P126: Mastering Percentages – Increase, Decrease and Reverse | P126：精通百分比

📚 P126: Mastering Percentages – Increase, Decrease and Reverse | P126：精通百分比 – 增加、减少与逆向计算

Page 126 of the Cambridge KS3 Mathematics textbook introduces a crucial set of skills: calculating percentage increases, decreases, and finding the original amount before a percentage change. These techniques are not only fundamental for your Checkpoint exam but also essential for real-life situations such as shopping discounts, tax calculations, and interpreting data. In this article, we will break down each concept step by step, provide clear examples, and highlight common pitfalls to avoid. By the end, you will be confident in tackling any percentage problem that comes your way.

剑桥初中数学教材第126页讲解了一组关键技能：计算百分比增加、减少，以及求百分比变化前的原始值。这些方法不仅是Checkpoint考试的基础，对现实生活中的购物折扣、税务计算和数据分析也至关重要。本文将逐步拆解每个概念，配以清晰的例子，并指出需要避免的常见错误。学完之后，你将自信地应对各类百分比问题。

1. What Are Percentages? | 什么是百分比？

A percentage is a way of expressing a number as a fraction of 100. The word itself comes from ‘per cent’, meaning ‘out of a hundred’. For example, 35% means 35 out of 100, or 35/100.

百分比是一种将数字表示为百分之几的方式。”percent”一词源自”per cent”，意为”每百”。例如，35% 表示 100 份中的 35 份，即 35/100。

We use percentages everywhere: in test scores (e.g. 78% on a maths test), in shop sales (‘20% off’), and in statistics. Understanding them allows you to compare proportions easily, even when the totals are different.

百分比无处不在：考试成绩（如数学测验得78%）、商店促销（”打八折”）以及统计数据中都会用到。理解了百分比，即使总量不同，也能轻松比较比例。

To convert a percentage to a decimal, divide by 100 (move the decimal point two places to the left). To convert a decimal or fraction to a percentage, multiply by 100.

将百分比转换为小数时，除以100（小数点左移两位）。将小数或分数转换为百分比时，乘以100。

2. Percentage Increase | 百分比增加

A percentage increase happens when an amount goes up by a certain percent. To calculate the new amount after an increase, first find the increase amount, then add it to the original.

百分比增加指某个数值以一定百分比上升。计算增加后的新值时，先求出增加量，再加到原始值上。

Formula: Increase = Original × (Percentage ÷ 100). New amount = Original + Increase.

公式：增加量 = 原始值 × (百分比 ÷ 100)。新值 = 原始值 + 增加量。

New = Original × (1 + r)

where r is the percentage as a decimal (e.g. 15% becomes r = 0.15).

其中 r 是百分比写成的小数（例如15% ⇒ r = 0.15）。

Example: A bike costing £240 is increased by 12%. Increase = 240 × 0.12 = £28.80. New price = 240 + 28.80 = £268.80. Using the multiplier: 240 × 1.12 = £268.80.

例子：一辆自行车原价240英镑，涨价12%。增加量 = 240 × 0.12 = 28.80英镑。新价格 = 240 + 28.80 = 268.80英镑。使用乘数法：240 × 1.12 = 268.80英镑。

3. Percentage Decrease | 百分比减少

A percentage decrease takes an amount down by a certain percent. The method is similar: find the decrease amount and subtract it from the original.

百分比减少指数值以一定百分比下降。方法类似：先求出减少量，再从原始值中减去。

Formula: Decrease = Original × (Percentage ÷ 100). New amount = Original – Decrease.

公式：减少量 = 原始值 × (百分比 ÷ 100)。新值 = 原始值 – 减少量。

New = Original × (1 – r)

Again, r is the decimal form of the percentage.

同样，r 是百分比的小数形式。

Example: A laptop priced at £650 is reduced by 18%. Decrease = 650 × 0.18 = £117. Sale price = 650 – 117 = £533. Or directly: 650 × (1 – 0.18) = 650 × 0.82 = £533.

例子：一台笔记本电脑标价650英镑，降价18%。减少量 = 650 × 0.18 = 117英镑。折后价 = 650 – 117 = 533英镑。或直接计算：650 × (1 – 0.18) = 650 × 0.82 = 533英镑。

4. Using Multipliers | 使用乘数

A multiplier is a single number that you multiply the original amount by to find the final amount after a percentage change. This is a very efficient method and is essential when dealing with reverse percentages and repeated changes.

