Percentage Increase and Decrease in IGCSE Mathematics | IGCSE数学：百分比的增减计算

📚 Percentage Increase and Decrease in IGCSE Mathematics | IGCSE数学：百分比的增减计算

Percentages are everywhere in IGCSE Mathematics: price rises, sales discounts, compound interest, population growth and depreciation. Understanding how to increase or decrease an amount by a percentage is a core skill that appears in nearly every exam paper.

百分比在 IGCSE 数学中无处不在：价格上升、打折促销、复利、人口增长和贬值。掌握如何按百分比增加或减少一个数值，是几乎每张考卷都会考查的核心技能。

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1. Percentages as Multipliers | 百分比作为乘数

A percentage simply means ‘out of 100’. To convert a percentage into a decimal, divide it by 100. For example, 15% = 15 ÷ 100 = 0.15. To convert a decimal into a percentage, multiply by 100.

百分比的意思就是“每100份中的多少”。将百分比化为小数，只需除以100。例如，15% = 15 ÷ 100 = 0.15。将小数化为百分比，则乘以100。

This idea becomes powerful when we think about percentage changes. If a price increases by 15%, the new price is 100% + 15% = 115% of the original. As a decimal, 115% = 1.15, so multiplying the original amount by 1.15 gives the new amount directly.

这一思路在处理百分比变化时非常有用。如果价格上涨15%，新价格是原价的 100% + 15% = 115%。化为小数后，115% = 1.15，因此直接用原价乘以1.15即可得到新价格。

Similarly, if a price decreases by 15%, the new price is 100% – 15% = 85% of the original. As a decimal, 85% = 0.85, so multiply the original amount by 0.85.

同理，如果价格下降15%，新价格是原价的 100% – 15% = 85%。化为小数后，85% = 0.85，因此用原价乘以0.85即可。

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2. The Percentage Change Formula | 百分比变化公式

The most important formula for this topic compares the size of the actual change with the original amount.

这个主题最重要的公式，是将实际变化量与原始数值进行比较。

Percentage change = (actual change ÷ original amount) × 100%

For an increase, actual change = new value – original value. For a decrease, actual change = original value – new value. The denominator is always the original value, not the new value.

对于增加，实际变化量 = 新值 – 原值；对于减少，实际变化量 = 原值 – 新值。分母永远是原值，而不是新值。

This formula works both when you know the original and new values, and when you only know the amount of change. Always state whether the change is an increase or a decrease in your final sentence.

无论你已知原值和新值，还是只知道变化量，这个公式都适用。在最终答案中，务必说明这是增加还是减少。

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3. Calculating Percentage Increase | 计算百分比增加

Let us examine a typical example. A train ticket rises from £24 to £30. Find the percentage increase.

我们来看一个典型例题。一张火车票从24英镑涨到30英镑，求上涨的百分比。

First calculate the actual increase: 30 – 24 = 6. Then use the formula: percentage increase = (6 ÷ 24) × 100% = 0.25 × 100% = 25%.

首先计算实际增加量：30 – 24 = 6。然后套用公式：百分比增加 = (6 ÷ 24) × 100% = 0.25 × 100% = 25%。

We can check this with the multiplier method. An increase of 25% means the multiplier is 1.25, and 24 × 1.25 = 30, so the answer is correct.

我们也可以用乘数法验证。增加25%意味着乘数为1.25，而 24 × 1.25 = 30，所以答案正确。

When the percentage increase is given but the new value is unknown, use the multiplier directly. For example, if a £40 item is increased by 12%, the new price is 40 × 1.12 = £44.80.

当已知百分比增加而未知新值时，可直接使用乘数。例如，一件40英镑的商品上涨12%，新价格为 40 × 1.12 = 44.80英镑。

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4. Calculating Percentage Decrease | 计算百分比减少

A laptop costs £800 but is reduced by 35% in a sale. There are two efficient methods to find the sale price.

一台笔记本电脑售价800英镑，促销时降价35%。有两种高效方法可以求出售价。

Method 1: find the decrease first. 35% of 800 = 0.35 × 800 = 280, so the sale price is 800 – 280 = £520.

方法一：先求出减少量。800 × 35% = 0.35 × 800 = 280，因此售价为 800 – 280 = 520英镑。

Method 2: use the multiplier. A decrease of 35% leaves 65%, so multiply by 0.65: 800 × 0.65 = £520.

