📚 KS3 Cambridge Maths: Fractions, Decimals and Percentages | 剑桥初中数学：分数、小数与百分数
In Cambridge KS3 mathematics, fractions, decimals and percentages are core tools for comparing quantities, solving problems and preparing for algebraic ratio work. This revision guide builds the skills you need to convert between forms, calculate accurately and apply percentages in real-life contexts.
在剑桥 KS3 数学中，分数、小数和百分数是比较数量、解决问题以及为代数比例学习做准备的核心工具。本复习指南帮助你建立相互转换、准确计算以及在现实情境中应用百分数所需的技能。
1. What Fractions, Decimals and Percentages Mean | 分数、小数与百分数的含义
A fraction describes a part of a whole using two numbers. The numerator (the top number) tells you how many equal parts you have, and the denominator (the bottom number) tells you how many equal parts the whole has been divided into. In 3/4, the whole is split into 4 equal parts and we take 3 of them.
分数用两个数描述整体的一个部分。分子（上面的数）表示你取了多少等份，分母（下面的数）表示整体被平均分成了多少份。在 3/4 中，整体被平均分成 4 份，我们取其中的 3 份。
A decimal records parts of a whole using place value. The first place after the decimal point is the tenths place, the second is the hundredths place and the third is the thousandths place, so 0.47 means 4 tenths and 7 hundredths.
小数用位值来记录整体的部分。小数点后第一位是十分位，第二位是百分位，第三位是千分位，因此 0.47 表示 4 个十分之一和 7 个百分之一。
A percentage is a fraction with denominator 100. The words "per cent" mean "out of one hundred", so 25% means 25 out of 100, which is the same as 25/100 or 0.25.
百分数是分母为 100 的分数。"per cent" 的意思是"每一百"，所以 25% 表示 100 份中的 25 份，这与 25/100 或 0.25 完全相同。
All three forms answer the same question: how much of the whole do we have? Choosing between them is a matter of convenience, not of meaning.
这三种形式回答的是同一个问题：我们拥有整体的多少？在它们之间做选择只是方便与否的问题，而不是含义上的差别。
2. Equivalent Forms of Numbers | 数的等价形式
A fraction, a decimal and a percentage can all describe exactly the same proportion. For example, 1/2, 0.5 and 50% represent the identical part of a whole.
分数、小数和百分数都可以描述完全相同的比例。例如，1/2、0.5 和 50% 表示整体中相同的部分。
Recognising equivalent forms helps you choose the most convenient representation when comparing or calculating. The table below lists the equivalences that are worth memorising because they appear again and again in KS3 questions.
识别等价形式有助于你在比较或计算时选择最方便的表示方式。下表列出了值得背下来的常见等价关系，因为它们在 KS3 题目中反复出现。
| 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% |
| 3/8 | 0.375 | 37.5% |
| 1/10 | 0.1 | 10% |
| 1/20 | 0.05 | 5% |
| 1/3 | 0.333... | 33.3...% |
| 2/3 | 0.666... | 66.7% (3 s.f.) |
Notice that some fractions give terminating decimals (they stop) while others give recurring decimals (a digit or block of digits repeats forever).
注意：有些分数得到的是有限小数（会终止），而另一些得到的是循环小数（某一位数字或数字组无限重复）。
3. Converting a Fraction to a Decimal | 分数化小数
To convert a fraction to a decimal, divide the numerator by the denominator. The fraction bar is simply a division sign, so 3/4 means 3 ÷ 4.
要把分数转换为小数，用分子除以分母。分数线本身就是一个除号，所以 3/4 表示 3 ÷ 4。
3/4 = 3 ÷ 4 = 0.75
owers of ten, which simply shifts the digits: 7/10 = 0.7, 7/100 = 0.07 and 7/1000 = 0.007.
剑桥 KS3 练习中经常出现的例子包括除以 10 的幂，这实际上只是移动数位：7/10 = 0.7，7/100 = 0.07，7/1000 = 0.007。
For fractions that do not terminate, keep dividing until you see the repeating pattern. For 5/6, the calculation gives 0.8333..., which can be written as 0.83 with a dot over the 3. You may also be asked to round it, for example to 0.833 (3 decimal places) or 83.3% (1 decimal place).
对于不能整除的分数，一直除到出现重复规律为止。对于 5/6，计算得到 0.8333...，可以写成 0.83 并在 3 上加循环点。题目也可能要求你取近似值，例如保留 3 位小数为 0.833，或保留 1 位小数为 83.3%。
will see 0.833333333. Never copy the whole display as your final answer - read the question to see how many decimal places or significant figures are required.
