📚 Calculating Percentage Increase and Decrease | 百分比增减计算
Percentages are everywhere – from shop discounts and tax to population growth and savings interest. Understanding how to calculate percentage increase and decrease is a core skill at KS3. This article will take you through using multipliers, working backwards to find the original amount, and applying percentage change in real-life situations. By the end, you'll be confident in handling all kinds of percentage change problems and avoiding common mistakes.
百分比无处不在——从商店折扣和税收，到人口增长和储蓄利息。掌握如何计算百分比增减是 KS3 阶段的重要技能。本文将带你学习使用乘数、逆向计算原始值，以及在现实情境中应用百分比变化。读完本文，你将能自信地处理各种百分比变化问题，并避开常见错误。
1. Understanding Percentage Change | 理解百分比变化
A percentage change tells us how much a quantity has increased or decreased relative to its original value, expressed as a percentage. If a quantity goes up, we have a percentage increase; if it goes down, we have a percentage decrease. The formula for percentage change is:
百分比变化表示某一数量相对于原始值增加或减少了多少，并以百分数表示。如果数量上升，就是百分比增加；如果下降，就是百分比减少。百分比变化的公式是：
Percentage Change = ( Change ÷ Original Amount ) × 100%
This means we first work out the difference between the new and original values, then divide by the original, and finally multiply by 100 to turn it into a percentage. The change could be positive (increase) or negative (decrease).
这意味着我们首先计算新值与原始值之间的差值，然后除以原始值，最后乘以 100 转化为百分数。变化可以是正值（增加）或负值（减少）。
ew from 50 cm to 70 cm, the change is 20 cm, so the percentage increase is (20 ÷ 50) × 100% = 40%. Remember that the denominator must always be the original amount, not the new one.
例如，一株植物从 50 厘米长到 70 厘米，变化量为 20 厘米，因而百分比增加为 (20 ÷ 50) × 100% = 40%。要记住分母必须是原始值，而不是新值。
2. The Multiplier Method | 乘数法
At KS3, one of the most powerful tools you'll learn is the multiplier method. Instead of working out the change and then adding or subtracting, you can simply multiply the original amount by a single number. Finding the right multiplier depends on whether there is an increase or a decrease:
在 KS3 阶段，你会学到的最强大的工具之一就是乘数法。不需要先算出变化量再加或减，你可以直接将原始值乘以一个数字。找到正确的乘数取决于是增加还是减少：
- For an increase: Multiplier = 1 + (percentage increase ÷ 100) → 增：乘数 = 1 + (百分比增加 ÷ 100)
- For a decrease: Multiplier = 1 − (percentage decrease ÷ 100) → 减：乘数 = 1 − (百分比减少 ÷ 100)
For example, a 15% increase means the new amount is 115% of the original, so the multiplier is 1.15. A 20% decrease means you keep 80%, so the multiplier is 0.80.
例如，增加 15% 意味着新值是原始值的 115%，乘数就是 1.15。减少 20% 意味着保留 80%，乘数就是 0.80。
This method is much faster when you have a list of changes or when working with money. It also makes reverse percentage calculations much easier, as we will see later.
当你有一系列变化或处理货币问题时，这种方法要快得多。它还能让逆向百分比计算变得更加简单，稍后我们会看到。
3. Calculating Percentage Increase | 计算百分比增加
To increase an amount by a given percentage, you can either find the increase and add it on, or use the multiplier. Both methods give the same answer. Let's try an example: A shop buys a jacket for £40 and wants to sell it with a 25% profit. Find the selling price.
要给一个数值增加某个百分比，你可以先算出增加量再加到原值上，或者使用乘数。两种方法结果相同。我们来试一个例子：一家商店以 40 英镑买入一件夹克，想以 25% 的利润出售，求售价。
The traditional method: 25% of £40 = 0.25 × 40 = £10, so selling price = £40 + £10 = £50. With the multiplier: 1 + 0.25 = 1.25, then new price = 1.25 × £40 = £50.
