Mastering Percentage Increase and Decrease | 掌握百分比增减计算

📚 Mastering Percentage Increase and Decrease | 掌握百分比增减计算

Percentages are everywhere in daily life, from discounts in shops to interest rates at the bank. In this revision guide, you will learn how to calculate percentage increases and decreases confidently, use multiplier methods, and solve reverse percentage problems. Whether you’re preparing for Cambridge Checkpoint or consolidating your skills, mastering these concepts will give you a solid foundation for future mathematics.

百分比在日常生活中无处不在，从商店折扣到银行利率。在本复习指南中，你将学会如何自信地计算百分比增加和减少，使用乘数法，以及解决反向百分比问题。无论你是在为剑桥 Checkpoint 考试做准备，还是在巩固技能，掌握这些概念将为未来的数学打下坚实基础。

1. Understanding Percentages: Basics | 理解百分比基础

A percentage means ‘out of 100’. For example, 25% is equivalent to 25/100, which simplifies to 1/4. Percentages help us compare proportions easily. To convert a percentage to a decimal, divide by 100; to convert a decimal to a percentage, multiply by 100.

百分比意味着“每一百份”，例如 25% 等于 25/100，化简为 1/4。百分比帮助我们轻松比较比例。要把百分比转换为小数，除以 100；要把小数转换为百分比，乘以 100。

Key conversions to remember: 50% = 0.5 = 1/2, 25% = 0.25 = 1/4, 10% = 0.1 = 1/10, 1% = 0.01 = 1/100. Being comfortable with these allows quick mental calculations.

需要记住的关键换算：50% = 0.5 = 1/2，25% = 0.25 = 1/4，10% = 0.1 = 1/10，1% = 0.01 = 1/100。熟练这些可以快速心算。

---

2. Calculating Percentage of a Quantity | 计算一个数量的百分比

To find a percentage of a number, first find 1% by dividing by 100, then multiply by the required percentage. For instance, to find 15% of £60: 1% is £0.60, so 15% is 15 × £0.60 = £9. Alternatively, multiply by the percentage as a decimal: 0.15 × £60 = £9.

要计算一个数的百分比，首先除以 100 求出 1%，然后乘以所需的百分比。例如，求 £60 的 15%：1% 是 £0.60，所以 15% 是 15 × £0.60 = £9。或者用小数乘：0.15 × £60 = £9。

This method works for any percentage, even those with decimals like 3.5%. Always convert the percentage to a decimal first to use a calculator efficiently.

这种方法适用于任何百分比，即使是像 3.5% 这样带小数的百分比。使用计算器时，总是先将百分比转换为小数，这样效率更高。

---

3. Percentage Increase | 百分比增加

A percentage increase occurs when a quantity grows by a certain percent of its original value. The formula is:

百分比增加是指一个量以其原始值的某个百分比增长。公式如下：

New Value = Original + (Percentage Increase % × Original)

新值 = 原始值 + (百分比增加% × 原始值)

For example, if a phone costing £200 increases in price by 20%, the increase is 20% of £200 = £40, so the new price is £240. You can also calculate directly: 120% of £200 = 1.20 × £200 = £240.

例如，一部售价 £200 的手机价格上涨 20%，增加额为 £200 的 20% = £40，所以新价格为 £240。你也可以直接计算：£200 的 120% = 1.20 × £200 = £240。

---

4. Percentage Decrease | 百分比减少

A percentage decrease subtracts a percentage of the original amount. The formula is:

百分比减少是从原始量中减去其百分比。公式如下：

New Value = Original − (Percentage Decrease % × Original)

新值 = 原始值 − (百分比减少% × 原始值)

Suppose a jacket originally priced at £80 is reduced by 15% in a sale. The discount is 15% of £80 = £12, so the sale price is £68. Using a multiplier: 85% of £80 = 0.85 × £80 = £68.

假设一件夹克原价 £80，在促销中降价 15%。折扣为 £80 的 15% = £12，所以售价为 £68。使用乘数：£80 的 85% = 0.85 × £80 = £68。

---

5. Using Multipliers for Percentage Change | 使用乘数表示百分比变化

A multiplier is a decimal factor that represents the overall percentage after a change. To increase by r%, the multiplier is 1 + r/100. To decrease by r%, the multiplier is 1 − r/100.

乘数是一个小数因子，表示变化后的整体百分比。增加 r%，乘数为 1 + r/100。减少 r%，乘数为 1 − r/100。

For example, a 7% increase multiplier is 1.07; a 30% decrease multiplier is 0.70. This method is faster and essential for compound changes. The table below shows common multipliers:

例如，增加 7% 的乘数是 1.07；减少 30% 的乘数是 0.70。这种方法更快，对于复合变化也必不可少。下表显示了常见乘数：

Percentage Change / 百分比变化	Multiplier / 乘数	Calculation (Original = 100) / 计算示例
Increase 10% (增加10%)	1.10	100 × 1.10 = 110
Decrease 25% (减少25%)	0.75	100 × 0.75 = 75
Increase 0.5% (增加0.5%)	1.005	100 × 1.005 = 100.5

Always convert the percentage to a decimal before adding or subtracting to get the multiplier.

在加或减之前总是先将百分比转换为小数，以得到乘数。

---

6. Finding the Original Amount After a Percentage Change | 百分比变化后求原始量

Sometimes you know the new value after a percentage change and need to find the original. This is called reverse percentage. If a quantity is increased by r% to a new value N, the original is N ÷ (1 + r/100). For a decrease: original = N ÷ (1 − r/100).

