📚 Mastering Percentages: From Fractions to Real-World Applications | 掌握百分比：从分数到实际应用
Percentages are everywhere – from store discounts and bank interest to test scores and weather forecasts. This KS3 revision guide will help you understand how percentages link to fractions and decimals, and how to apply them in real-life problems. By the end, you will feel confident converting between these forms and solving percentage questions without hesitation.
百分比无处不在——从商店折扣、银行利息到考试成绩和天气预报。本文将为 KS3 学生梳理百分比与分数、小数的关系，并讲解如何在实际问题中应用它们。学完之后，你将能自如地进行相互转换，并熟练解决各类百分比题目。
1. What is a Percentage? | 什么是百分比？
The word ‘percent’ comes from the Latin per centum, meaning ‘by the hundred’. A percentage is simply a fraction with a denominator of 100. For example, 25% means 25 out of 100, or 25/100. Percentages allow us to compare proportions easily on a common scale.
“百分比”一词源于拉丁语 per centum，意为“每一百”。百分比本质上就是以 100 为分母的分数。例如，25% 表示 100 份中的 25 份，即 25/100。百分比让我们能够在一个统一尺度上轻松比较比例。
2. Converting Fractions to Percentages | 分数转换为百分比
To convert a fraction to a percentage, divide the numerator by the denominator and then multiply the result by 100. If the denominator is already a factor of 100, you can simply find an equivalent fraction with denominator 100. For 3/5, divide 3 by 5 to get 0.6, then multiply by 100 to get 60%.
将分数转换为百分比，可以把分子除以分母，然后将商乘以 100。如果分母已经是 100 的因数，还可以直接通分成分母为 100 的等价分数。对于 3/5，用 3 除以 5 得 0.6，再乘以 100，即得到 60%。
Fraction → Percentage: (Numerator ÷ Denominator) × 100%
分数 → 百分比：(分子 ÷ 分母) × 100%
3. Converting Decimals to Percentages | 小数转换为百分比
Moving from a decimal to a percentage is straightforward: multiply the decimal by 100, or simply move the decimal point two places to the right. For instance, 0.35 × 100 = 35%, and 0.075 becomes 7.5%. Remember to add the percent symbol after the number.
从小数转换为百分比非常简单：将小数乘以 100，或直接将小数点向右移动两位。例如，0.35 × 100 = 35%；0.075 移动小数点后就是 7.5%。记得在数字后面加上百分号。
Decimal → Percentage: Multiply by 100 and add the % sign
小数 → 百分比：乘以 100，再添加百分号
4. Converting Percentages to Fractions and Decimals | 百分比转换为分数和小数
To turn a percentage into a fraction, write it over 100 and simplify if possible. For example, 45% = 45/100 = 9/20. If the percentage is a decimal, such as 12.5%, write it as 12.5/100, then multiply numerator and denominator by 10 to clear the decimal: 125/1000 = 1/8. To get a decimal, simply divide the percentage by 100.
将百分比转换为分数时，可先写成分母为 100 的分数，再约分。例如 45% = 45/100 = 9/20。如果百分比本身是小数，例如 12.5%，可先写成 12.5/100，再将分子分母同乘 10 消去小数点，得到 125/1000 = 1/8。若求小数，直接将百分数除以 100 即可。
Percentage → Fraction: %/100, then simplify
百分比 → 分数：先写为 %/100，再约分
5. Finding a Percentage of a Quantity | 求一个数量的百分之几
To find a percentage of an amount, convert the percentage to a decimal or fraction, then multiply by the amount. For 30% of 200, turn 30% into 0.3 and multiply: 0.3 × 200 = 60. Alternatively, find 10% first by dividing by 10, then scale up: 10% of 200 is 20, so 30% is 3 × 20 = 60.
求一个数量的百分之几，可以先把百分比转化为小数或分数，再乘以该数量。比如求 200 的 30%，将 30% 化为 0.3，再乘 200 得 60。另一种方法是先除以 10 算出 10%，再以此为基础扩大：200 的 10% 是 20，所以 30% 就是 3 × 20 = 60。
Percentage of Quantity = (Percentage/100) × Total
数量的百分比 = (百分数/100) × 总数
6. Percentage Increase and Decrease | 百分比增加和减少
When a value increases by a percentage, you add that percentage of the original to the original. For a 15% increase on £40, find 15% of £40 (£6) and add: £40 + £6 = £46. A multiplier approach is faster: an increase of 15% is 100% + 15% = 115% = 1.15, so new value = £40 × 1.15 = £46.
当一个数值增加了某个百分比，要用原值加上原值乘以该百分比的部分。比如 40 英镑增加 15%，先算出 40 英镑的 15% 是 6 英镑，然后 40 + 6 = 46 英镑。更快速的方法是使用乘数：增加 15% 意味着新值是原值的 100% + 15% = 115%，即 1.15，因此新值 = 40 × 1.15 = 46 英镑。
Increase: New = Original × (1 + %/100)
增加：新值 = 原值 × (1 + %/100)
For a decrease of 20% on £50, the multiplier is 100% − 20% = 80% = 0.8, so £50 × 0.8 = £40.
