# Percentages

> Edexcel International GCSE Mathematics A · 4MA1 (2016)
> 来源: https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s1-percentages/

This guide covers all percentage content for Edexcel IGCSE Maths A (4MA1) foundation and higher tiers, including conversions, multipliers, change calculations, reverse percentages, compound interest and repeated change.

**先修:** [Fraction and decimal conversion](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s1-fractions-decimals/); [Basic multiplication and division of decimals](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s1-arithmetic-operations/)

## 学习目标

- Define percentages as parts per 100 and convert between percentages, fractions and decimals
- Calculate percentages of amounts and solve simple percentage increase/decrease problems using multipliers
- Solve reverse percentage problems to find original values after a percentage change
- Calculate compound interest and depreciation using repeated multipliers
- Compute total percentage change for multi-step repeated percentage change scenarios (Higher only)

## Percentage Fundamentals and Conversions

A percentage means 'number of parts per 100', and is a common way to represent proportions, changes and comparisons in everyday life and exam problems.

**Percentage** — A ratio that represents a number of parts per 100, denoted with the % symbol

*例:* 25% means 25 parts out of 100, equal to $\frac{25}{100}$ or 0.25

- To convert a percentage to a decimal: divide by 100
- To convert a decimal to a percentage: multiply by 100
- To convert a percentage to a fraction: write the value over 100 and simplify fully

**例题:** Convert 35% to a simplified fraction and a decimal, and convert 0.04 to a percentage.

1. Write 35% as a fraction over 100 and simplify:

   $$35\% = \frac{35}{100} = \frac{7}{20}$$
2. Divide 35 by 100 to get the decimal equivalent:

   $$35 \div 100 = 0.35$$
3. Multiply 0.04 by 100 to get the percentage:

   $$0.04 \times 100 = 4\%$$

**考试命令词**

- **Express X as a percentage of Y** — Calculate $(X/Y) \times 100$, give answer with % symbol *(Express 15 as a percentage of 60: $(15/60) \times 100 = 25\%$)*

- **Find the percentage change** — Calculate $\frac{(new - original)}{original} \times 100$, state if increase or decrease

> **考试提示:** Always simplify fractions fully when asked to express a percentage as a fraction to gain full marks.

*计算器:* allowed

## Percentage of Amounts and Simple Change

The multiplier method is the fastest, most accurate way to calculate percentages of amounts and percentage increases or decreases. A multiplier is the decimal equivalent of the percentage you are applying.

**Percentage Multiplier** — Decimal value used to calculate percentage operations in one step: <br> - Percentage of an amount: multiplier = $r/100$ for r% <br> - Increase by r%: multiplier = $1 + r/100$ <br> - Decrease by r%: multiplier = $1 - r/100$

**例题:** Calculate 18% of £240, then find the new price of a £240 jacket after an 18% price increase.

1. Calculate the 18% multiplier:

   $$18/100 = 0.18$$
2. Find 18% of £240:

   $$0.18 \times 240 = £43.20$$
3. Calculate the increase multiplier:

   $$1 + 0.18 = 1.18$$
4. Find the new jacket price:

   $$240 \times 1.18 = £283.20$$

> **tip**
>
> Using multipliers avoids separate addition or subtraction steps, reducing the risk of arithmetic errors in exams.

*计算器:* allowed

## Reverse Percentages

Reverse percentage questions ask you to find the original value of a quantity before a known percentage change was applied. The key rule is to divide by the change multiplier, do not add or subtract the percentage from the final value.

**考试命令词**

- **Find the original value** — Reverse percentage question, divide the final value by the percentage change multiplier

**例题:** A shirt is on sale for £42 after a 30% discount. Find the original price of the shirt.

1. Calculate the discount multiplier for 30% off:

   $$1 - 30/100 = 0.7$$
2. Set up the relationship between original and sale price:

   $$Original \times 0.7 = 42$$
3. Rearrange to find the original price:

   $$Original = 42 \div 0.7 = £60$$

> **考试提示:** Always check your reverse percentage answer by applying the given percentage change to your result to confirm you get the stated final value.

*计算器:* allowed

## Compound Interest and Depreciation

Compound interest and depreciation use repeated multipliers to calculate value change over multiple time periods. For n periods, you raise the annual multiplier to the power of n. No formula is provided for this topic, so you must recall this method.

