📚 AQA Year 10 Maths: Case Study Problem-Solving Practice | AQA 数学：案例分析问题求解实战
In AQA Year 10 Mathematics, case study problems are designed to assess your ability to apply mathematical skills in real-world scenarios. These multi-step questions often combine topics like number, algebra, geometry, ratio, and statistics. Developing a systematic approach is key to success.
在 AQA 九年级数学中，案例分析题旨在考查你将数学技能应用于实际场景的能力。这些多步骤问题常结合数字、代数、几何、比例和统计等知识。掌握系统的解题方法是关键。
1. Understanding the Case Study Question | 理解案例分析问题
A case study question presents a realistic situation, such as organising a school trip, designing a product, or analysing survey data. You must read the entire scenario carefully before attempting any calculations.
案例分析题会呈现一个真实情境，如组织学校旅行、设计产品或分析调查数据。在开始计算前，必须仔细阅读整个背景描述。
Identify exactly what the question is asking. Look for command words like ‘calculate’, ‘compare’, ‘justify’, or ‘recommend’. These tell you the type of response expected.
准确识别问题要求。查看如“计算”“比较”“证明”或“建议”等指令词，它们提示了预期的回答类型。
2. Extracting Key Data from the Scenario | 从情景中提取关键数据
Highlight or list all numerical values, units, and conditions. For example, a problem about a mobile phone contract might include monthly fees, data limits, and extra charges per gigabyte.
圈出或列出所有数值、单位和条件。例如，手机套餐问题可能包含月费、流量上限以及每额外吉字节的收费。
Organise the data in a table or bullet points. This helps prevent missing important details later.
用表格或要点整理数据，有助于后续避免遗漏重要细节。
| Condition | Value |
|---|---|
| Monthly fee | £15 |
| Included data | 5 GB |
| Extra data cost per GB | £2.50 |
A table like the one above keeps your information organised and ready for calculation.
像上表这样整理信息，可以让数据井井有条，便于后续计算。
3. Identifying the Relevant Mathematical Topics | 识别相关的数学知识点
Case studies often blend multiple topics. Recognise which areas of maths are involved: percentages for discounts, ratio for mixing ingredients, Pythagoras’ theorem for distances, or averages for survey results.
案例分析常融合多个知识点。辨认出涉及哪些数学领域：折扣中的百分比，配料混合中的比率，距离计算中的勾股定理，调查结果的均值等。
Refresh the key formulas or methods you will need. For example, area of a rectangle is length × width, and percentage change is (difference ÷ original) × 100%.
复习你需要的关键公式或方法。例如，矩形面积 = 长 × 宽，百分比变化 = （差值 ÷ 原值）× 100%。
Percentage change = (new – original) ÷ original × 100%
百分比变化 = （新值 – 原值）÷ 原值 × 100%
4. Breaking Down the Problem into Steps | 将问题分解为步骤
Write a step-by-step plan before you calculate. For instance: Step 1 – find total ingredients needed for 120 guests; Step 2 – work out cost per ingredient; Step 3 – add fixed costs.
在计算前，写下逐步计划。例如：步骤1 – 求出为120位客人所需的总配料量；步骤2 – 计算每种配料的成本；步骤3 – 加上固定费用。
Numbering each step reduces confusion and allows you to check each part independently.
给每个步骤编号可减少混淆，并让你能单独检查每一步。
5. Converting Units and Managing Scale | 单位换算与比例管理
Many case study mistakes come from incorrect units. Check if you need to convert between millilitres and litres, grams and kilograms, or centimetres and metres. Use conversion factors confidently.
许多案例分析错误源于单位不当。检查是否需要在毫升和升、克和千克、厘米和米之间转换。熟练使用换算因子。
If the scenario involves a scale drawing, remember: map scale 1 : 50 000 means 1 cm on the map represents 50 000 cm in real life (0.5 km).
如果场景涉及比例图，牢记：地图比例尺 1 : 50 000 表示图上 1 cm 代表实际 50 000 cm（即 0.5 km）。
Actual distance = measured distance × scale factor
实际距离 = 测量距离 × 比例因子
6. Performing Calculations Accurately | 准确进行计算
Use a calculator where allowed, but write down intermediate values with sensible rounding. Do not round until the final answer, unless the question specifies. Show all working out.
