📚 Percentage Increase and Decrease: Real-Life Applications | 百分数增减：现实应用
Percentages are used to compare changes, work out discounts, and understand data in everyday life. In this revision guide, you will learn how to convert between fractions, decimals and percentages, calculate percentage increase and decrease, and use multipliers to solve problems quickly. We will also look at reverse percentages and real-life examples such as VAT, discounts and interest rates.
百分数用于比较变化、计算折扣以及理解日常生活中的数据。在本篇复习指南中，你将学习如何在分数、小数和百分数之间互化，计算百分数增加和减少，并利用倍数快速解题。我们还将学习逆百分数，以及增值税、折扣和利率等实际例子。
1. Understanding Percentages | 理解百分数
A percentage is a number out of 100. The word ‘per cent’ means ‘per hundred’, so 25% means 25 out of 100. Percentages do not have units; they describe a relative amount compared with a whole.
百分数是以 100 为基准的数。’per cent’ 的意思是 ‘每一百’，所以 25% 表示每一百中有 25。百分数没有单位；它描述的是相对于整体的多少。
Because percentages are always out of 100, they are easy to write as fractions with denominator 100. For example, 7% = 7/100 and 130% = 130/100. This idea is the foundation for all percentage calculations.
由于百分数总是以 100 为基准，因此它们很容易写成分母为 100 的分数。例如，7% = 7/100，130% = 130/100。这一思路是所有百分数计算的基础。
Percentages can describe parts of a whole, changes over time, or comparisons between two quantities. A result of 80% on a test means 80 marks out of 100, but it can also mean a relative score if the test is marked differently.
百分数可以描述整体的一部分、随时间的变化或两个数量之间的比较。测试中得 80% 意味着满分 100 分中获得 80 分，但如果试卷总分不同，它也可以表示相对得分。
- 25% means 25 out of 100 / 25% 表示每一百中有 25
- 100% means the whole amount / 100% 表示整体
- Percentages can be greater than 100% / 百分数可以大于 100%
2. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分数的互化
To convert a percentage to a decimal, divide by 100. To convert a decimal to a percentage, multiply by 100. To convert a fraction to a percentage, first divide the numerator by the denominator, then multiply by 100.
将百分数化为小数，除以 100；将小数化为百分数，乘以 100。将分数化为百分数时，先用分子除以分母，再乘以 100。
The table below shows some useful conversions that you should remember for KS3 Maths. These appear frequently in non-calculator papers and in multi-step problems.
下表列出了一些你应该在 KS3 数学中记住的常用互化。它们经常出现在非计算器试卷和多步问题中。
| Percentage | 百分数 | Fraction | 分数 | Decimal | 小数 |
|---|---|---|
| 50% | 1/2 | 0.50 |
| 25% | 1/4 | 0.25 |
| 75% | 3/4 | 0.75 |
| 20% | 1/5 | 0.20 |
| 10% | 1/10 | 0.10 |
| 5% | 1/20 | 0.05 |
| 1% | 1/100 | 0.01 |
Using these conversions can make mental calculations faster. For example, to find 25% of an amount, just divide by 4; to find 10%, divide by 10.
使用这些互化可以加快心算速度。例如，求一个数的 25%，只需除以 4；求 10%，只需除以 10。
3. Finding a Percentage of an Amount | 求一个数的百分之几
To find a percentage of an amount, write the percentage as a fraction over 100 and multiply. For example, 30% of 200 = (30/100) × 200 = 60.
求一个数的百分之几，先把百分数写成分数分母 100，再乘。例如，200 的 30% = （30/100）× 200 = 60。
A quicker method is to convert the percentage to a decimal first: 30% = 0.30, so 0.30 × 200 = 60. This is useful when using a calculator or when the percentage is not a simple fraction.
更快的方法是先把百分数化为小数：30% = 0.30，因此 0.30 × 200 = 60。在使用计算器或百分数不是简单分数时，这种方法很有用。
For non-calculator work, use benchmarks. To find 35% of 80, you can calculate 10% = 8, 30% = 24, and 5% = 4, then add: 24 + 4 = 28. This method helps you check your answer carefully.
在非计算器题目中，可以使用基准值。求 80 的 35% 时，可以先计算 10% = 8，30% = 24，5% = 4，然后相加：24 + 4 = 28。这种方法有助于仔细检查答案。
part = (percentage ÷ 100) × whole
部分 = （百分数 ÷ 100）× 整体
4. What Is Percentage Increase? | 什么是百分数增加
Percentage increase measures how much a quantity has grown compared with its original value. It is always calculated using the original amount, not the new amount. The formula is:
百分数增加衡量一个量相对于原始值增长了多少。计算时始终使用原数，而不是新数。公式为：
percentage increase = (increase ÷ original amount) × 100%
百分数增加 = （增加量 ÷ 原数）× 100%
If a price rises from £40 to £50, the increase is £10. The percentage increase is (10 ÷ 40) × 100% = 25%. This means the new price is 125% of the original price.
如果价格从 40 英镑上涨到 50 英镑，增加量为 10 英镑。百分数增加为 （10 ÷ 40）× 100% = 25%。这意味着新价格是原价的 125%。
Always subtract the original from the new amount first, then divide by the original. Avoid using the new amount as the denominator, because that gives a different and incorrect answer.
始终先用新数减去原数，再除以原数。不要用新数作分母，因为那样会得到不同且错误的答案。
5. What Is Percentage Decrease? | 什么是百分数减少
Percentage decrease uses the same logic as percentage increase. The formula is:
百分数减少使用与百分数增加相同的思路。公式为：
percentage decrease = (decrease ÷ original amount) × 100%
百分数减少 = （减少量 ÷ 原数）× 100%
If a shop reduces a jacket from £80 to £60, the decrease is £20. The percentage decrease is (20 ÷ 80) × 100% = 25%. The new price is 75% of the original price.
如果商店把一件夹克从 80 英镑降到 60 英镑，减少量为 20 英镑。百分数减少为 （20 ÷ 80）× 100% = 25%。新价格是原价的 75%。
Percentage decrease cannot be more than 100% when the quantity starts positive, because you cannot lose more than the whole original amount. A decrease of 100% means the value becomes zero.
当初始数量为正时，百分数减少不可能超过 100%，因为你失去的量不能超过原始总量。减少 100% 意味着数值变为零。
6. Using Multipliers for Quick Calculations | 使用倍数快速计算
A multiplier is a single decimal number that changes an amount in one step. To increase by r%, multiply by (1 + r/100). To decrease by r%, multiply by (1 – r/100).
倍数是一个一步改变数量的单个小数。增加 r% 时，乘以（1 + r/100）；减少 r% 时，乘以（1 – r/100）。
For a 15% increase, the multiplier is 1 + 0.15 = 1.15. For a 20% decrease, the multiplier is 1 – 0.20 = 0.80. This method is much faster for multi-step problems and for calculator work.
增加 15% 的倍数是 1 + 0.15 = 1.15；减少 20% 的倍数是 1 – 0.20 = 0.80。对于多步问题和使用计算器时，这种方法要快得多。
- Increase by 5% = multiply by 1.05 / 增加 5% = 乘以 1.05
- Increase by 25% = multiply by 1.25 / 增加 25% = 乘以 1.25
- Decrease by 10% = multiply by 0.90 / 减少 10% = 乘以 0.90
- Decrease by 75% = multiply by 0.25 / 减少 75
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