What is Ratio? 什么是比例？ | What is Ratio? What Is Ratio?

What is Ratio? 什么是比例？

Ratio is a way of comparing two or more quantities. In AQA GCSE Mathematics, ratio questions appear in both Foundation and Higher tier papers — and they are consistently among the most frequently tested topics.

比例 (Ratio) 是比较两个或多个数量的一种方式。在AQA GCSE数学考试中，比例题目同时出现在基础卷和高级卷中，是最常考的核心知识点之一。

A ratio of 3 : 5 means that for every 3 of the first quantity, there are 5 of the second. Crucially, this does not mean the total is 8 — it means the parts are in that proportion. Understanding this distinction is the foundation of all ratio problems.

Simplifying Ratios 化简比例

To simplify a ratio, divide all parts by their highest common factor (HCF). For example, 12 : 18 : 24 simplifies to 2 : 3 : 4 (divide by HCF = 6). This is a skill tested in nearly every exam series.

化简比例：将比例的所有部分除以最大公约数（HCF）。例如 12 : 18 : 24 ⟶ 2 : 3 : 4（除以 HCF = 6）。这是几乎每次考试都会考察的基础技能。

Key tip 关键技巧: Always express ratios in their simplest integer form. If decimals appear (e.g., 1.5 : 2.5), multiply through by 10 to get 15 : 25, then simplify to 3 : 5. If fractions appear (e.g., 1/2 : 1/3), multiply by the LCM of denominators (6), giving 3 : 2.

Sharing in a Given Ratio 按比例分配

This is one of the most common exam question types. The method is: find the total number of parts, then divide the total amount by that number of parts, and finally multiply each part by the unit value.

按比例分配是最常见的考试题型。解题方法：先求总份数，再用总量除以总份数得到每份的值，最后乘以各部分对应的份数。

Example: Share £120 in the ratio 2 : 3. Total parts = 2 + 3 = 5. One part = £120 ÷ 5 = £24. So the shares are 2 × £24 = £48 and 3 × £24 = £72. Always check: £48 + £72 = £120 ✓

Ratio and Proportion 比例与正反比

Direct proportion (正比例): When two quantities increase or decrease together at the same rate. If y ∝ x, then y = kx. Graphically, this is a straight line through the origin.

Inverse proportion (反比例): When one quantity increases as the other decreases. If y ∝ 1/x, then y = k/x. Graphically, this is a hyperbola.

AQA examiners love testing the distinction between these two. Be prepared to identify proportionality from tables, graphs, and word problems.

Common Exam Question Types 常见考试题型

1. Given one part, find others (已知一部分，求其他): If a : b = 3 : 7 and a = 15, then 1 part = 15 ÷ 3 = 5, so b = 7 × 5 = 35.

2. Given the total, find each part (已知总数，求各部分). Total parts method as shown above.

3. Given the difference, find each part (已知差，求各部分): If a : b = 5 : 2 and a is 18 more than b, then difference in parts = 5 − 2 = 3 parts = 18, so 1 part = 6, giving a = 30, b = 12.

4. Changing ratios (变化比例): When quantities are added or removed, set up equations. These are Higher-tier favourites.

5. Map scales and unit conversion (地图比例尺与单位换算): A scale of 1 : 50000 means 1 cm on the map represents 50000 cm (500 m) in reality. These appear on both Foundation and Higher papers.

Top 5 Common Mistakes 五大常见错误

❌ Mistake 1: Forgetting to check that simplified ratios are in integer form. Fix: Always multiply through to clear decimals and fractions.

❌ Mistake 2: Mixing up parts and totals — using the total (e.g., 5) when you need the ratio part (e.g., 2 or 3). Fix: Clearly label "total" vs "part" in your working.

❌ Mistake 3: Confusing direct and inverse proportion. Fix: Check: when one quantity doubles, does the other double (direct) or halve (inverse)?

❌ Mistake 4: Not writing units in ratio problems involving different units. Fix: Convert all quantities to the same unit before forming the ratio.

❌ Mistake 5: Ignoring the order — the ratio a : b is NOT the same as b : a. Fix: Read the question carefully and maintain the given order throughout.

Exam Tips for AQA GCSE 考试提分技巧

时间管理 Time Management: Ratio questions in Foundation papers are usually 2–4 marks — aim for 2–3 minutes each. Higher-tier multi-step ratio problems can be worth 4–6 marks — allocate 5–7 minutes.

展示解题过程 Show Your Working: AQA examiners award method marks even if the final answer is wrong. Write down: total parts → value of one part → each share. This structured approach consistently earns marks.

检查答案 Verify Your Answer: Add up your parts — do they sum to the total? Substitute back into the original ratio — does it simplify correctly? These 30-second checks can save 4–6 marks per paper.

Practice Makes Perfect 熟能生巧

Ratio is a high-yield topic — mastering it guarantees marks across multiple question types. We recommend working through at least 20 past paper ratio questions before your exam. Focus especially on AQA Paper 2 and Paper 3 from 2022–2025, as these contain excellent ratio problem-solving questions.

比例是GCSE数学中回报率最高的知识点之一——掌握它意味着你可以在多种题型中稳定拿分。建议在考前至少练习20道历年真题中的比例题目，重点关注AQA 2022-2025年Paper 2和Paper 3中的综合应用题。

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