传统方法：40 英镑的 25% = 0.25 × 40 = 10 英镑，所以售价 = 40 + 10 = 50 英镑。用乘数法：1 + 0.25 = 1.25，然后新价格 = 1.25 × 40 = 50 英镑。
Always check that your answer makes sense. An increase of 25% should give a price higher than the original, and indeed £50 is greater than £40.
始终检查答案是否合理。增加 25% 应该给出比原价高的价格，而 50 英镑确实比 40 英镑大。
4. Calculating Percentage Decrease | 计算百分比减少
For a percentage decrease, the process is similar. Subtract the percentage from 100% to find what fraction remains, then multiply. Suppose a tennis racket is reduced by 30% from £120 in a sale. What is the sale price?
对于百分比减少，过程类似。从 100% 中减去百分比，得出剩余的比例，然后相乘。假设网球拍原价 120 英镑，打折 30%，售价是多少？
With the multiplier: 100% − 30% = 70%, so multiplier = 0.70. Sale price = 0.70 × £120 = £84. Alternatively, 30% of 120 = £36, then 120 − 36 = £84. Again, both methods match.
使用乘数：100% − 30% = 70%，乘数为 0.70。售价 = 0.70 × 120 = 84 英镑。或者，120 的 30% 是 36 英镑，120 − 36 = 84 英镑。同样，两种方法一致。
An important check: the sale price must be lower than £120, which it is. Using the multiplier also helps prevent common mistakes like adding the percentage when you should subtract.
关键检查：售价必须低于 120 英镑，确实如此。使用乘数还能防止类似本应减去却加成百分比的常见错误。
5. Using Multipliers in a Table | 用表格呈现乘数
Organising percentage changes in a table can help you spot patterns and work more systematically. Below is a table showing common percentage increases and decreases with their multipliers:
用表格整理百分比变化有助于发现规律，更系统地解题。以下表格展示了常见的百分比增减及其乘数：
| Percentage Change | Increase/Decrease | Multiplier |
|---|---|---|
| +10% | Increase | 1.10 |
| −15% | Decrease | 0.85 |
| +5.5% | Increase | 1.055 |
| −2.5% | Decrease | 0.975 |
| +100% | Increase (doubling) | 2.00 |
Notice that the multiplier for an increase is always greater than 1, and for a decrease it is between 0 and 1. This is a useful way to check your work: if your multiplier for an increase is less than 1, you've made a mistake.
注意，增加对应的乘数总是大于 1，而减少对应的乘数介于 0 和 1 之间。这是一个检查错误的好方法：如果求增加时的乘数却小于 1，那你就算错了。
6. Finding the Original Amount After a Percentage Increase | 在百分比增加后求原始值
Sometimes we know the final amount after a percentage increase and need to work backwards to find the original. This is called a reverse percentage problem. For example: A bike now costs £345 after a 15% increase. What was the original price?
有时我们知道百分比增加后的最终值，需要逆向求出原始值。这被称为逆向百分比问题。例如：一辆自行车涨价 15% 后现价为 345 英镑，原来的价格是多少？
The key is to recognise that £345 represents 115% of the original (100% + 15%). So 115% = £345. To find 1%, divide by 115: £345 ÷ 115 = £3. Then 100% = £3 × 100 = £300. Using the multiplier approach: original × 1.15 = 345, so original = 345 ÷ 1.15 = £300.
关键在于认识到 345 英镑是原始值的 115%（100% + 15%）。因此 115% = 345 英镑。要得到 1%，除以 115：345 ÷ 115 = 3 英镑。那么 100% = 3 × 100 = 300 英镑。利用乘数法：原值 × 1.15 = 345，所以原值 = 345 ÷ 1.15 = 300 英镑。
Always check: 15% of £300 = £45, and £300 + £45 = £345. It works!