使用计算器时，输入 5 ÷ 6 =，屏幕上会显示 0.833333333。不要把整个显示结果抄成最终答案——先看题目要求保留几位小数或几位有效数字。
4. Converting Between Decimals and Percentages | 小数与百分数的互化
To convert a decimal to a percentage, multiply by 100. To convert a percentage to a decimal, divide by 100. Because 100 has two zeros, this is equivalent to moving the decimal point two places.
要把小数转换为百分数，乘以 100。要把百分数转换为小数，除以 100。因为 100 有两个零，这相当于把小数点移动两位。
0.75 × 100 = 75% and 75% ÷ 100 = 0.75
A quick check is always possible: a number less than 1 as a decimal must become a percentage smaller than 100%, and a percentage greater than 100% must become a decimal greater than 1.
随时可以做快速检查：小于 1 的小数转换成百分数后一定小于 100%，而大于 100% 的百分数转换成小数后一定大于 1。
- 0.4 → 40% | 0.4 → 40%
- 0.06 → 6% | 0.06 → 6%
- 1.25 → 125% | 1.25 → 125%
- 8% → 0.08 | 8% → 0.08
- 150% → 1.5 | 150% → 1.5
- 0.5% → 0.005 | 0.5% → 0.005
The last example catches many students: 0.5% is half of one per cent, so it is a very small number, not 50%. Read the percent sign carefully.
最后一个例子难倒了很多学生：0.5% 是百分之一的二分之一，因此它是一个非常小的数，而不是 50%。读题时要特别留意见到百分号的位置。
5. Converting a Percentage to a Fraction | 百分数化分数
To convert a percentage to a fraction, write the percentage over 100 and simplify if possible. Remember that the denominator is always 100, no matter how large or small the percentage is.
要把百分数转换为分数，把百分数写成分母为 100 的分数，然后尽可能约分。记住无论百分数是大是小，分母永远是 100。
35% = 35/100 = 7/20
For percentages that are not whole numbers, multiply the numerator and denominator by 10 (or 100) to clear the decimal before simplifying.
对于不是整数的百分数，先给分子和分母同时乘以 10（或 100）去掉小数，然后再约分。
12.5% = 12.5/100 = 125/1000 = 1/8
- 20% = 20/100 = 1/5 | 20% = 20/100 = 1/5
- 75% = 75/100 = 3/4 | 75% = 75/100 = 3/4
- 150% = 150/100 = 3/2 = 1½ | 150% = 150/100 = 3/2 = 1½
- 0.4% = 0.4/100 = 4/1000 = 1/250 | 0.4% = 0.4/100 = 4/1000 = 1/250
Always check whether the question asks for a fraction in its simplest form, a mixed number, or a decimal. Cambridge mark schemes usually require the simplest form.
一定要看清题目要求的是最简分数、带分数还是小数。剑桥的评分标准通常要求写出最简形式。
6. Converting a Fraction to a Percentage | 分数化百分数
There are two reliable routes from a fraction to a percentage. Route one: divide the numerator by the denominator to get a decimal, then multiply by 100. Route two: multiply the fraction by 100 directly and simplify.
从分数到百分数有两条可靠路径。路径一：用分子除以分母得到小数，再乘以 100。路径二：直接把分数乘以 100 并化简。
3/5 = 3 ÷ 5 = 0.6 = 0.6 × 100 = 60%
7/8 = 7 ÷ 8 = 0.875 = 87.5%
on converts to a recurring decimal, the percentage also recurs. For example, 1/3 = 33.3...%, which is normally rounded to 33.3% (1 d.p.) or 33% (nearest whole number).
当分数转换成循环小数时，百分数也会循环。例如 1/3 = 33.3...%，通常写作 33.3%（保留 1 位小数）或 33%（取整）。
If the denominator is already a factor or multiple of 100, an elegant method is to scale up. For 7/20, multiply top and bottom by 5 to obtain 35/100, which reads directly as 35%.
如果分母本身是 100 的因数或倍数，有一个更优雅的方法：通分放大。对于 7/20，把分子分母同时乘以 5 得到 35/100，直接读作 35%。
7. Comparing and Ordering Numbers in Different Forms | 不同形式的数比较与排序
You can only compare numbers fairly when they are written in the same form. The safest strategy is to convert everything to decimals, because decimals are easy to line up by place value.
