📚 Finding a Percentage of a Quantity | 求一个数的百分比
In this IGCSE Mathematics revision session, we build the core skill of finding a percentage of a quantity. You will learn how to convert percentages into decimals or fractions, use a multiplier, and apply the method to real exam problems.
在这节 IGCSE 数学复习课中，我们建立“求一个数的百分比”这一核心技能。你将学习如何把百分数转换为小数或分数、使用乘法因子，并把方法应用到真实考试题目中。
1. What Is a Percentage? | 什么是百分比
A percentage is a ratio whose second term is 100. The symbol % means 'out of 100', so 37% means 37 out of 100 equal parts.
百分数是一个第二项为 100 的比。符号 % 表示“每一百份中”，因此 37% 表示 100 等份中的 37 份。
- 50% = 50 ÷ 100 = 1/2 | 50% = 50 ÷ 100 = 1/2
- 25% = 25 ÷ 100 = 1/4 | 25% = 25 ÷ 100 = 1/4
- 10% = 10 ÷ 100 = 1/10 | 10% = 10 ÷ 100 = 1/10
2. Converting Percentages to Decimals and Fractions | 百分数、小数与分数互换
To change a percentage into a decimal, divide by 100. This is the same as moving the decimal point two places to the left.
要把百分数转换为小数，除以 100，也就是把小数点向左移动两位。
p% = p ÷ 100
For example, 7% = 0.07, 45% = 0.45, and 125% = 1.25.
例如，7% = 0.07，45% = 0.45，125% = 1.25。
To change a percentage into a fraction, write it over 100 and simplify if possible.
要把百分数转换为分数，先写成分母为 100 的分数，再尽可能化简。
40% = 40/100 = 2/5
3. The Core Formula: Part = Percentage × Whole | 核心公式：部分 = 百分数 × 整体
When you need to find a percentage of a quantity, the quantity is the whole, and the answer is the part.
当你需要求一个量的百分数时，这个量就是整体，答案就是部分。
Part = percentage × whole
Because a percentage must be written as a decimal or fraction before multiplying, the working formula is:
因为百分数在相乘前必须写成小数或分数，所以实际使用的公式是：
Part = (p ÷ 100) × W
Here p is the number in front of the % sign, and W is the whole quantity.
这里的 p 是 % 前面的数，W 是整体量。
4. Method 1: Decimal Multiplier Method | 方法一：小数乘法因子法
The decimal method is usually the fastest method when a calculator is allowed.
当可以使用计算器时，小数法通常是最快的方法。
Example: Find 35% of 80.
例题：求 80 的 35%。
- Write 35% as a decimal: 35% = 0.35 | 把 35% 写成小数：35% = 0.35
- Multiply the whole by the decimal: 0.35 × 80 | 用整体乘以小数：0.35 × 80
- Answer: 28 | 答案：28
35% of 80 = 0.35 × 80 = 28
5. Method 2: Fraction Method | 方法二：分数法
The fraction method is useful for simple percentages such as 50%, 25%, 20%, 10%, and 5%.
分数法对于 50%、25%、20%、10%、5% 等简单百分数非常有用。
Example: Find 25% of 64.
例题：求 64 的 25%。
- Write 25% as a fraction: 25% = 1/4 | 把 25% 写成分数：25% = 1/4
- Multiply: 64 × 1/4 = 16 | 相乘：64 × 1/4 = 16
25% of 64 = 64 × 1/4 = 16
6. Using 1% as a Quick Benchmark | 以 1% 作为快速基准
If a percentage is not a familiar fraction, find 1% first and then scale up to the required percentage.
如果某个百分数不是熟悉的分数，可以先求 1%，再扩大到所需的百分数。
Example: Find 6% of 150.
例题：求 150 的 6%。
- 1% of 150 = 150 ÷ 100 = 1.5 | 150 的 1% = 150 ÷ 100 = 1.5
- 6% = 6 × 1.5 = 9 | 6% = 6 × 1.5 = 9
6% of 150 = 9
7. Common Percentages and Mental Shortcuts | 常见百分数与心算技巧
Memorising the following conversions will speed up mental calculation and reduce errors.
记住以下转换可以加快心算速度并减少错误。
| Percentage | Decimal | Fraction |
|---|---|---|
| 1% | 0.01 | 1/100 |
| 5% | 0.05 | 1/20 |
| 10% | 0.10 | 1/10 |
| 20% | 0.20 | 1/5 |
| 25% | 0.25 | 1/4 |
| 50% | 0.50 | 1/2 |
| 75% | 0.75 | 3/4 |
| 100% | 1.00 | 1/1 |
8. Percentage Increase and Decrease | 百分数增加与减少
For percentage increase, add the percentage to 100% before using the multiplier. For percentage decrease, subtract the percentage from 100%.
百分数增加时，先将百分数加到 100% 再使用乘法因子；百分数减少时，从 100% 中减去该百分数。
Increase: new = original × (1 + p ÷ 100)
Decrease: new = original × (1 − p ÷ 100)
Example: Increase 120 by 15%. 15% as a decimal is 0.15, so the multiplier is 1.15.
例题：把 120 增加 15%。15% 写成小数是 0.15，因此乘法因子为 1.15。
120 × 1.15 = 138
Example: Decrease 120 by 15%. The multiplier is 1 − 0.15 = 0.85.
例题：把 120 减少 15%。乘法因子为 1 − 0.15 = 0.85。
120 × 0.85 = 102
9. IGCSE-Style Worked Examples | IGCSE 考试型例题
Worked Example 1: A bicycle costs $240 before tax. VAT of 17.5% is added. Find the total price.
例题 1：一辆自行车税前价格为 $240，加收 17.5% VAT。求总价。
- Find 17.5% of 240: 0.175 × 240 = 42 | 求 240 的 17.5%：0.175 × 240 = 42
- Add the tax to the original price: 240 + 42 = 282 | 将税额加到原价：240 + 42 = 282
Total price = $282
Worked Example 2: Ahmad scored 18 out of 25 in a test. Find his percentage score.
例题 2：Ahmad 在一次测验
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