📚 Percentages: Working with 100 | 百分数：与 100 打交道
Percentages are one of the most practical tools in mathematics because they describe a number as a fraction of 100. In the IGCSE Year 7 journey, a solid grasp of percentages links directly to fractions, decimals, ratio and everyday problems such as discounts, interest and data comparisons. This article will guide you through the key ideas step by step, from the meaning of the word 'percent' to reverse percentages and exam technique.
百分数是数学中最实用的工具之一，因为它用 100 作为基准来描述一个数。在 IGCSE 7 年级的学习中，牢固掌握百分数与分数、小数、比以及日常问题（如折扣、利息和数据比较）直接相关。本文将逐步带你梳理核心概念，从 'percent' 一词的含义一直到逆向百分数和考试技巧。
1. Understanding Percentages | 理解百分数
The word 'percent' comes from the Latin phrase 'per centum', meaning 'by the hundred'. Therefore, 37% means 37 out of every 100, or 37/100. Percentages give us a standard way to compare different quantities, because every percentage is measured against the same base of 100.
'百分数'（percent）一词来自拉丁语 'per centum'，意思是'每一百'。因此，37% 表示每一百中有 37，即 37/100。百分数为我们提供了一种比较不同数量的标准方法，因为每一个百分数都是以相同的基准 100 来衡量的。
00, they are closely related to fractions with denominator 100 and to decimals. The symbol % itself is simply a shorthand way of writing "/100", so 37% = 37/100 = 0.37.
由于百分数总是以 100 为基准，它与分母为 100 的分数以及小数密切相关。符号 % 本身就是 '/100' 的简写形式，所以 37% = 37/100 = 0.37。
Some benchmark percentages are worth memorising, because they appear in almost every question and let you estimate answers quickly:
有一些基准百分数非常值得记住，因为它们几乎出现在每一道题中，并且能让你快速估算答案：
- 100% = the whole amount = 1
- 50% = one half = ½ = 0.5
- 25% = one quarter = ¼ = 0.25
- 75% = three quarters = ¾ = 0.75
- 10% = one tenth = 1/10 = 0.1
- 1% = one hundredth = 1/100 = 0.01
- 20% = one fifth = 1/5 = 0.2
If you know that 10% means "divide by 10" and 1% means "divide by 100", you can build almost any other percentage by combining them, without needing a calculator at all.
如果你知道 10% 意味着'除以 10'、1% 意味着'除以 100'，那么你几乎可以把它们组合出任何其他百分数，完全不需要计算器。
2. Percentages, Fractions and Decimals: The Three-Way Link | 百分数、分数与小数：三方联系
Every percentage can be written as a fraction and as a decimal, and every fraction or decimal can be written as a percentage. This three-way link is the foundation of the whole topic, and examiners test it constantly in both directions.
每一个百分数都可以写成分数和小数，每一个分数或小数也都可以写成百分数。这种三方联系是整个专题的基础，考官会不断地从两个方向考查它。
To change a percentage into a fraction, write it over 100 and simplify. To change it into a decimal, divide by 100, which moves the decimal point two places to the left.
要把百分数化成分数，就把它写在 100 的上面然后化简；要化成小数，就除以 100，也就是把小数点向左移动两位。
Percentage → Fraction: 45% = 45/100 = 9/20
Percentage → Decimal: 45% = 45 ÷ 100 = 0.45
Going the other way, multiply a decimal by 100 to get a percentage, and multiply a fraction by 100 (or divide the numerator by the denominator and then multiply by 100) to get a percentage.
反方向运算时，把小数乘以 100 就得到百分数；把分数乘以 100（或者先用分子除以分母，再乘以 100）就得到百分数。
Decimal → Percentage: 0.08 × 100 = 8%
Fraction → Percentage: 3/8 = 3 ÷ 8 × 100 = 37.5%
A quick way to check your work is to ask whether the answer is sensible. A decimal smaller than 1 always gives a percentage smaller than 100%, and a percentage larger than 100% always comes from a number greater than the whole.
检查答案的一个快捷方法就是问自己结果是否合理：小于 1 的小数一定给出小于 100% 的百分数，而大于 100% 的百分数一定来自比整体还大的数量。
3. Converting Between the Three Forms | 在三种形式之间转换
Conversion becomes fast with practice, and a conversion table is a useful revision aid. Read it left to right to go from fractions to percentages, and right to left to go the other way.
转换在练习之后会变得很快，而一张转换表是很有用的复习工具。从左往右读是从分数到百分数，从右往左读则是反向转换。
| Fraction 分数 | Decimal 小数 | Percentage 百分数 |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 1/3 | 0.333… | 33⅓% |
| 2/3 | 0.666… | 66⅔% |
Notice that some fractions such as 1/3 and 1/6 give recurring decimals. In these cases the percentage must be written as a fraction of a percent (33⅓%) or rounded, and the question will normally tell you how many decimal places to use.