只有当数字写成同一种形式时，才能公平地比较它们。最稳妥的策略是把所有数都转换成小数，因为小数很容易按位值对齐比较。
Example: arrange 3/5, 0.62, 58% and 5/8 in ascending order.
例题：把 3/5、0.62、58% 和 5/8 按从小到大排列。
- 3/5 = 0.6 | 3/5 = 0.6
- 0.62 = 0.62 | 0.62 = 0.62
- 58% = 0.58 | 58% = 0.58
- 5/8 = 0.625 | 5/8 = 0.625
So the ascending order is 58%, 3/5, 0.62, 5/8. Always write the numbers in the form the question used in your final answer, unless the question says otherwise.
所以从小到大为 58%、3/5、0.62、5/8。除非题目另有要求，最终答案要使用题目原本给出的形式来书写这些数。
Remember the inequality symbols: < means "is less than" and > means "is greater than". A useful sentence is 0.62 < 5/8 because 0.620 < 0.625.
记住不等号：< 表示"小于"，> 表示"大于"。例如 0.62 < 5/8，因为 0.620 < 0.625。
8. Finding a Fraction or a Percentage of an Amount | 求一个量的分数或百分数
To find a fraction of an amount, divide by the denominator and multiply by the numerator. This works because dividing by the denominator finds one part, and multiplying by the numerator takes the number of parts you need.
要求一个量的分数，先用分母去除，再乘以分子。这样做的道理是：除以分母得到一份的大小，乘以分子就取到所需的份数。
3/8 of 240: 240 ÷ 8 = 30, then 30 × 3 = 90
To find a percentage of an amount, convert the percentage to a decimal and multiply. This is called the multiplier method and it is the most reliable approach at KS3.
要求一个量的百分数，把百分数转换为小数再相乘。这叫做乘数法，是 KS3 阶段最可靠的方法。
15% of 60 = 0.15 × 60 = 9
Mental methods are still useful for friendly percentages: 10% is found by dividing by 10, 5% is half of 10%, 1% is found by dividing by 100, and 20% is double 10%. To find 35% of 80: 10% = 8, so 30% = 24, and 5% = 4, giving 28.
对于好算的百分数，心算方法仍然有用：10% 是除以 10，5% 是 10% 的一半，1% 是除以 100，20% 是 10% 的两倍。求 80 的 35%：10% = 8，所以 30% = 24，5% = 4，合计 28。
Always decide whether the answer should be bigger or smaller than the original amount. A percentage greater than 100% gives a bigger answer; a proper fraction gives a smaller answer.
作答前先判断答案应该比原量大还是小。大于 100% 的百分数会得到更大的结果；真分数则得到更小的结果。
9. Percentage Increase and Decrease | 百分数的增加与减少
To increase an amount by a percentage, add the percentage to 100% and use the resulting multiplier. To increase by 15%, use the multiplier 1.15. To decrease by 15%, subtract from 100% and use the multiplier 0.85.
要把一个量增加一个百分数，先把该百分数与 100% 相加，再用得到的乘数。增加 15% 就用乘数 1.15。减少 15% 则从 100% 中减去，用乘数 0.85。
80 × 1.15 = 92 and 80 × 0.85 = 68
Percentage change compares the size of the change with the original value. The formula is:
百分数变化是把变化量与原来的值进行比较。公式是：
percentage change = (change ÷ original) × 100%
sold for 300 pounds. The change is 50 pounds, so the percentage increase is 50 ÷ 250 × 100% = 20%.
例题：一辆自行车原价 250 英镑，售价 300 英镑。变化量是 50 英镑，所以百分数增加为 50 ÷ 250 × 100% = 20%。
A common trap is adding percentages that are applied one after another. A 20% rise followed by a 10% s not return the original value; the multipliers 1.2 × 0.9 = 1.08 give a net increase of 8%.
一个常见的陷阱是把连续施加的百分数直接相加。先涨 20% 再降 10% 并不会回到原值；乘数 1.2 × 0.9 = 1.08，净增长为 8%。
10. Reverse Percentages: Finding the Original Amount | 逆百分数：求原来的量
When you know the value after a change and you need the original value, do not simply add or subtract the percentage. Work backwards using the multiplier.
当你知道变化之后的数值而需要求原来的数值时，不要简单地加或减那个百分数。要用乘数倒推回去。
Example: after a 20% discount, a coat costs 64 pounds. The sale price is 80% of the original, so the multiplier is 0.8.
例题：一件外套打八折（降价 20%）后售价 64 英镑。售价是原价的 80%，所以乘数是 0.8。
original = 64 ÷
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图