- Compound interest: interest is added to the principal each period, so interest is earned on previous interest
- Depreciation: the value of an asset decreases by a fixed percentage each period

**例题:** Calculate the value of a £1500 investment after 3 years with 4% annual compound interest. Give your answer to the nearest penny.

1. Calculate the annual interest multiplier:

   $$1 + 4/100 = 1.04$$
2. Calculate the total multiplier for 3 years:

   $$1.04^3 = 1.124864$$
3. Calculate the final value and round to 2 decimal places:

   $$1500 \times 1.124864 = £1687.30$$

> **warning**
>
> Keep the unrounded value of $multiplier^n$ in your calculator for intermediate steps to avoid rounding errors that can cost you marks.

*计算器:* allowed

## Repeated Percentage Change (Higher Only)

When multiple percentage changes are applied sequentially, the total change is found by multiplying all individual change multipliers together. You cannot add or subtract the percentage values directly, as each change applies to a different base value.

**例题:** A shop increases all prices by 30%, then runs a 20% off sale for all items. What is the total percentage change in price?

1. Calculate the 30% increase multiplier:

   $$1 + 30/100 = 1.3$$
2. Calculate the 20% decrease multiplier:

   $$1 - 20/100 = 0.8$$
3. Find the total multiplier by multiplying the two values:

   $$1.3 \times 0.8 = 1.04$$
4. Convert the multiplier to percentage change: 1.04 = 104%, so total change is a 4% increase.

> **考试提示:** The order of percentage changes does not affect the final total multiplier, so you can multiply them in any order to cross-check your answer.

*计算器:* allowed

## 常见错误

- **错误做法:** Converting percentage to decimal by multiplying by 100 instead of dividing
  - 原因: Mixes up conversion directions, leading to values 100x larger than correct
  - 正确做法: Divide percentages by 100 to get decimal multipliers, multiply decimals by 100 to get percentages
- **错误做法:** Adding the discount percentage to the sale price for reverse percentage questions, e.g. £42 + 30% of 42 = £54.60 for the earlier shirt example
  - 原因: The 30% discount applies to the original price, not the sale price
  - 正确做法: Divide the final value by the percentage change multiplier to get the original value
- **错误做法:** Rounding intermediate multiplier powers for compound interest, e.g. rounding $1.04^3$ to 1.12 early
  - 原因: Rounding errors accumulate, leading to incorrect final answers
  - 正确做法: Keep all unrounded intermediate values in your calculator, only round the final answer
- **错误做法:** Calculating simple interest instead of compound interest by multiplying by n instead of raising to power n
  - 原因: Confuses simple and compound growth, leading to understated final values
  - 正确做法: Raise the annual multiplier to the power of the number of periods for compound growth or decay
- **错误做法:** Adding sequential percentage changes directly, e.g. 30% increase then 20% decrease = 10% total increase
  - 原因: Each percentage change applies to a different base value, so they cannot be added directly
  - 正确做法: Multiply all individual percentage change multipliers together to get the total change multiplier

## 速查表

| Task | Method | Example |
| --- | --- | --- |
| Convert % to decimal | Divide by 100 | 12% = 0.12 |
| Convert decimal to % | Multiply by 100 | 0.65 = 65% |
| % increase by r% | × $(1 + r/100)$ | 10% increase: × 1.1 |
| % decrease by r% | × $(1 - r/100)$ | 15% decrease: × 0.85 |
| Reverse % (find original) | Divide by change multiplier | Sale price £56 after 20% off: $56 ÷ 0.8 = £70$ |
| Compound interest n years | × $(multiplier)^n$ | £1000 at 5% for 2 years: $1000 × 1.05² = £1102.50$ |
| Repeated % change | Multiply all multipliers | 20% increase then 10% decrease: $1.2 × 0.9 = 1.08$ (8% increase) |

## 下一步

Now that you have mastered percentage calculations for Edexcel IGCSE Maths A, you can apply these skills to more advanced topics in the numbers unit, including ratio and proportion, and practical problem-solving questions involving money and finance. Percentages are also frequently tested alongside statistical topics such as data interpretation and probability, where you will need to calculate percentage changes from graphs and tables. Make sure to practice both foundation and higher tier past paper questions to solidify your understanding, paying close attention to rounding requirements for money answers and showing all multiplier steps to gain full marks.

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