📚 KS3 Maths: Percentage Increase and Decrease | KS3 数学：百分数的增加与减少
Percentage increase and decrease are essential skills in the Cambridge KS3 Mathematics curriculum. They appear in word problems about prices, discounts, interest, profit, tax and data changes. This article explains the core methods, the key formulas and the exam strategies you need, step by step, with worked examples that you can copy straight into your notes.
百分数增加与减少是剑桥 KS3 数学课程中的基本技能。它们经常出现在关于价格、折扣、利息、利润、税收和数据变化的文字题中。本文将逐步讲解核心方法、关键公式和考试策略，并配有可以直接抄进笔记的例题。
1. Understanding Percentages | 理解百分数
A percentage is a number expressed as a fraction of 100. The symbol % means 'out of 100', so 45% means 45 out of 100, or 45/100. In decimal form, 45% is 0.45. Every percentage question in KS3 rests on this one idea, so make sure it is completely secure before moving on.
百分数是以 100 为分母的分数。符号 % 表示'每一百份中'，所以 45% 表示 100 份中的 45 份，即 45/100。写成小数就是 0.45。KS3 中每一道百分数题目都建立在这个概念之上，所以在继续学习之前，一定要把它彻底弄懂。
| Percentage | Decimal | Fraction | 中文含义 |
|---|---|---|---|
| 25% | 0.25 | 1/4 | 四分之一 |
| 50% | 0.5 | 1/2 | 二分之一 |
| 75% | 0.75 | 3/4 | 四分之三 |
| 10% | 0.1 | 1/10 | 十分之一 |
| 5% | 0.05 | 1/20 | 二十分之一 |
To convert a percentage to a decimal, divide by 100, which moves the decimal point two places to the left. To convert a decimal to a percentage, multiply by 100. To convert a fraction to a percentage, divide the numerator by the denominator and then multiply by 100.
把百分数化成小数要除以 100，也就是把小数点向左移动两位；把小数化成百分数要乘以 100；把分数化成百分数，先用分子除以分母，再乘以 100。
Being able to switch quickly between percentages, decimals and fractions makes all percentage calculations easier, and it allows you to handle many KS3 questions without a calculator at all.
能够快速在百分数、小数和分数之间转换，会让所有百分数计算变得更加简单，也能让你在完全不用计算器的情况下解决许多 KS3 题目。
2. Finding a Percentage of a Quantity | 求一个数量的百分之几
To find p% of a quantity N, first write p% as a decimal by dividing by 100. Then multiply the decimal by N. For example, 15% of 240 is 0.15 × 240 = 36.
求某个量 N 的 p%，先把 p% 除以 100 写成小数，然后用这个小数乘以 N。例如，240 的 15% 是 0.15 × 240 = 36。
p% of N = (p ÷ 100) × N
Always check that your answer makes sense. A useful benchmark is 10%. For 240, 10% is 24, so 15% should be 24 + 12 = 36. Since 15% is slightly more than 10%, the answer must be slightly more than 24, and 36 fits.
始终检查答案是否合理。10% 是一个很有用的基准。对于 240，10% 是 24，因此 15% 应该是 24 + 12 = 36。因为 15% 略大于 10%，答案就必须略大于 24，而 36 正好符合。
These mental building blocks are worth memorising because they make non-calculator questions fast:
这些心算分解值得记牢，因为它们能让非计算器题目做得很快：
10% = divide by 10
10% = 除以 10
5% = 10% ÷ 2
5% = 10% 的一半
1% = 10% ÷ 10
1% = 10% 再除以 10
20% = 10% × 2, and 25% = divide by 4
20% = 10% 乘以 2，25% = 除以 4
For example, to find 35% of 80: 10% is 8, so 30% is 24, and 5% is 4. Adding gives 24 + 4 = 28.