减少 20% 时，乘数为 100% − 20% = 80% = 0.8，比如 50 英镑 × 0.8 = 40 英镑。
Decrease: New = Original × (1 − %/100)
减少：新值 = 原值 × (1 − %/100)
7. Percentage Change | 百分比变化
Percentage change compares the difference between a new and an original amount to the original amount. The formula is: ((New − Original) ÷ Original) × 100%. If a phone’s price drops from £320 to £272, the change is (£272 − £320) = −£48. Percentage change = (−48/320) × 100% = −15%, so it is a 15% decrease.
百分比变化用于比较新值与原值的差异占原值的比例。公式为：((新值 − 原值) ÷ 原值) × 100%。如果一部手机的价格从 320 英镑降至 272 英镑，变化量为 272 − 320 = −48 英镑。百分比变化 = (−48/320) × 100% = −15%，即价格下降了 15%。
Percentage Change = (Change ÷ Original) × 100%
百分比变化 = (变化量 ÷ 原值) × 100%
8. Reverse Percentages | 逆向百分比
Reverse percentages are used when you know the final amount after a percentage increase or decrease and need to find the original. If a jacket costs £84 after a 40% discount, £84 represents 60% of the original price (100% − 40%). To find 100%, divide £84 by 0.6, giving £140.
当已知增减后的最终结果，需要反推原值时，就要用到逆向百分比。若一件夹克在享受 40% 折扣后售价为 84 英镑，那么 84 英镑对应原价的 60%（100% − 40%）。要计算原价 100%，可用 84 除以 0.6，得到 140 英镑。
Original = Final Amount ÷ Multiplier
原值 = 最终值 ÷ 乘数
9. Real-Life Applications: Discounts and Sales | 实际应用：折扣与销售
Shops often advertise discounts like ‘25% off’. To work out the sale price, first find the discount amount or use a multiplier. If a £60 video game has 25% off, the multiplier for the discounted price is 75% = 0.75. The sale price is £60 × 0.75 = £45. You can also calculate the saving: £60 × 0.25 = £15, then subtract.
商店常打出“25% off”之类的广告。计算折扣价时，可先求出折扣金额，也可直接使用乘数。如果一款 60 英镑的游戏打七五折（即减价 25%），折扣价的计算乘数为 75% = 0.75，因此售价为 60 × 0.75 = 45 英镑。也可以先算出节省金额 60 × 0.25 = 15 英镑，再从原价中扣除。
- Discount percentage: 100% − discount% = multiplier
- 折扣百分比：100% − 折扣百分比 = 乘数
- Save: Original × discount%
- 节省：原价 × 折扣百分比
10. Real-Life Applications: Interest and Tax | 实际应用：利息与税
Simple interest is a percentage of money earned or paid on a principal amount over time. If you invest £500 at a simple interest rate of 4% per year, the interest after one year is £500 × 0.04 = £20. After three years, it is £20 × 3 = £60. Tax calculations work similarly: 20% VAT on a £200 item adds £200 × 0.20 = £40, making the total £240.
单利是指在一定时间内，本金按固定百分率获得的利息。若将 500 英镑按年利率 4% 投资，一年的利息为 500 × 0.04 = 20 英镑，三年利息为 20 × 3 = 60 英镑。税收计算同理：一件 200 英镑的商品加收 20% 增值税，税额为 200 × 0.20 = 40 英镑，含税总价为 240 英镑。
Simple Interest = Principal × Rate × Time
单利 = 本金 × 利率 × 时间
11. Common Mistakes and Tips | 常见错误与提示
One typical mistake is confusing percentage points with percentages. A rise from 10% to 15% is an increase of 5 percentage points, but the percentage increase is (5/10) × 100 = 50%. Another error is forgetting to convert a decimal multiplier correctly for a decrease: 3% decrease is 0.97, not 0.3. Always double-check whether you are working with the multiplier for increase (1 + p) or decrease (1 − p).
一个常见错误是混淆百分点与百分比增长。从 10% 升到 15% 是增加了 5 个百分点，但百分比增长为 (5/10) × 100 = 50%。另一个错误是弄错减少时的乘数：减少 3% 对应的乘数是 0.97，而不是 0.3。务必区分清楚是在计算增加 (1 + p) 还是减少 (1 − p) 的乘数。
- Always check if the question asks for an amount, a percentage, or an original value.
- 始终检查题目要求的是数量、百分比还是原始值。
- Use simple checks: a 50% increase gives a multiplier of 1.5, and a 50% decrease gives 0.5.
- 用简单数值检验：增加 50% 乘数是 1.5，减少 50% 乘数是 0.5。
12. Practice Problems Summary | 练习总结
To master percentages, practise converting between forms and solving word problems. Here is a quick review: convert 7/20 to a percentage (35%), find 18% of 250 (45), and determine the original price of a laptop sold for £680 after a 15% discount (£800). Regular practice will build speed and accuracy for your KS3 assessments.
要掌握百分比，需多加练习形式转换和应用题。回顾一下：将 7/20 转换成百分比（35%），计算 250 的 18%（45），以及求一件打完 85 折后售价 680 英镑的笔记本电脑的原价（800 英镑）。经常练习可以帮助你在 KS3 考试中更快速、更准确地作答。
Key formulas: Fraction → % = (part/whole) × 100% | % change = (difference/original) × 100%
核心公式：分数 → % = (部分/整体) × 100% | % 变化 = (差值/原值) × 100%
Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com
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