Percentages | 百分比

📚 Percentages | 百分比

Percentages are one of the most practical and frequently used mathematical tools in everyday life. From calculating discounts during a sale to understanding interest rates in a bank account, percentages help us compare and interpret numbers in a standardised way. In the Cambridge KS3 curriculum, mastering percentages involves understanding what a percentage is, converting between fractions, decimals, and percentages, finding a percentage of a quantity, and solving problems involving percentage increase and decrease. This revision guide will walk you through each key skill step by step, with clear examples and common pitfalls to avoid.

百分比是日常生活中最实用、最常用的数学工具之一。从计算打折促销的折扣到理解银行账户的利率，百分比帮助我们以标准化的方式比较和理解数字。在剑桥 KS3 课程中，掌握百分比需要理解百分比的含义，学会分数、小数和百分比之间的转换，求一个数量的百分之几，以及解决涉及百分比增加和减少的问题。这本复习指南将一步步带你学习每一项关键技能，配有清晰的示例和需要避免的常见错误。

1. Understanding Percentages | 理解百分比

The word ‘percent’ comes from the Latin ‘per centum’, meaning ‘out of one hundred’. A percentage is simply a fraction with a denominator of 100. For example, 25% means 25 out of 100, or 25/100. Thinking in terms of hundredths makes comparisons much easier. You will often see percentages used in statistics, finance, and even weather forecasts.

“百分数”（percent）这个词源自拉丁语“per centum”，意思是“每一百”。百分比就是一个分母为 100 的分数。例如，25% 表示 100 份中的 25 份，也就是 25/100。用百分之一百来思考会让比较变得容易得多。你经常会看到百分比被用在统计、金融甚至天气预报中。

2. Converting Between Percentages and Fractions | 百分比与分数的转换

To change a percentage to a fraction, write the percentage number over 100 and simplify if possible. For example, 40% = 40/100 = 2/5 after dividing numerator and denominator by 20. To convert a fraction to a percentage, find an equivalent fraction with denominator 100, or multiply by 100%. For instance, 3/5 = (3 × 20)/(5 × 20) = 60/100 = 60%. Alternatively, 3/5 × 100% = 60%. Remember to simplify fractions fully and always write percentages with the % symbol.

把百分比转换为分数时，把百分数的数字写在 100 上方，如果可以的话再进行化简。例如，40% = 40/100，分子分母同时除以 20 得到 2/5。把分数转换为百分比时，要么找一个等值的分母为 100 的分数，要么直接乘以 100%。例如，3/5 = (3 × 20)/(5 × 20) = 60/100 = 60%。或者 3/5 × 100% = 60%。记住要把分数彻底化简，并始终在百分比后面加上 % 符号。

3. Converting Between Percentages and Decimals | 百分比与小数的转换

Converting a percentage to a decimal is straightforward: divide by 100, which moves the decimal point two places to the left. For example, 67% = 0.67, and 8% = 0.08. To convert a decimal to a percentage, multiply by 100, moving the decimal point two places to the right. So 0.45 = 45%, and 1.2 = 120%. This skill is essential when using a calculator for percentage problems, as the percentage button often works by converting percentages to decimals automatically.

把百分比转换为小数非常简单：除以 100，也就是把小数点向左移动两位。例如，67% = 0.67，8% = 0.08。把小数转换为百分比时，乘以 100，把小数点向右移动两位。所以 0.45 = 45%，1.2 = 120%。在用计算器解决百分比问题时，这项技能非常重要，因为百分号按钮通常会自动把百分比转换为小数。

4. Common Conversions to Memorise | 需要记住的常见转换

Some percentage-fraction-decimal equivalents appear so often that memorising them will save you time. For instance, 50% = 1/2 = 0.5, 25% = 1/4 = 0.25, 75% = 3/4 = 0.75, 10% = 1/10 = 0.1, 20% = 1/5 = 0.2, and 33¹⁄₃% = 1/3 ≈ 0.333. Being able to recall these instantly helps with mental arithmetic and checking your answers in exams.