注意，像 1/3 和 1/6 这样的分数会得到循环小数。在这种情况下，百分数必须写成几分之一个百分点（33⅓%），或者四舍五入，而题目通常会告诉你保留几位小数。
4. Finding a Percentage of an Amount | 求一个量的百分数
Finding a percentage of an amount is the most common calculation in this topic. The core idea is that the word 'of' means multiply, and the percentage must first be turned into a decimal or a fraction.
求一个量的百分数是本专题中最常见的计算。核心思路是：'of'（的）表示相乘，而百分数必须先化成小数或分数。
Percentage of an amount = (percentage ÷ 100) × amount
or, the method is direct: change the percentage to a decimal and multiply. For example, 15% of 240 = 0.15 × 240 = 36.
使用计算器时，方法很直接：把百分数化成小数再相乘。例如，240 的 15% = 0.15 × 240 = 36。
Without a calculator, build the percentage from simple parts. For 15% of 240, split 15% into 10% and 5%: 10% of 240 = 24, 5% is half of that = 12, and 24 + 12 = 36.
不使用计算器时，就用简单的部分拼出所需的百分数。求 240 的 15%，把 15% 拆成 10% 和 5%：240 的 10% = 24，5% 是它的一半 = 12，于是 24 + 12 = 36。
Useful mental building blocks include:
有用的心算积木包括：
- 10%: divide by 10
- 5%: find 10% and halve it
- 1%: divide by 100
- 25%: divide by 4
- 50%: divide by 2
- 20%: find 10% and double it
- 75%: find 25% and multiply by 3
Always sanity-check the size of your answer. A percentage less than 100 gives a result smaller than the original amount, and a percentage greater than 100 gives a result larger than the original amount.
一定要检查答案的大小是否合理：小于 100 的百分数给出的结果小于原量，大于 100 的百分数给出的结果大于原量。
5. Percentage Increase and Decrease | 百分数的增加与减少
A percentage increase means the amount grows by that percentage of its original value; a percentage decrease means it shrinks. There are two reliable methods, and both give the same answer.
百分数增加意味着该量按原值的某个百分比增长；百分数减少则意味着它缩小。有两种可靠的方法，结果完全相同。
Method 1: find the change and add or subtract it. To increase 80 by 15%: 15% of 80 = 12, and 80 + 12 = 92.
方法一：先求出变化量，再进行加减。把 80 增加 15%：80 的 15% = 12，于是 80 + 12 = 92。
Method 2: use a single multiplier. An increase of 15% gives the multiplier 1 + 0.15 = 1.15, and a decrease of 12% gives 1 − 0.12 = 0.88.
方法二：使用单一乘数。增加 15% 对应的乘数是 1 + 0.15 = 1.15，减少 12% 对应的乘数是 1 − 0.12 = 0.88。
Increase by r%: multiply by (1 + r ÷ 100)
Decrease by r%: multiply by (1 − r ÷ 100)
So increasing 80 by 15% gives 80 × 1.15 = 92, and decreasing 250 by 12% × 0.88 = 220. The multiplier method is faster and is the one most examiners expect to see.
所以把 80 增加 15% 得 80 × 1.15 = 92，把 250 减少 12% 得 250 × 0.88 = 220。乘数法更快，也是大多数考官期望看到的方法。
6. Multipliers: The Fast Method | 乘数法：最快的方法
A multiplier is the single number you multiply by to apply a percentage change in one step. Learning to read a question and write the multiplier immediately saves a great deal of time in exams.
乘数就是你用来一步完成百分数变化的那个数。学会读题后立刻写出乘数，能在考试中节省大量时间。
| Change 变化 | Multiplier 乘数 |
|---|---|
| Increase by 5% | 1.05 |
| Decrease by 5% | 0.95 |
| Increase by 12.5% | 1.125 |
| Decrease by 30% | 0.7 |
| Increase by 100% | 2 |
| Decrease by 100% | 0 |
anges easy, because you simply multiply the multipliers together. A price that rises by 10% and then by 20% is multiplied by 1.1 × 1.2 = 1.32, which is a total increase of 32%, not 30%.
乘数也让连续变化变得简单，因为只需把乘数相乘。一个价格先上涨 10% 再上涨 20%，等于乘以 1.1 × 1.2 = 1.32，总涨幅是 32%，而不是 30%。
Similarly, a 20% discount followed by a further 10% discount gives 0.8 × 0.9 = 0.72, which means the customer pays 72% of the original price, a total discount of 28%.
同样地，先打 20% 的折扣、再打 10% 的折扣，得到 0.8 × 0.9 = 0.72，意思是顾客支付原价的 72%，总折扣为 28%。
7. Discounts, VAT and Sale Prices | 折扣、增值税与售价
Discounts and taxes are the everyday face of percentage change. In a discount question, "20% off" means you pay 80% of the original price, so the multiplier is 0.8.
