📚 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
a fraction, write the percentage number over 100 and simplify if possible. For example, 40% tion 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
ward: divide by 100, which moves the decimal point two places to the left. For example, 67% = 0.67, and 8% ercentage, multiply by 100, moving the decimal point two places to the right. So 0.45 = 45%, and 1.2 = 120% 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
imal 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% 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
e 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% 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
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
e 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— 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
age 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% portions, 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
o the original value. The formula is: percentage change = (change ÷ original amount) × 100%. For example, if a town' m 20,000 to 23,500, the change is 3,500, so the percentage increase = (3500 ÷ 20000) × 100% = 17.5% e 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
, 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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