有些百分比、分数、小数的等值转换出现得非常频繁，把它们记住可以节省你的时间。例如，50% = 1/2 = 0.5，25% = 1/4 = 0.25，75% = 3/4 = 0.75，10% = 1/10 = 0.1，20% = 1/5 = 0.2，以及 33¹⁄₃% = 1/3 ≈ 0.333。能够迅速想起这些转换有助于心算，也便于在考试中检查答案。

5. Finding a Percentage of a Quantity | 求一个数量的百分之几

To find a percentage of a quantity without a calculator, break it into simpler parts. For example, to find 15% of 80, you can find 10% (8) and 5% (half of 10% = 4), then add them: 8 + 4 = 12. With a calculator, convert the percentage to a decimal and multiply: 15% of 80 = 0.15 × 80 = 12. You can also use the fraction method: 15% = 15/100, so (15/100) × 80 = 1200/100 = 12. Always check your answer is reasonable—15% is a bit less than a fifth, and a fifth of 80 is 16, so 12 makes sense.

不用计算器求一个数量的百分之几时，可以把它拆成更简单的部分。例如，要求 80 的 15%，可以先求 10%（8）和 5%（10% 的一半是 4），然后把它们相加：8 + 4 = 12。有计算器时，把百分比转换成小数再乘以数量：80 的 15% = 0.15 × 80 = 12。你也可以用分数方法：15% = 15/100，所以 (15/100) × 80 = 1200/100 = 12。一定要检查答案是否合理——15% 比五分之一略少，而 80 的五分之一是 16，所以 12 是合理的。

6. Percentage Increase | 百分比增加

Percentage increase means adding a percentage of the original amount to the original amount. For example, increasing £200 by 30% means finding 30% of £200 = £60, then adding: £200 + £60 = £260. Using a multiplier is faster: add the percentage increase to 100% to get the new total percentage, then convert to a decimal. For a 30% increase, the multiplier is 100% + 30% = 130% = 1.3. Multiply the original amount by 1.3: £200 × 1.3 = £260. Multipliers are very useful for repeated percentage changes.

百分比增加指的是把原始数量的百分之几加到原始数量上。例如，把 200 英镑增加 30%，就是先求出 200 英镑的 30% 为 60 英镑，然后相加：200 英镑 + 60 英镑 = 260 英镑。使用乘数会更快：把增加的百分比与 100% 相加，得到新的总百分比，再转换成小数。对于 30% 的增加，乘数是 100% + 30% = 130% = 1.3。用原始金额乘以 1.3：200 英镑 × 1.3 = 260 英镑。乘数对于重复的百分比变化非常有用。

7. Percentage Decrease | 百分比减少

Percentage decrease works similarly, but you subtract the percentage from the original. To decrease £200 by 30%, find 30% of £200 = £60, then subtract: £200 − £60 = £140. The multiplier method: subtract the percentage decrease from 100%. For 30% decrease, 100% − 30% = 70% = 0.7. Multiply: £200 × 0.7 = £140. Be careful not to use the wrong multiplier—a common mistake is using 0.3 instead of 0.7, which would give you just the amount taken away, not the reduced amount.

百分比减少的计算方法类似，只是从原始数量中减去相应的百分比。要把 200 英镑减少 30%，先求 200 英镑的 30% 为 60 英镑，然后相减：200 英镑 − 60 英镑 = 140 英镑。乘数法则是：用 100% 减去减少的百分比。对于减少 30%，100% − 30% = 70% = 0.7。相乘：200 英镑 × 0.7 = 140 英镑。注意不要用错乘数——一个常见的错误是用了 0.3 而不是 0.7，那样你得到的只是被减去的部分，而不是减少后的数量。

8. Reverse Percentages | 逆向百分比

Reverse percentage problems ask you to find the original amount before a percentage change, when you know the final amount. For example, if a price of a jacket after a 20% decrease is £64, what was the original price? A 20% decrease means the final price is 80% of the original (100% − 20%). So 80% of the original = £64. Written as an equation: 0.8 × original = 64, so original = 64 ÷ 0.8 = £80. Always check by applying the forward percentage to your answer: 20% off £80 is £64. Using the unitary method (finding 1% first) is another reliable approach.