折扣和税费是百分数变化在日常生活中的面貌。在折扣题中，'打八折（20% off）'意味着你支付原价的 80%，所以乘数是 0.8。
Example: a jacket costs £64 and is reduced by 25%. The sale price is 64 × 0.75 = £48, and the saving is 64 × 0.25 = £16.
例题：一件夹克售价 64 英镑，降价 25%。售价为 64 × 0.75 = 48 英镑，节省了 64 × 0.25 = 16 英镑。
ue Added Tax (VAT) works in the opposite direction: it is added to the price. If VAT is 20%, a price of £45 before tax becomes 45 × 1.2 = £54 after tax.
增值税（VAT）的方向相反：它是在价格上加收的。如果 VAT 为 20%，那么税前价 45 英镑在加税后变为 45 × 1.2 = 54 英镑。
Two habits keep these questions safe:
两个好习惯能保证这类题目不出错：
- Write down the multiplier before you calculate, so you can see whether it is above or below 1.
- Remember that "x% off" always means the multiplier is (1 − x ÷ 100), never just x ÷ 100.
- Give money answers to two decimal places and include the currency symbol.
If a question asks for the total saving instead of the new price, calculate the original amount minus the new amount, or simply multiply by the discount rate expressed as a decimal.
如果题目问的是总共省了多少钱而不是新价格，就用原价减去新价，或者直接用折扣率的小数形式乘以原价。
8. Simple Interest, Profit and Loss | 单利、利润与亏损
Simple interest is money paid for the use of money, calculated only on the original amount each year. It is a classic application of percentages and appears frequently in IGCSE papers.
单利是为使用资金而支付的利息，每年只按最初的本金计算。它是百分数的经典应用，在 IGCSE 试卷中经常出现。
Simple interest I = (P × R × T) ÷ 100
Here P is the principal (the amount invested or borrowed), R is the annual rate as a percentage, and T is the time in years.
这里 P 是本金（存入或借入的金额），R 是以百分数表示的年利率，T 是以年为单位的时间。
Example: £800 is invested at 3.5% simple interest per year for 4 years. Then I = (800 × 3.5 × 4) ÷ 100 = £112, and the total value of the investment is 800 + 112 = £912.
例题：800 英镑按每年 3.5% 的单利投资 4 年。则 I = (800 × 3.5 × 4) ÷ 100 = 112 英镑，投资总额为 800 + 112 = 912 英镑。
Profit and loss are also percentage ideas, but the percentage is always measured against the cost price, not the selling price:
利润和亏损同样是百分数概念，但百分比始终是相对于成本价来衡量的，而不是售价：
Percentage profit = (profit ÷ cost price) × 100%
So a shop that buys an item for £40 and sells it for £50 makes a profit of £10, and 10 ÷ 40 × 100% = 25% profit.
所以一家商店以 40 英镑进货、以 50 英镑卖出，赚了 10 英镑，10 ÷ 40 × 100% = 25% 的利润率。
9. Expressing One Quantity as a Percentage of Another | 用一个量表示另一个量的百分数
Sometimes the question gives you two quantities and asks what percentage one is of the other. The golden rule is: the quantity that follows the word 'of' goes on the bottom of the fraction.
有时题目给出两个量，问其中一个量是另一个量的百分之几。黄金法则是：紧跟在 'of'（的）后面的那个量放在分数的分母上。
Percentage = (part ÷ whole) × 100%
Example: a student scores 18 out of 45 in a test. The percentage score is 18 ÷ 45 × 100% = 0.4 × 100% = 40%.
例题：某学生在满分 45 分的测验中得了 18 分。百分制得分为 18 ÷ 45 × 100% = 0.4 × 100% = 40%。
lass has 12 girls and 20 boys, so the fraction of girls is 12 out of 32, and 12 ÷ 32 × 100% = 37.5%.
再举一例：一个班有 12 名女生和 20 名男生，那么女生占 32 人中的 12 人，12 ÷ 32 × 100% = 37.5%。
A common slip is to divide by the wrong number. Read the sentence carefully: "what percentage of the total is spent on rent?" means the total goes on the bottom; "what percentage of the rent is spent on heating?" means the rent goes on the bottom.
常见的失误是除错了数。仔细读句子：'租金占总支出的百分之几？'意味着总支出在分母；'取暖费占租金的百分之几？'意味着租金在分母。
10. Percentage Change and Reverse Percentages | 百分数变化与逆向百分数
Percentage change tells you how much a quantity has grown or shrunk compared with its original value, and it is always measured against the original, never the new, value.
百分数变化告诉你一个量相对于原值增长或缩小了多少，它始终相对于原值来衡量，绝不是相对于新值
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