方法二：使用乘数。减少35%后剩下65%，所以乘以0.65：800 × 0.65 = 520英镑。

If we are asked for the percentage decrease between two known values, the same formula applies. For example, from 80 kg to 60 kg: decrease = 20, so percentage decrease = (20 ÷ 80) × 100% = 25%.

如果题目让我们求两个已知数值之间的下降百分比，同样使用该公式。例如，从80千克降到60千克：减少量为20，所以下降百分率 = (20 ÷ 80) × 100% = 25%。

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5. Multipliers for Increase and Decrease | 增加与减少的乘数

The multiplier method is often the fastest and safest way to solve percentage change questions. For an increase of r%, multiply by (1 + r/100). For a decrease of r%, multiply by (1 – r/100).

乘数法往往是解决百分比变化问题最快、最稳妥的方法。增加 r% 时，乘以 (1 + r/100)；减少 r% 时，乘以 (1 – r/100)。

Percentage change	Multiplier
Increase by 10%	1.10
Decrease by 10%	0.90
Increase by 25%	1.25
Decrease by 25%	0.75
Increase by 100%	2.00
Decrease by 100%	0.00

A multiplier greater than 1 indicates an increase; a multiplier between 0 and 1 indicates a decrease. A multiplier of exactly 1 means no change.

乘数大于1表示增加，乘数在0与1之间表示减少，乘数正好等于1则表示没有变化。

Using multipliers is especially useful when a question involves several consecutive changes, because you can combine the multipliers neatly.

乘数法在连续多次变化的问题中尤其有用，因为你可以将多个乘数简洁地合并使用。

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6. Repeated Percentage Change | 重复百分比变化

When a value changes by the same percentage repeatedly, such as compound interest or annual depreciation, use a power of the multiplier.

当一个数值按相同百分比反复变化时，例如复利或每年贬值，需要使用乘数的幂。

Final amount = Original amount × (multiplier)ⁿ

Here n is the number of times the change is applied. For example, a population of 5000 grows by 2% each year. After 3 years, the population is 5000 × 1.02³ = 5000 × 1.061208 = 5306.04, so about 5306.

其中 n 是变化次数。例如，某地区人口为5000，每年增长2%。3年后人口为 5000 × 1.02³ = 5000 × 1.061208 = 5306.04，也就是大约5306人。

For depreciation, the multiplier is less than 1. If a car worth £18,000 loses 12% of its value each year, its value after 4 years is 18000 × 0.88⁴ ≈ £10,794.50.

对于贬值，乘数小于1。假设一辆价值18,000英镑的汽车每年贬值12%，4年后的价值为 18000 × 0.88⁴ ≈ 10,794.50英镑。

On a calculator, apply the power first, then multiply by the original amount. Do not round the multiplier too early, or the final answer may lose accuracy.

在计算器上，应先计算乘方的值，再乘原值。不要过早将乘数四舍五入，否则最终答案可能不够准确。

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7. Reverse Percentages: Finding the Original Amount | 反向百分比：求原值

In reverse percentage questions, you know the final value after a percentage change and you need to find the original amount. The key is to divide by the multiplier, not to subtract a percentage.

在反向百分比问题中，已知经过百分比变化后的最终值，要求原始值。关键在于用最终值除以乘数，而不是直接减掉一个百分比。

Example: after a 20% increase, a salary is £36,000. The multiplier is 1.20, so the original salary is 36000 ÷ 1.20 = £30,000.

例如：工资上涨20%后为36,000英镑。乘数为1.20，所以原工资为 36000 ÷ 1.20 = 30,000英镑。

Example: after a 15% decrease, a phone costs £510. The multiplier is 0.85, so the original price is 510 ÷ 0.85 = £600.

例如：手机降价15%后价格为510英镑。乘数为0.85，所以原价为 510 ÷ 0.85 = 600英镑。

A common error is to find 15% of £510 and add it on, but that would give £586.50, which is wrong. The 15% must be calculated on the original amount, not on the final amount.