逆向百分比问题要求你在知道最终数量的情况下，找出发生百分比变化之前的原始数量。例如，一件夹克降价 20% 后售价 64 英镑，原价是多少？降价 20% 意味着最终价格是原来的 80%（100% − 20%）。所以原价的 80% = 64 英镑。写成等式：0.8 × 原价 = 64，所以原价 = 64 ÷ 0.8 = 80 英镑。一定要把得到的答案做正向百分比验算：80 英镑打八折正好是 64 英镑。使用归一法（先求 1%）也是一个可靠的方法。

9. Expressing One Quantity as a Percentage of Another | 一个量占另一个量的百分比

To write one number as a percentage of another, form a fraction and multiply by 100%. For instance, if you scored 18 out of 25 in a test, the percentage is (18/25) × 100% = 0.72 × 100% = 72%. If comparing a part to a whole, make sure the part is the numerator and the whole is the denominator. This skill is extremely useful for comparing proportions, such as finding what percentage of a class are girls or what percentage of your income you spend on food.

要把一个数表示成另一个数的百分比，先写成分数，再乘以 100%。例如，如果你在一次考试中得了 25 分里的 18 分，百分比就是 (18/25) × 100% = 0.72 × 100% = 72%。如果是部分与整体比较，一定要把部分作为分子，整体作为分母。这项技能对于比较比例非常有用，比如求一个班级里女生占百分之多少，或者你的收入中有百分之多少花在食物上。

10. Percentage Change | 百分比变化

Percentage change is used to describe how much a quantity has increased or decreased compared to the original value. The formula is: percentage change = (change ÷ original amount) × 100%. For example, if a town’s population grows from 20,000 to 23,500, the change is 3,500, so the percentage increase = (3500 ÷ 20000) × 100% = 17.5%. If it decreases, the change is negative, but we usually describe it as a percentage decrease. Be careful with the order: it is always new value minus original value over original value.

百分比变化用于描述一个量相对于原始值增加或减少了多少。公式是：百分比变化 = （变化量 ÷ 原始数量） × 100%。例如，如果一个城镇的人口从 20,000 增长到 23,500，变化量是 3,500，那么百分比增长 = (3500 ÷ 20000) × 100% = 17.5%。如果数量减少，变化量就是负数，但我们通常用百分比减少来描述。注意顺序：始终是（新值 − 原值）除以原值。

11. Simple Interest | 单利

Simple interest is a way of calculating interest earned or paid on a lump sum of money, where the interest is a fixed percentage of the original principal each year. The formula is I = P × R × T, where I is the interest, P is the principal, R is the rate per annum as a decimal, and T is the time in years. For example, if you invest £500 at 4% per annum simple interest for 3 years, the interest is 500 × 0.04 × 3 = £60. The total amount after 3 years is £500 + £60 = £560. This concept reinforces multiplying percentages and handling decimal rates.

单利是一种计算一笔钱赚取或支付的利息的方法，每年的利息是原始本金的一个固定百分比。公式是 I = P × R × T，其中 I 是利息，P 是本金，R 是年利率（以小数表示），T 是时间（以年为单位）。例如，如果你以年利率 4% 的单利投资 500 英镑，为期 3 年，利息就是 500 × 0.04 × 3 = 60 英镑。3 年后的总金额是 500 英镑 + 60 英镑 = 560 英镑。这个概念巩固了百分比相乘以及处理小数利率的能力。

12. Percentages in Real-Life Contexts | 现实生活中的百分比

Percentages appear in countless real-life scenarios: VAT (value-added tax) is often a percentage added to prices, tips in restaurants are often calculated as a percentage of the bill, and statistics such as pass rates or population growth are communicated in percentages. Interpreting newspaper headlines like ‘crime fell by 15%’ requires a solid understanding of percentage decrease. When shopping, a 20% discount followed by an extra 10% off the reduced price is not the same as a 30% discount off the original, because the second discount is taken on a smaller amount. Always think critically about how percentages are applied.

百分比出现在无数现实生活场景中：增值税（VAT）通常是加在价格上的一个百分比，餐厅小费常按账单的百分比来计算，考试通过率或人口增长等统计数据也用百分比来传达。理解像“犯罪率下降 15%”这样的报纸标题需要对百分比减少有扎实的理解。购物时，先打八折、再在折后价上打九折，这和原价打七折是不一样的，因为第二个折扣是在较小的金额上计算的。对于百分比的应用方式，要始终进行批判性思考。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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Percentages | 百分比