检查一下：300 英镑的 15% 是 45 英镑，300 + 45 = 345 英镑。正确！
7. Finding the Original Amount After a Percentage Decrease | 在百分比减少后求原始值
Similarly, after a decrease you can find the original by dividing by the multiplier. Suppose a laptop is in a 20% off sale and now costs £512. What was the original price?
类似地，在减少之后，你可以通过除以乘数来求原值。假设笔记本电脑打八折后售价为 512 英镑，原价是多少？
Here £512 is 80% of the original (100% − 20%). So multiplier = 0.80. Original = £512 ÷ 0.80 = £640. Or break it down: 80% = 512, so 1% = 512 ÷ 80 = £6.40, then 100% = £640. Both methods are fine, but dividing by the multiplier is quicker.
这里的 512 英镑是原价的 80%（100% − 20%），乘数为 0.80。原价 = 512 ÷ 0.80 = 640 英镑。或者逐步来：80% = 512，所以 1% = 512 ÷ 80 = 6.40 英镑，100% = 640 英镑。两种方法都可以，但除以乘数更快。
Reverse percentage questions often confuse students because they are tempted to add 20% to £512, which gives £614.40 – wrong! You must always remember the percentage relates to the unknown original, not the known final amount.
逆向百分比问题常让学生困惑，因为他们容易给 512 加上 20%，得到 614.40 英镑——这是错误的！必须牢记百分比对应的是未知的原始值，而不是已知的最终值。
8. Repeated Percentage Change | 连续百分比变化
ce, such as investment growth over several years. If a population of 20,000 increases by 5% every year, after three years it is not simply 20,000 + 3 × 5%. Instead, you multiply by 1.05 each year:
有时百分比变化会发生多次，比如几年间的投资增长。如果 20,000 的人口每年增长 5%，三年后并不是简单的 20,000 + 3 × 5%。相反，你需要每年乘以 1.05：
Year 1: 20,000 × 1.05 = 21,000
Year 2: 21,000 × 1.05 = 22,050
Year 3: 22,050 × 1.05 = 23,152.50
5³ = 20,000 × 1.157625 = 23,152.50. This is the foundation of compound interest, which you'll study later.
或者更高效地，n 年后，最终值 = 原始值 × (乘数)ⁿ。这里 20,000 × 1.05³ = 20,000 × 1.157625 = 23,152.50。这就是复利的基础，你以后会学到。
This method also works for repeated decreases. A car value of £15,000 that depreciates by 12% per year: multiplier = 0.88. After 2 years, value = 15,000 × 0.88² = 15,000 × 0.7744 = £11,616.
这种方法同样适用于连续减少。一辆价值 15,000 英镑的汽车每年折旧 12%：乘数 = 0.88。两年后，价值 = 15,000 × 0.88² = 15,000 × 0.7744 = 11,616 英镑。
9. Simple Interest vs Compound Growth | 单利与复利增长
At KS3, you may encounter simple interest as a simpler version of percentage increase. With simple interest, you calculate the interest once on the original amount and then multiply by the number of years. The formula is:
在 KS3 阶段，你可能会遇到单利，它比复利简单。在单利下，你只对原始本金计算一次利息，然后乘以年数。公式是：
Simple Interest = P × r × t
nd t is the time in years. The total amount after t years is P + (P × r × t). For example, £2000 at 4% simple interest for 3 years gives interest = 2000 × 0.04 × 3 = £240. Total = £2240.
其中 P 是本金，r 是年利率的小数形式，t 是以年为单位的时间。t 年后的总金额为 P + (P × r × t)。例如，2000 英镑按 4% 单利存 3 年，利息 = 2000 × 0.04 × 3 = 240 英镑。总额 = 2240 英镑。
This is linear growth, while repeated percentage increase (compound interest) is exponential. It's important to know which one a problem is asking for. Usually, if the question says "simple interest", use the formula above; if it says "compound" or "increases by x% each year", use the multiplier repeatedly.