一个常见错误是先求出510英镑的15%再加回去，这样会得到586.50英镑，但这是错的。15%必须基于原值计算，而不是基于最终值。

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8. Common Mistakes and Exam Tips | 常见错误与考试技巧

One frequent mistake is trying to add or subtract the percentage directly. For example, increasing 60 by 10% does not mean 60 + 10; it means 60 × 1.10 = 66.

一个常见错误是直接加减百分比。例如，将60增加10%并不等于60加10，而是 60 × 1.10 = 66。

Always use the original value as the denominator in the percentage change formula. Using the new value as the denominator will produce a different and usually incorrect answer.

在百分比变化公式中，永远将原值作为分母。若把新值作为分母，会得到不同的结果，而且通常不正确。

Do not confuse percentage change with the actual amount of change. A change from 10 to 20 is a 100% increase, even though the actual increase is 10.

不要将百分比变化与实际变化量混为一谈。从10变到20是增加100%，尽管实际增加量只是10。

In reverse percentage questions, check whether the multiplier should be greater than or less than 1. If the amount increased, divide by a multiplier greater than 1; if it decreased, divide by a multiplier less than 1.

在反向百分比问题中，先判断乘数应大于1还是小于1。如果数值增加了，就除以大于1的乘数；如果数值减少了，就除以小于1的乘数。

Write down your multiplier and your calculation steps clearly. Even if the final answer is wrong, the examiner can award method marks for correct working.

把乘数和计算步骤清晰写出来。即使最终答案错了，阅卷老师也会根据正确的过程给方法分。

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9. Worked Exam-Style Questions | 考试题型精解

Question 1: A shop increases all prices by 8%. A jacket originally costs £45. Find the new price.

例题1：某商店将所有商品价格提高8%。一件夹克原价45英镑，求新价格。

Solution: multiplier = 1.08, so new price = 45 × 1.08 = £48.60.

解答：乘数 = 1.08，所以新价格 = 45 × 1.08 = 48.60英镑。

Question 2: In a sale, all items are reduced by 30%. A dress costs £56 in the sale. What was the original price?

例题2：促销期间所有商品降价30%。一条连衣裙的促销价是56英镑，原价是多少？

Solution: multiplier = 0.70, so original price = 56 ÷ 0.70 = £80.

解答：乘数 = 0.70，所以原价 = 56 ÷ 0.70 = 80英镑。

Question 3: An investment of £4000 earns 3% compound interest each year. Calculate the total value after 2 years.

例题3：一笔4000英镑的投资每年获得3%的复利。计算2年后的总价值。

Solution: total value = 4000 × 1.03² = 4000 × 1.0609 = £4243.60.

解答：总价值 = 4000 × 1.03² = 4000 × 1.0609 = 4243.60英镑。

Notice that the second year’s interest is calculated on £4120, not on the original £4000. This is exactly why compound interest uses a power.

注意，第二年的利息是在4120英镑的基础上计算的，而不是在原来的4000英镑基础上。这正是复利需要使用幂的原因。

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10. Summary and Practice Advice | 总结与练习建议

To succeed with percentage increase and decrease questions, remember the following key ideas.

要成功解答百分比增减问题，请牢记以下核心要点。

• Percentage change = (actual change ÷ original amount) × 100%

  百分比变化 = (实际变化量 ÷ 原始值) × 100%

• Increase by r% uses multiplier (1 + r/100); decrease by r% uses multiplier (1 – r/100).

  增加 r% 使用乘数 (1 + r/100)；减少 r% 使用乘数 (1 – r/100)。

• New amount = original amount × multiplier; original amount = new amount ÷ multiplier.

  新值 = 原值 × 乘数；原值 = 新值 ÷ 乘数。

• For repeated changes, use (multiplier)ⁿ over n time periods.

  对于重复变化，在 n 个时间段内使用 (乘数)ⁿ。

Practice by turning each percentage word problem into a multiplication problem. Compare your answer with a rough estimate to catch simple errors before the exam finishes.

练习时，尽量把每个百分比文字题转化为乘法问题。将答案与估算结果对比，可以在考试结束前发现简单错误。

Once you are confident with single-step changes, try multi-step and reverse percentage questions. These appear frequently in Edexcel IGCSE papers and reward clear, systematic working.

当你对单步变化充满信心后，多练习多步变化和反向百分比问题。这类题目在 Edexcel IGCSE 试卷中频繁出现，清晰、有条理的步骤会带来理想分数。

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