例：一项调查询问了 240 名学生最喜欢的运动。饼图显示足球 150°、网球 60°、篮网球 75°，其余为“其他”。求选择网球的人数：60°/360° × 240 = 40 人。计算“其他”的扇区角度：360 − (150+75+60) = 75°。
When comparing two groups, refer to the mean for an overall impression and the range for consistency. A lower range indicates more consistent results. Use precise language: 'On average, boys spent 15 minutes more on homework than girls, but the girls' times were more consistent because the range was smaller.'
在比较两组数据时，用平均数反映总体印象，用极差反映一致性。极差越小说明结果越稳定。使用精确语言：“平均而言，男生做作业的时间比女生多 15 分钟，但女生的时间更稳定，因为极差更小。”
7. Speed, Distance, Time: Planning a Journey | 速度、距离、时间：规划行程
Travel case studies require you to use the relationship speed = distance ÷ time, rearranged as needed. You must also handle time units (hours and minutes) and often convert between km/h and m/s, or between mph and minutes per mile.
行程类案例需要运用关系式 速度 = 距离 ÷ 时间，并根据需要进行变形。你还必须处理时间单位（小时和分钟），经常需要在 km/h 和 m/s 之间，或 mph 与每英里分钟数之间进行换算。
A coach travels 150 miles at an average speed of 50 mph. How long does the journey take? dditional time = 60 ÷ 40 = 1.5 hours = 1 hour 30 minutes. Total time = 4 hours 30 minutes.
一辆长途汽车以平均时速 50 英里行驶 150 英里。行程耗时多久？时间 = 距离 ÷ 速度 = 150 ÷ 50 = 3 小时。如果之后以 40 mph 再行驶 60 英里，额外时间 = 60 ÷ 40 = 1.5 小时 = 1 小时 30 分钟。总时间为 4 小时 30 分钟。
5 seconds: speed in m/s = 400 ÷ 55 ≈ 7.27 m/s; in km/h = 7.27 × 3.6 = 26.2 km/h (approx.).
注意混合单位。将分钟转换为小时，除以 60；将 m/s 转换为 km/h，乘以 3.6。一名跑步者用 55 秒跑完 400 米：速度（m/s）= 400 ÷ 55 ≈ 7.27 米/秒；换算成 km/h = 7.27 × 3.6 = 26.2 公里/小时（约值）。
8. Algebra in Context: Mobile Phone Tariffs | 情境代数：手机资费
Algebraic modelling appears when you need to create formulas to represent costs, revenues or other relationships. A classic case is comparing two mobile phone contracts: Plan A costs £15 per month plus 5p per minute of calls; Plan B costs £20 per month but includes 200 free minutes and then charges 8p per minute.
当需要建立公式来表示成本、收入或其他关系时，就会出现代数建模。一个经典案例是对比两种手机套餐：方案 A 每月 15 英镑，另加每分钟 5 便士通话费；方案 B 每月 20 英镑，包含 200 分钟免费通话，超出部分每分钟 8 便士。
0 + 0.08(x − 200). You might be asked to find the number of minutes for which the two costs are equal, or to advise which plan is cheaper for a given usage.
设 x 为每月通话分钟数。方案 A 成本 = 15 + 0.05x。方案 B：若 x ≤ 200，成本 = 20；若 x > 200，成本 = 20 + 0.08(x − 200)。你可能会被要求找出两种方案费用相同时的通话时长，或针对特定使用量给出建议。
. Below 366 minutes Plan B could be cheaper; above, Plan A is cheaper. Interpret the break‑even point in context.
当 x > 200 时，令两者相等：15 + 0.05x = 20 + 0.08(x − 200)。化简：15 + 0.05x = 20 + 0.08x − 16 → 15 + 0.05x = 4 + 0.08x → 11 = 0.03x → x = 366.67 分钟。在 366 分钟以下方案 B 可能更便宜；超过则方案 A 更划算。在情境中解读这个损益平衡点。
s: minutes are whole numbers, so you might test x = 366 and x = 367 to confirm the switch.
还应考虑实际限制：分钟数是整数，因此可测试 x = 366 和 x = 367 来确认切换点。
9. Probability and Risk Assessment | 概率与风险评估
Probability case studies often involve combined events, tree diagrams or two‑way tables. A scenario might describe a bag containing coloured counters, or the risk of rain affecting an outdoor event. You need to calculate single probabilities, 'and' probabilities (multiplying) and 'or' probabilities (adding, but being careful of overlap).
概率案例通常涉及组合事件、树形图或双向表。情境可能描述一个装有彩色筹码的袋子，或下雨影响户外活动的风险。你需要计算单一概率、“且”概率（相乘）和“或”概率（相加，但注意重叠）。
arbles. If two marbles are drawn without replacement, what is the probability both are red? irst red) = 6/12 = ½. P(second red given first red) = 5/11. P(both red) = ½ × 5/11 = 5/22.
示例：一个盒子里装有 6 个红球、4 个蓝球和 2 个绿球。每次取出一个球且不放回，求两次都取到红球的概率。P(第一次为红) = 6/12 = ½。P(第二次为红|第一次为红) = 5/11。P(两次均为红) = ½ × 5/11 = 5/22。
Risk assessment may ask: 'Is the event likely to happen?' Compare the probability to benchmarks such as 0.5, 0.1, etc. Use language like 'unlikely', 'even chance' or 'very likely'. Also relate to relative frequency: ' a simulation of 200 trials, we would expect about 5/22 × 200 ≈ 45 occurrences of two reds.'
风险评估可能会问：“事件发生的可能性大吗？”将概率与 0.5、0.1 等基准进行比较。使用“不太可能”、“发生的机会均等”或“非常可能”等措辞。也可以联系相对频率：“在 200 次模拟试验中，预计大约会出现 5/22 × 200 ≈ 45 次两次都取到红球的情况。”
10. Checking and Validating Answers | 检查与验证答案
After solving a case study, always validate your answers. Ask yourself: does the result make sense in the real‑world context? Could a fence be longer than the perimeter of the field? Could the cost per person be negative? Sanity checks catch many mistakes.
解完案例分析后，一定要验证答案。问问自己：结果在现实情境中合理吗？围栏可能比场地的周长还长吗？人均成本会是负数吗？合理性检查可以揪出许多错误。
A powerful technique is estimation. Round numbers to one significant figure and recalculate mentally. If the exact answer is far from the estimate, re‑examine your working. Also review whether you have answered all parts of the question and used the correct units.
一个强有力的技巧是估算。将数字四舍五入到一位有效数字，再心算一遍。如果精确答案与估算相去甚远，重新审视解题过程。还要检查是否回答了问题的所有子部分，以及单位是否使用正确。
Finally, present your working logically so that an examiner can follow your reasoning. Number each step, label diagrams and write a short concluding sentence if the question asks for a decision or comparison.
最后，将解题步骤条理清晰地呈现出来，让考官能跟上你的思路。给每一步编号，标注图表，如果题目要求作出决策或比较，写一句简短的结论性语句。
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