# Ratio and Proportion

> Edexcel International GCSE Mathematics A · 4MA1 S1
> 来源: https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s1-ratio-and-proportion/

This guide covers all Edexcel IGCSE Maths A S1 ratio and proportion content: simplifying ratios, sharing quantities by ratio, direct proportion scaling, and map/scale diagram word problems with exam-aligned examples.

**先修:** [Fraction simplification and HCF calculation](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s1-factors-multiples/); [Unit conversion for length, mass, and currency](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s1-units-conversion/)

## 学习目标

- Simplify ratios to integer simplest form or 1:n format, linking to fraction notation
- Divide a given quantity across 2+ parts in a specified ratio
- Use unitary scaling to solve numerical direct proportion problems
- Solve word problems including map scales and scale diagram ratio questions

## Ratio Notation & Simplification

**Simplified Ratio** — A ratio written with integers that have a highest common factor (HCF) of 1, or in the required 1:n format for unit comparison.

*例:* 12:18 simplifies to 2:3 (integer simplest form) or 1:1.5 (1:n form).

Ratios compare relative sizes of 2+ quantities, written with colons between parts. Always align units across all parts before simplifying, then divide every part by their shared HCF to get integer simplest form. If asked for 1:n format, divide all parts by the first value to set the first part equal to 1.

**例题:** Simplify the ratio 240g : 1.2kg, then express it in the form 1:n.

1. 1. Align units: convert 1.2kg to 1200g, so the ratio is 240:1200
2. 2. Calculate HCF of 240 and 1200 = 240, divide both parts by 240 to get 1:5
3. 3. The first part is already 1, so the 1:n form is 1:5

> **考试提示:** Always check units first when simplifying ratios; mismatched units are the top error for this question type.

*计算器:* allowed

## Sharing Quantities in a Given Ratio

To split a quantity into parts matching a given ratio, first calculate the total number of parts in the ratio, find the value of 1 single part, then multiply by the number of parts for each share. Always sum your final shares to confirm they match the original total.

**例题:** Share £416 between three people in the ratio 4:3:1.

1. 1. Calculate total parts: 4 + 3 + 1 = 8 parts
2. 2. Value of 1 part = £416 ÷ 8 = £52
3. 3. Calculate each share: 4 × £52 = £208, 3 × £52 = £156, 1 × £52 = £52
4. 4. Verify total: £208 + £156 + £52 = £416, matching the original amount

> **考试提示:** Spend 10 seconds adding up your final shares to catch calculation errors; this check will save you easy lost marks.

*计算器:* allowed

## Direct Proportion Using Scaling

**Unitary Method for Proportion** — A numerical method to solve direct proportion problems by first calculating the value of 1 unit, then scaling up or down to find the required quantity.

*例:* If 3 pens cost £1.20, 1 pen costs £0.40, so 7 pens cost 7 × £0.40 = £2.80.

Two quantities are directly proportional if when one increases, the other increases by the same factor. For S1 questions, use only unitary or factor scaling methods, not algebraic proportionality constants (assessed in later topics).

**例题:** The mass of 12 identical wooden blocks is 3.6kg. Find the mass of 20 such blocks.

1. 1. Find mass of 1 block: 3.6kg ÷ 12 = 0.3kg per block
2. 2. Scale to 20 blocks: 20 × 0.3kg = 6kg
3. 3. Verify with factor scaling: 20/12 = 5/3, so 3.6kg × 5/3 = 6kg, confirming the result

> **考试提示:** Use whichever method is faster for the numbers given: unitary method is easiest for small values, factor scaling works well for clean multiples.

*计算器:* allowed

## Ratio in Maps & Scale Diagrams

Map scales are written as ratios, where the first number is the distance on the map, and the second is the actual distance in the same unit. Memorize that 1km = 100,000cm to simplify unit conversion for scale questions.

**例题:** A map has a scale of 1:25,000. The distance between two villages on the map is 8cm. Calculate the actual distance between the villages in kilometres.

1. 1. 1cm on map = 25,000cm actual distance
2. 2. 8cm × 25,000 = 200,000cm total actual distance
3. 3. Convert cm to km: 200,000 ÷ 100,000 = 2km

> **考试提示:** The first value in any scale ratio always refers to the diagram/map measurement, never the real-world measurement. This is a common mix-up in exams.

*计算器:* allowed

## 常见错误

- **错误做法:** Simplifying ratios with mismatched units (e.g. 200m : 1km simplified to 200:1)
  - 原因: Units must be identical to compare relative sizes of quantities
  - 正确做法: Convert all quantities to the same unit before simplifying ratios
- **错误做法:** Dividing total quantity by the first ratio part instead of total parts when sharing
  - 原因: The ratio represents parts of the whole, so you need the sum of all parts to find the value of 1 unit
  - 正确做法: Add all parts of the ratio to get total parts, then divide the total quantity by this sum
- **错误做法:** Mixing up map scale parts (e.g. assuming 25,000cm map = 1cm actual)
  - 原因: Map scales are always written as [map distance]:[actual distance]
  - 正确做法: Remember the first number in any scale ratio refers to the diagram or map measurement
- **错误做法:** Leaving ratios as decimals when asked for simplest integer form
  - 原因: Standard simplest form requires integer values with HCF 1, unless 1:n format is explicitly requested
  - 正确做法: Check question instructions: use integers for standard simplified ratios, decimals only for 1:n form
- **错误做法:** Using algebraic proportionality constant k for S1 proportion questions
  - 原因: This topic assesses numerical scaling only, and algebraic methods introduce unnecessary error risk
  - 正确做法: Use unitary or factor scaling methods for all S1 ratio and proportion questions

## 速查表

| Task | Key Steps | Quick Example |
| --- | --- | --- |
| Simplify ratio | 1. Align units 2. Divide all parts by HCF | 240g:1.2kg = 240:1200 = 1:5 |
| Share quantity in ratio | 1. Sum ratio parts for total 2. Find 1 part value 3. Calculate each share | Share £416 in 4:3:1: 8 parts = £52 each, shares £208, £156, £52 |
| Solve direct proportion | 1. Find value of 1 unit 2. Scale to required quantity | 12 blocks = 3.6kg, 20 blocks = 6kg |
| Calculate map scale distance | 1. Multiply map distance by scale factor 2. Convert to required units | 1:25000 scale, 8cm map = 2km actual |

## 下一步

Now that you have mastered basic ratio and proportion for Section S1, you can move on to related number topics including percentages and compound interest, before progressing to algebraic proportion (Section 2.5) for Higher tier content. Practicing past paper ratio questions will help you build speed and avoid common unit conversion errors, which are frequently tested in both calculator and non-calculator papers. These ratio skills also appear frequently in multi-part geometry and statistics questions, so mastering them now will boost your performance across the full exam syllabus.

- [Percentages (Edexcel IGCSE Maths A S1)](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s1-percentages/)

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