乘数是一个单独的数，原始值只需乘以它就可以直接得到百分比变化后的最终值。这种方法非常高效，处理逆向百分比和重复变化时尤为重要。

For an increase of p%, the multiplier is (1 + p/100). For a decrease, it is (1 – p/100).

对于增加 p%，乘数为 (1 + p/100)。对于减少，乘数为 (1 – p/100)。

Change	Percentage	Multiplier
Increase	5%	1.05
Increase	40%	1.40
Decrease	10%	0.90
Decrease	75%	0.25

Always check: if the percentage is more than 100%, the multiplier for an increase will be greater than 2 (e.g. 150% increase ⇒ multiplier = 2.5). This helps you spot mistakes quickly.

注意检查：如果百分比大于100%，增加情况的乘数会大于2（例如150% 增加 ⇒ 乘数为2.5）。这能帮你迅速发现错误。

5. Finding the Original Amount (Reverse Percentages) | 逆向求原值

Sometimes we know the final amount after a percentage change and need to work backwards to find the original. This is called a reverse percentage calculation. The key is to identify the multiplier that has already been applied and then divide instead of multiply.

有时候我们已知百分比变化后的最终值，需要倒推求出原始值。这就是逆向百分比计算。关键是识别出已经用过的乘数，然后用除法代替乘法。

If the final amount (New) was obtained by multiplying the Original by a multiplier m, then Original = New ÷ m.

如果最终值（新值）是由原始值乘以乘数 m 得到的，那么原始值 = 新值 ÷ m。

Example: A coat is sold for £102 after a 20% increase. What was the original price? A 20% increase gives a multiplier of 1.20. So Original = 102 ÷ 1.20 = £85.

例子：一件外套涨价20%后售价102英镑。原价是多少？20%的增加对应的乘数是1.20。因此原价 = 102 ÷ 1.20 = 85英镑。

For a decrease: A phone now costs £270 after a 10% discount. Multiplier = 1 – 0.10 = 0.90. Original = 270 ÷ 0.90 = £300.

减少的例子：一部手机折后售价270英镑（打了10%折扣）。乘数 = 1 – 0.10 = 0.90。原价 = 270 ÷ 0.90 = 300英镑。

Common mistake: Students often try to subtract 20% from the new amount, but that is incorrect because the percentage is applied to the original, not the final amount. Always use division by the multiplier.

常见错误：不少学生试图从新值中直接减去20%，但这是错的，因为百分比是针对原始值的，而非最终值。务必使用除以乘数的方法。

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

When a quantity undergoes the same percentage change more than once, we can apply the multiplier repeatedly. Do not simply add or subtract the percentages; use powers of the multiplier.

当一个量经历多次相同的百分比变化时，我们可以重复使用乘数。切忌直接把百分比相加或相减，而是使用乘数的幂。

For n years of the same percentage increase, Final = Original × (multiplier)ⁿ.

经过 n 次相同的百分比增加，最终值 = 原始值 × (乘数)ⁿ。

Example: A population of 10,000 grows by 3% each year for 4 years. Multiplier = 1.03. Final population = 10,000 × (1.03)⁴ ≈ 10,000 × 1.1255 = 11,255.

例子：一个10000的人口每年增长3%，连续增长4年。乘数 = 1.03。最终人口 = 10,000 × (1.03)⁴ ≈ 10,000 × 1.1255 = 11,255。

If the change alternates, apply the respective multipliers in sequence. For instance, a 10% increase followed by a 10% decrease does not return you to the original, because the base changes. The overall multiplier is 1.10 × 0.90 = 0.99, a net 1% decrease.

如果变化交替，则依次应用相应的乘数。例如先增加10%再减少10%并不会回到原值，因为基数变了。总乘数为 1.10 × 0.90 = 0.99，相当于净减少1%。

7. Common Mistakes to Avoid | 常见错误

Mistake 1: Confusing ‘percentage of’ with ‘percentage change’. Finding 20% of a number is different from increasing a number by 20%. Always read the problem carefully.