允许使用计算器时就用，但要写下中间值并进行合理舍入。除非题目明确要求，否则不要在最终答案前舍入。写出所有计算过程。
For multi-part questions, label each part (a), (b), etc. This helps the examiner follow your logic and award method marks even if the final answer is wrong.
对于多部分题目，标注 (a)、(b) 等。这有助于阅卷老师理解你的思路，即使最终答案错误，也可能得到方法分。
7. Checking for Reasonableness and Validation | 检查合理性并验证
After obtaining a result, ask yourself: does this answer make sense in the context? For example, if you calculate that a train journey takes 1.2 hours, that feels plausible (1 hour 12 minutes).
得出结果后，问问自己：这个答案在情境中合理吗？例如，如果你算出一段火车旅行需1.2小时，那看起来是合理的（1小时12分钟）。
Use estimation to verify. Round numbers to the nearest whole or ten and re-calculate mentally to see if your answer is in the right ballpark.
用估算来验证。把数字四舍五入到最近的整数或十位，心算一下，看看答案是否大致正确。
8. Presenting Your Final Solution | 呈现最终解答
Write a clear conclusion in words, not just a number. If the question asks for a recommendation, state it explicitly with supporting calculations.
用文字写出清晰的结论，而不仅仅是数字。如果问题要求给出建议，要明确陈述并用计算数据支持。
For example: ‘Based on the comparison, Package B offers better value, saving £2.40 per month compared to Package A.’ This shows full understanding.
例如：“比较结果显示，套餐 B 性价比更高，相比套餐 A 每月节省 £2.40。” 这展示了完整的理解。
9. Common Pitfalls in AQA Case Studies | AQA 案例分析中的常见陷阱
Watch out for these frequent errors: misreading compound units (e.g., m/s instead of km/h), forgetting to include tax or delivery fees, and mixing up diameter and radius.
警惕这些常见错误：混淆复合单位（如 m/s 与 km/h），忘记包含税费或运费，以及弄混直径和半径。
Another trap is failing to convert percentages to decimals before multiplying (for example, 15% = 0.15). Always double-check your input when using a calculator.
另一个陷阱是在乘法前未将百分比转换为小数（例如15% = 0.15）。使用计算器时务必再次检查输入。
10. Worked Example: Planning a School Fair Stall | 实战示例：规划学校义卖摊位
Scenario: You are planning a smoothie stall. Ingredients cost £0.80 per cup, and you plan to sell each cup for £2.00. The stall hire fee is £15. You want to make a profit of at least £50.
情景： 你计划开设一个果汁摊位。每杯原料成本为 £0.80，计划每杯售价 £2.00。摊位租金 £15。你希望至少盈利 £50。
Step 1: Find the profit per cup: £2.00 – £0.80 = £1.20.
步骤1： 求每杯利润：£2.00 – £0.80 = £1.20。
Step 2: Let n be the number of cups needed. Total profit = 1.20n – 15. Set this ≥ 50.
步骤2： 设所需杯数为 n。总利润 = 1.20n – 15。令其 ≥ 50。
Step 3: Solve inequality: 1.20n – 15 ≥ 50 → 1.20n ≥ 65 → n ≥ 54.166… So you need to sell at least 55 cups.
步骤3： 解不等式：1.20n – 15 ≥ 50 → 1.20n ≥ 65 → n ≥ 54.166… 因此至少需要卖出55杯。
Conclusion: You should prepare ingredients for at least 55 cups. Break even point would be when 1.20n – 15 = 0, which gives n = 12.5, so 13 cups to cover costs.
结论： 需准备至少55杯的原料。盈亏平衡点为 1.20n – 15 = 0 时，得 n = 12.5，即13杯可覆盖成本。
This simple demonstration shows how everyday decisions can be modelled using inequalities and proportional reasoning.
这个简单示例展示了如何用不等式和比例推理来模拟日常决策。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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