这是线性增长，而连续百分比增加（复利）是指数增长。识别题目要求哪种很重要。通常，如果题目说“单利”，就用上述公式；如果说“复利”或“每年增长 x%”，就重复使用乘数。
10. Real-Life Applications | 现实应用
nt errors, and even biology (bacteria growth). Understanding multipliers helps you make quick decisions, like comparing discounts: is a 20% reduction followed by an extra 10% off the same as a 30% reduction? (It's not – 20% off then 10% off gives a multiplier of 0.8 × 0.9 = 0.72, i.e., a 28% overall discount.)
百分比增减出现在许多现实情境中：打折促销、增值税和服务费、加薪、通货膨胀、测量误差，甚至生物学（细菌繁殖）。理解乘数能帮助你快速决策，比如比较折扣：先打八折再打九折相当于打七折吗？（不是——两次折扣的乘数为 0.8 × 0.9 = 0.72，即总共打了二八折。）
Another common trap is adding percentages when combining changes. If a value goes up by 50% then down by 50%, it does not return to the original. From 100, up 50% gives 150, down 50% gives 75. Always use multipliers to get the correct final value.
另一个常见陷阱是在组合变化时将百分比相加。如果一个值先涨 50% 再跌 50%，它并不会回到原值。比如从 100 开始，涨 50% 得 150，再跌 50% 得 75。一定要用乘数来得出正确的最终值。
11. Common Mistakes and How to Avoid Them | 常见错误与如何避免
Here are the most frequent errors students make with percentage change, and tips to steer clear of them:
下面是学生在处理百分比变化时最常犯的错误，以及避免它们的技巧：
- Using the wrong denominator: Always divide the change by the original amount, not the new one. → 用错分母：始终用变化量除以原始值，而不是新值。
- Confusing percentage points with percentage change: A rise from 10% to 15% is a 5 percentage point increase, but a 50% increase relative to 10%. → 混淆百分点与百分比变化：从 10% 涨到 15%，是增加了 5 个百分点，但相对于 10% 是增长了 50%。
- Adding instead of multiplying for repeated change: Use the power method, not addition. → 在连续变化时相加而不是相乘：用幂的方法，而不是加法。
- Reversing the multiplier incorrectly: To go from final to original, divide by the multiplier, not multiply. → 错误地逆向使用乘数：要从最终值求原始值，应除以乘数，而不是乘。
- Forgetting that percentages are out of 100: Always convert to a decimal by dividing by 100 before using in calculations. → 忘记百分比是以 100 为基准的：在计算中使用前，一定要先除以 100 转化为小数。
Writing down each step and checking with estimation can catch many mistakes before they become final answers.
写下每一步并用估算检查，可以在答案定型前发现许多错误。
12. Summary and Key Points | 总结与要点
To master percentage increase and decrease, remember these essentials:
要掌握百分比增减，请记住这些要点：
- Percentage change = (change ÷ original) × 100% → 百分比变化 = (变化量 ÷ 初始值) × 100%
- Increase multiplier = 1 + (percentage ÷ 100) → 增加乘数 = 1 + (百分比 ÷ 100)
- Decrease multiplier = 1 − (percentage ÷ 100) → 减少乘数 = 1 − (百分比 ÷ 100)
- Original amount after change = final value ÷ multiplier → 变化后的原始值 = 最终值 ÷ 乘数
- For repeated change, raise the multiplier to the power of the number of periods → 对于连续变化，将乘数乘以周期次数的幂
Practice with a variety of problems, including word problems, and you will soon find percentage change becomes second nature. Use these methods not only in your maths classroom but whenever you shop or read the news – being percentage-smart is a life skill.
通过练习各种题型，包括应用题，你很快就会发现百分比变化变得自然而然。不仅在数学课堂上使用这些方法，在购物或看新闻时也要用——精通百分比是一项生活技能。
Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com
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