例如，求 80 的 35%：10% 是 8，所以 30% 是 24，5% 是 4。相加得到 24 + 4 = 28。
3. Percentage Increase | 百分数增加
Percentage increase describes how much a value grows compared with its original amount. The increase is always measured against the original value, not the new value. This is the single most important rule in the whole topic, and it is the source of most lost marks.
百分数增加描述一个值相对于原始值增长了多少。增加量始终是相对于原始值来衡量的，而不是相对于新值。这是整个专题中最重要的一条规则，也是大多数失分点的来源。
Percentage increase = (increase ÷ original) × 100%
find a new value directly, add the increase to the original. If a quantity increases by r% be written as original × (1 + r/100). The expression (1 + r/100) is called the multiplier.
要直接求新值，把增加量加到原始值上。如果一个量增加了 r%，它的新值也可以写成 原始值 × (1 + r/100)。其中 (1 + r/100) 称为乘数。
e: a school club had 40 members last year. This year the number of members increased by 15% 46. Using the multiplier: 40 × 1.15 = 46. The percentage increase check is (6 ÷ 40) × 100% = 15%, which confirms the answer.
例：某学校社团去年有 40 名成员，今年成员人数增加了 15%。增加量为 0.15 × 40 = 6，所以现在有 40 + 6 = 46 名成员。用乘数法计算：40 × 1.15 = 46。再用百分数增加公式检验：(6 ÷ 40) × 100% = 15%，答案得到确认。
4. Percentage Decrease | 百分数减少
Percentage decrease describes how much a value falls compared with its original amount. The decrease, like the increase, is measured against the original value.
百分数减少描述一个值相对于原始值下降了多少。与增加量一样，减少量也是相对于原始值来衡量的。
Percentage decrease = (decrease ÷ original) × 100%
If a quantity decreases by r%, its new value is original × (1 - r/100). For a 25% reduction the multiplier is 0.75, and for a 40% reduction it is 0.6.
如果一个量减少了 r%，它的新值是 原始值 × (1 - r/100)。减少 25% 时乘数是 0.75，减少 40% 时乘数是 0.6。
Example: a jacket costs £80 and is reduced by 25% in a sale. The decrease is 0.25 × 80 = £20, so the sale price is 80 - 20 = £60. Using the multiplier: 80 × 0.75 = £60. Note that a 25% rice is three quarters of the original by coincidence; it is exactly the same calculation.
例：一件夹克售价 80 英镑，促销时降价 25%。减少量为 0.25 × 80 = 20 英镑，所以售价是 80 - 20 = 60 英镑。用乘数法计算：80 × 0.75 = 60 英镑。注意，减少 25% 后价格正好是原价的四分之三，并不是巧合，而是同一个计算。
5. The Multiplier Method | 乘数法
The multiplier method turns a two-step calculation into a single multiplication, which reduces errors and saves time in exams. An increase of r% uses the multiplier (1 + r/100); a decrease of r% uses the multiplier (1 - r/100).
乘数法把两步计算变成一次乘法，既减少错误又能在考试中节省时间。增加 r% 使用乘数 (1 + r/100)，减少 r% 使用乘数 (1 - r/100)。
| Change | 变化 | Multiplier | 乘数 | Example | 示例 |
|---|---|---|
| Increase 5% | 增加 5% | 1.05 | 200 → 210 |
| Increase 30% | 增加 30% | 1.3 | 50 → 65 |
| Decrease 12% | 减少 12% | 0.88 | 150 → 132 |
| Decrease 45% | 减少 45% | 0.55 | 80 → 44 |
| Increase 100% | 增加 100% | 2 | 35 → 70 |
| Decrease 100% | 减少 100% | 0 | 35 → 0 |
When two changes happen one after another, multiply the multipliers together. A 20% increase followed by a 10% decrease gives 1.2 × 0.9 = 1.08, which is a net increase of 8%, not 10%.
当两个变化先后发生时，把乘数相乘。先增加 20% 再减少 10%，得到 1.2 × 0.9 = 1.08，即净增加 8%，而不是 10%。
This is why successive percentages cannot simply be added or subtracted. Each change acts on the new value produced by the previous change, so the base keeps moving.