错误1：混淆”求百分比”与”百分比变化”。求一个数的20%与将这个数增加20%完全不同。务必仔细读题。

Mistake 2: Using the wrong multiplier for reverse problems. When working backwards, remember to divide by the multiplier, not multiply. If a price including 20% tax is £120, dividing by 1.20 gives £100, but multiplying £120 by 0.80 gives £96 – which is wrong.

错误2：逆向问题中用错乘数。倒推时记住是除以乘数，而不是乘。若含20%税的价款为120英镑，除以1.20得100英镑，但如果用120乘以0.80则得到96英镑——这是错的。

Mistake 3: Adding or subtracting percentages directly when dealing with successive changes. A 50% rise followed by a 50% cut does not cancel out; the overall effect is a 25% loss.

错误3：在处理连续变化时直接将百分比相加或相减。先上涨50%再下降50%并不抵消，总效果是损失了25%。

Mistake 4: Ignoring the decimal place when converting percentages. 7% is 0.07, not 0.7. Misplacing the decimal can lead to answers ten times too large or too small.

错误4：转换百分比时忽略小数位。7% 是 0.07，不是 0.7。小数点放错位置会导致答案放大或缩小十倍。

8. Real-Life Applications | 实际应用

Percentage change skills are used daily: calculating tips, understanding interest rates, interpreting news about stock market changes, and figuring out sale prices. In science, percentage error and yield are also important concepts.

百分比变化的技能天天都会用到：计算小费、理解利率、解读股市波动新闻，以及算出折后价格。在科学中，百分比误差和产率也是重要概念。

For investments, compound interest uses repeated percentage increase. If you save £500 at an annual compound interest rate of 4%, after 3 years the amount is 500 × (1.04)³ ≈ £562.43.

在投资中，复利就是用重复百分比增加。如果你以4%的年复利存入500英镑，3年后的金额为 500 × (1.04)³ ≈ 562.43英镑。

Retailers often use percentage change to set profit margins. If a shop buys an item for £40 and wants a 30% profit on the cost price, the selling price should be 40 × 1.30 = £52.

零售商常用百分比变化设定利润率。如果商店以40英镑购入商品，希望有30%的成本利润率，售价应为 40 × 1.30 = 52英镑。

9. Practice Problems | 练习题

Test your understanding with these problems. Try to use multipliers where possible and check your answers using both methods.

用以下题目检测你的理解。尽量使用乘数法，并用两种方法核对你的答案。

A video game originally costs £56. In a sale, it is reduced by 15%. Find the sale price. (Answer: £47.60)

一款电子游戏原价56英镑，促销降价15%。求折后价。（答案：47.60英镑）
A restaurant bill of £80 includes a 12.5% service charge. What was the bill before the service charge? (Answer: £71.11 to the nearest penny)

一张餐馆账单80英镑，已含12.5%的服务费。服务费前的账单金额是多少？（答案：71.11英镑，保留到便士）
The population of a town is 24,500. It decreases by 6% one year and then increases by 8% the next year. Find the new population. (Answer: approximately 24,915)

某镇人口24,500。第一年减少6%，第二年增加8%。求新人口。（答案：约24,915）
A bicycle value depreciates by 20% each year. If it is worth £180 now, how much was it worth 2 years ago? (Answer: £281.25)

一辆自行车每年贬值20%。若现在价值180英镑，两年前的价值是多少？（答案：281.25英镑）

Remember: always identify whether you are finding a new amount or going back to an original, and choose the correct operation.

记住：始终分清你是要求新值还是倒推原值，并选择正确的运算。

10. Summary and Key Points | 总结与要点

Mastering percentage change means understanding multipliers, working forwards and backwards confidently, and avoiding common pitfalls. The three essential formulae are:

掌握百分比变化意味着理解乘数，能正向、逆向从容计算，并避免常见误区。三个核心公式是：

New = Original × (1 ± r) | Original = New ÷ (1 ± r)

With these tools, page 126 of your textbook no longer needs to be a challenge. Keep practising with real-world examples, and soon percentage calculations will become second nature.

掌握了这些工具，你教材的第126页不再有难题。多用现实例子练习，百分比计算很快就会变得得心应手。

Published by TutorHao | Maths Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

P126: Mastering Percentages – Increase, Decrease and Reverse | P126：精通百分比 – 增加、减少与逆向计算