有时你知道百分比变化后的新值，需要求原始值。这称为反向百分比。如果一个量增加了 r% 变成新值 N，原始值为 N ÷ (1 + r/100)。对于减少：原始值 = N ÷ (1 − r/100)。

Example: A bike’s price after a 20% increase is £360. The multiplier was 1.20, so original price = £360 ÷ 1.20 = £300.

例子：一辆自行车涨价 20% 后价格为 £360。乘数为 1.20，所以原始价格 = £360 ÷ 1.20 = £300。

Beware: a 20% increase followed by a 20% decrease does not return to the original value, because the decrease is taken from a larger amount. This is a common pitfall.

注意：先增加 20% 然后再减少 20% 不会回到原始值，因为减少是在更大的量上进行的。这是一个常见的陷阱。

---

7. Repeated Percentage Changes | 连续百分比变化

When a percentage change is applied multiple times, multiply the multipliers. For example, compound interest: £1000 invested at 5% p.a. for 3 years becomes £1000 × 1.05 × 1.05 × 1.05 = £1000 × (1.05)^3 ≈ £1157.63.

当百分比变化多次应用时，将乘数相乘。例如，复利：£1000 按照年利率 5% 投资 3 年，变为 £1000 × 1.05 × 1.05 × 1.05 = £1000 × (1.05)³ ≈ £1157.63。

The general formula for compound growth is: Final Amount = Original × (1 + r/100)^n, where n is the number of periods. For depreciation (decrease), use (1 − r/100)^n.

复合增长的通用公式为：最终量 = 原始量 × (1 + r/100)ⁿ，其中 n 是期数。对于贬值（减少），使用 (1 − r/100)ⁿ。

Be careful: if a value is increased by 10% then decreased by 10%, the overall multiplier is 1.10 × 0.90 = 0.99, resulting in a 1% loss, not the original.

请注意：如果一个值先增加 10% 再减少 10%，总乘数为 1.10 × 0.90 = 0.99，导致净损失 1%，而不是回到原值。

---

8. Real-World Applications | 实际应用

Percentage increase and decrease are widely used in finance, shopping, and statistics. Examples include: salary raises, tax calculations, population growth, and discounts. Understanding these helps you make informed decisions.

百分比增加和减少广泛应用于金融、购物和统计中。例子包括：加薪、税款计算、人口增长和折扣。理解这些有助于你做出明智的决定。

When comparing offers, such as a 10% off store-wide versus a ‘buy one get one half price’, you can calculate the effective percentage saving to find the best deal.

在比较优惠时，比如全场九折与“买一件第二件半价”，你可以计算实际节省的百分比，找到最划算的交易。

VAT (Value Added Tax) in the UK is currently 20%. If a product costs £50 excluding VAT, the final price is £50 × 1.20 = £60. This is a percentage increase application.

英国目前的增值税 (VAT) 为 20%。如果一件商品不含税价格为 £50，最终价格为 £50 × 1.20 = £60。这是百分比增加的应用。

---

9. Common Misconceptions | 常见误区

Mistake 1: Adding or subtracting percentages directly as numbers. For instance, increasing 80 by 50% does not give 130; it gives 80 + 40 = 120. Percentages must be converted to a proportion of the original.

误区 1：将百分比当作数字直接加减。例如，80 增加 50% 不是 130，而是 80 + 40 = 120。百分比必须转换为相对于原始值的比例。

Mistake 2: Confusing a percentage point change with a percentage change. If an interest rate rises from 4% to 5%, it is a 1 percentage point increase, but a 25% increase (since 1/4 = 0.25).

误区 2：混淆百分点变化和百分比变化。如果利率从 4% 升至 5%，是上升了 1 个百分点，但百分比增加了 25%（因为 1/4 = 0.25）。

Mistake 3: Reversing a percentage change incorrectly. A 20% decrease followed by a 20% increase does not restore the original; the net effect is a 4% loss.

误区 3：错误地逆转百分比变化。先降 20% 再涨 20% 不会恢复原值；净效果是损失 4%。

---

10. Practice and Summary | 练习与总结

To master percentages, practice with a variety of problems: calculate simple percentages, use multipliers, find original values, and work with compound changes. Always check if your answer is reasonable.

要掌握百分比，请练习各种题型：计算简单百分比、使用乘数、求原始值、处理复合变化。始终检查你的答案是否合理。

Summary of key formulas:

关键公式总结：

• Increase: New = Original × (1 + r/100) | 增加：新值 = 原始值 × (1 + r/100)
• Decrease: New = Original × (1 − r/100) | 减少：新值 = 原始值 × (1 − r/100)
• Reverse increase: Original = New ÷ (1 + r/100) | 反向增加：原始值 = 新值 ÷ (1 + r/100)
• Reverse decrease: Original = New ÷ (1 − r/100) | 反向减少：原始值 = 新值 ÷ (1 − r/100)
• Compound change: Final = Original × (1 ± r/100)^n | 复合变化：最终值 = 原始值 × (1 ± r/100)ⁿ

Remember: percentages are just fractions out of 100, and keeping this in mind will help you tackle any problem. Keep practising, and you’ll soon see how useful percentages are in real life.

记住：百分比只是分母为 100 的分数，牢记这一点将帮助你解决任何问题。坚持练习，你很快会发现百分比在现实生活中多么有用。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）