这就是为什么连续的百分数变化不能直接相加减。每一次变化都作用在前一次变化产生的新值上，所以基准一直在变化。
6. Worked Examples: Prices, Discounts and VAT | 例题：价格、折扣与增值税
Example 1: a bicycle costs £240 and is sold with a 15% discount. The multiplier is 0.85, so the sale price is 240 × 0.85 = £204. Check: 10% of 240 is 24, 5% is 12, and 24 + 12 = 36, so the discount is £36 and 240 - 36 = £204.
例 1：一辆自行车售价 240 英镑，打 15% 折扣出售。乘数是 0.85，所以售价为 240 × 0.85 = 204 英镑。检验：240 的 10% 是 24，5% 是 12，合计 36，所以折扣是 36 英镑，240 - 36 = 204 英镑。
Example 2: a laptop is priced at £450 and VAT of 20% must be added. The multiplier is 1.2, so the total price is 450 × 1.2 = £540. The VAT itself is 450 × 0.2 = £90.
例 2：一台笔记本电脑标价 450 英镑，需要加上 20% 的增值税。乘数是 1.2，所以总价为 450 × 1.2 = 540 英镑。其中增值税是 450 × 0.2 = 90 英镑。
Example 3: a train ticket rises from £32 to £36. The increase is 36 - 32 = £4, so the percentage increase is (4 ÷ 32) × 100% = 12.5%. Notice that the original price, not the new price, is the denominator.
例 3：一张火车票从 32 英镑涨到 36 英镑。增加量是 36 - 32 = 4 英镑，所以百分数增加为 (4 ÷ 32) × 100% = 12.5%。注意分母是原价，而不是新价。
7. Finding the Original Amount (Reverse Percentages) | 求原值（逆向百分数）
Sometimes the question gives you the final value and the percentage change, and asks for the original value. Do not subtract the percentage from the final value; instead divide by the multiplier.
有时候题目给出最终值和百分数变化，要求求出原值。此时不要用最终值减去百分数，而要用最终值除以乘数。
Original = final value ÷ multiplier
Example: after a 20% .2 = 80 000. A common wrong answer is 96 000 × 0.8 = 76 800, which is incorrect because 20% of 76 800 is not 19 200.
例：某人口在增加 20% 后为 96 000。乘数是 1.2，所以原人口为 96 000 ÷ 1.2 = 80 000。常见错误答案是 96 000 × 0.8 = 76 800，这是不对的，因为 76 800 的 20% 并不等于 19 200。
The same idea works for decreases. If a price falls by 30% to £63, the multiplier is 0.7, so the original price was 63 ÷ 0.7 = £90. Always check by applying the percentage change to your answer.
同样的思路也适用于减少的情况。如果某价格下降 30% 后为 63 英镑，乘数是 0.7，所以原价是 63 ÷ 0.7 = 90 英镑。一定要把百分数变化代回你的答案进行检验。
8. Percentage Change in Data and Science Contexts | 数据与科学情境中的百分数变化
KS3 questions often present percentage change as a data comparison, for example comparing test scores, temperatures, masses or populations. The formula is the same, but the words change: the original value may be called the 'initial value', the 'starting amount' or the 'previous reading'.
KS3 题目常把百分数变化放在数据比较的情境中，例如比较考试成绩、温度、质量或人口。公式是一样的，只是说法变了：原值可能被称为'初始值''起始量'或'上一次读数'。
Percentage change = ((new value - original value) ÷ original value) × 100%
result means an increase and a negative result means a decrease. For example, if a student's score goes from 45 to 54, the change is 54 - 45 = 9, and (9 ÷ 45) × 100% = +20%. If the temperature falls from 25 °C to 20 °C, the change is -5, and (-5 ÷ 25) × 100% = -20%.
结果为正表示增加，结果为负表示减少。例如，某学生的成绩从 45 提到 54，变化量是 54 - 45 = 9，(9 ÷ 45) × 100% = +20%。如果气温从 25 °C 降到 20 °C，变化量是 -5，(-5 ÷ 25) × 100% = -20%。
Be careful with 'percentage points' versus 'per cent'. If a pass rate rises from 40% to 50%, that is an increase of 10 percentage points but a percentage increase of (10 ÷ 40) × 100% = 25%. Examiners love this distinction.
要注意'百分点'与'百分之几'的区别。如果及格率从 40% 升到 50%，那是上升了 10 个百分点，但百分数增加为 (10 ÷ 40) × 100% = 25%。命题人非常喜欢考查这个区别。
9. Profit, Loss, Tax and Simple Interest | 利润、亏损、税收与单利
Percentage increase and decrease sit behind several money topics in KS3. Profit percentage and loss percentage are always calculated on the cost price, never on the selling price.
百分数增加与减少是 KS3 中多个金钱类专题的基础。利润百分数和亏损百分数始终按成本价计算，绝不按售价计算。
Profit % = (profit ÷ cost price) × 100%
Example: a shop buys an item for £40 and sells it for £50. The profit is £10, so the profit percentage is (10 ÷ 40) × 100% = 25%. If the same item were sold for £34, the loss would be £6 and the loss percentage would be (6 ÷ 40) × 100% = 15%.
例：商店以 40 英镑进货，以 50 英镑售出。利润是 10 英镑，所以利润率为 (10 ÷ 40) × 100% = 25%。如果该商品只卖到 34 英镑，亏损为 6 英镑，亏损率为 (6 ÷ 40) × 100% = 15%。
Simple interest also uses a percentage of the original amount. If £500 is invested at 4% simple interest per year for 3 years, the interest is 500 × 0.04 × 3 = £60, and the total is £560. Because it is simple interest, the 4% is always calculated on the original £500, not on the growing balance.
单利同样是按原始金额的百分数计算。如果 500 英镑以每年 4% 的单利存 3 年，利息为 500 × 0.04 × 3 = 60 英镑，总额为 560 英镑。因为是单利，4% 始终按原始 500 英镑计算，而不是按不断增长的余额计算。
10. Common Mistakes to Avoid | 需要避免的常见错误
Most errors in percentage increase and decrease questions come from a small number of habits. Knowing them in advance is the fastest way to raise your accuracy.
百分数增加与减少题目中的大多数错误都源自少数几个习惯。提前认识它们，是提高准确率最快的方法。
Using the new value as the denominator. Percentage change is always measured against the original value.
把新值当作分母。百分数变化始终以原值为基准。
Adding or subtracting successive percentages. Use multipliers and multiply them instead.
把连续的百分数直接相加减。应该改用乘数并相乘。
Confusing reverse percentage questions. Divide by the multiplier; do not apply the percentage to the final value.
混淆逆向百分数题。要除以乘数，而不是把百分数作用在最终值上。
Rounding too early. Keep full accuracy in the middle of a calculation and round only the final answer.
过早取近似值。计算过程中要保留完整精度，只在最后答案处取近似。
Forgetting units or the % sign. A percentage increase has no units, but a money answer does.
忘记单位或百分号。百分数增加本身没有单位，但金额答案有单位。
11. Exam Strategies and Checking | 考试策略与检查
Read the question twice and underline the original value before you start. Write down the multiplier explicitly, because method marks are often awarded for a correct multiplier even if the arithmetic slips.
题目要读两遍，动笔前先给原值划线。把乘数明确写出来，因为即使算术出错，正确的乘数往往仍能拿到方法分。
Estimate first. If a price of £200 increases by 10%, your answer must be near £220; if you write £22 or £2 200, the estimate immediately reveals the error. This single habit catches most mistakes before you hand the paper in.
先估算。如果 200 英镑的价格增加 10%，
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