📚 Percentages, Fractions and Decimals: Cambridge KS3 Maths | 百分数、分数与小数的转换与运算
In the Cambridge Lower Secondary Mathematics course, being able to move confidently between fractions, decimals and percentages is a core skill. These three forms describe the same idea: a part of a whole. This article explains the key methods you need for KS3, including conversion, ordering, percentage change and reverse percentages.
在剑桥初中数学课程中，能够自信地在分数、小数和百分数之间进行转换是一项核心技能。这三种形式描述的是同一个概念：整体的一部分。本文讲解 KS3 阶段需要掌握的关键方法，包括转换、排序、百分比变化和逆百分数。
1. Equivalent Forms: Fraction, Decimal and Percentage | 等价形式：分数、小数和百分数
A fraction such as 3/4, a decimal such as 0.75 and a percentage such as 75% all represent the same proportion. When you see a fraction, think of division: the numerator divided by the denominator. A percentage is simply a fraction with denominator 100, so 75% means 75/100.
分数（如 3/4）、小数（如 0.75）和百分数（如 75%）都表示相同的比例。看到分数时，可以把它想成除法：分子除以分母。百分数就是分母为 100 的分数，所以 75% 表示 75/100。
The table below shows common equivalent forms you should memorise. Being able to recall these quickly will save time in both calculator and non-calculator questions.
下表列出了你应该记住的常见等价形式。快速回忆起这些内容可以节省计算器题和非计算器题的时间。
| Fraction | Decimal | Percentage |
|---|---|---|
| 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% |
| 1/10 | 0.1 | 10% |
Understanding equivalence is not just about memorising. The key idea is that fractions, decimals and percentages are different ways of writing the same proportion of a whole.
理解等价形式不只是死记硬背。核心概念是：分数、小数和百分数是同一个整体比例的不同写法。
2. Converting Fractions to Decimals | 分数转小数
To convert a fraction to a decimal, divide the numerator by the denominator. This works for both proper fractions and improper fractions. For example, 3/8 means 3 ÷ 8.
要把分数转换为小数，用分子除以分母。这对真分数和假分数都适用。例如，3/8 表示 3 ÷ 8。
3/8 = 3 ÷ 8 = 0.375
.. . In KS3, you should be able to identify whether a decimal is terminating or recurring.
有些分数会得到有限小数，例如 1/4 = 0.25 或 3/8 = 0.375。另一些分数会得到循环小数，例如 1/3 = 0.333... 或 2/7 = 0.285714285714... 。在 KS3 阶段，你需要能判断一个小数是有限小数还是循环小数。
. In these cases, the decimal point moves to the left by the number of zeros in the denominator.
如果分母是 10、100 或 1000，转换会非常快。例如，47/100 = 0.47，3/1000 = 0.003。在这些情况下，小数点向左移动的位数等于分母中零的个数。
3. Converting Decimals to Percentages | 小数转百分数
To convert a decimal to a percentage, multiply the decimal by 100. In practice, this is the same as moving the decimal point two places to the right. Remember that a percentage is just a decimal multiplied by 100.
要把小数转换为百分数，将小数乘以 100。实际操作中，这相当于把小数点向右移动两位。请记住，百分数就是小数乘以 100 的结果。
decimal × 100 = percentage
For example, 0.07 × 100 = 7%, and 1.25 × 100 = 125%. Notice that decimals greater than 1 give percentages greater than 100% This is important when you work with increases or amounts larger than the original whole.
例如，0.07 × 100 = 7%，1.25 × 100 = 125%。注意，大于 1 的小数会得到大于 100% 的百分数。当你处理增加量或大于原始整体的数量时，这一点很重要。
ve the point two places to the right and add zeros as needed. For example, 0.8 = 0.80 = 80%, and 0.025 = 2.5%.
当小数的位数少于两位时，你可以直接把小数点向右移动两位，并根据需要补零。例如，0.8 = 0.80 = 80%，0.025 = 2.5%。
4. Converting Percentages to Fractions | 百分数转分数
To convert a percentage to a fraction, write the percentage number over 100 and simplify if possible. Since 'per cent' means 'out of one hundred', this conversion is very natural.
要把百分数转换为分数，把百分数的数字写在 100 上，然后尽可能化简。因为 'per cent' 的意思是 '每一百'，所以这种转换非常自然。
45% = 45/100 = 9/20
If the percentage includes a decimal, such as 12.5% 00 to remove the decimal point. In this case, 12.5/100 = 125/1000 = 1/8 after simplifying.
如果百分数包含小数，例如 12.5%，你可以先写成 12.5/100。然后把分子和分母同时乘以 10、100 或 1000，以去掉小数点。在这个例子中，12.5/100 = 125/1000，化简后得到 1/8。
Percentages greater than 100% become improper fractions or mixed numbers. For example, 140% = 140/100 = 7/5 = 1 2/5. This shows that the value is larger than the original whole.
大于 100% 的百分数会变成假分数或带分数。例如，140% = 140/100 = 7/5 = 1 2/5。这表明该值大于原始的整数。
5. Ordering Mixed Forms | 混合形式的排序
To compare or order a mixture of fractions, decimals and percentages, first convert all values to the same form. Decimals are often the easiest common form because they are simple to compare place by place.
要比较或排序分数、小数和百分数的混合形式，首先把所有值转换成同一种形式。小数通常是最容易使用的共同形式，因为可以逐位进行比较。
For example, order these values from smallest to largest: 0.45, 3/5, 38% and 5/8. Convert each one to a decimal: 0.45 is already a decimal; 3/5 = 0.60; 38% = 0.38; 5/8 = 0.625. Now the order is clear.
例如，将这些值从小到大排列：0.45、3/5、38% 和 5/8。把每个值都转换为小数：0.45 已经是小数；3/5 = 0.60；38% = 0.38；5/8 = 0.625。现在顺序就清晰了。
0.38 < 0.45 < 0.60 < 0.625
So the original order is 38%, 0.45, 3/5, 5/8. Always write the final answer using the original forms, not the converted forms, unless the question tells you otherwise.
所以原始顺序是 38%、0.45、3/5、5/8。除非题目另有要求，否则最终答案要用原始形式书写，而不是转换后的形式。
6. Finding a Percentage of an Amount | 求一个数的百分之几
To find a percentage of an amount, convert the percentage to a decimal and multiply. The formula below works for any percentage, including percentages greater than 100%.
要求一个数的百分之几，把百分数转换为小数，然后相乘。下面的公式适用于任何百分数，包括大于 100% 的百分数。
P% of amount = (P ÷ 100) × amount
For example, find 30% of 240. Since 30% = 30/100 = 0.30, the calculation is 0.30 × 240 = 72. So 30% of 240 is 72.
例如，求 240 的 30%。因为 30% = 30/100 = 0.30，计算为 0.30 × 240 = 72。所以 240 的 30% 是 72。
A useful non-calculator method is to find 10%, 5% and 1% first, then combine them. For 30% of 240, find 10% = 24, then multiply by 3 to get 72. For harder amounts, you can combine 10%, 5% and 1%. For example, 17% of 300 = 10% (30) + 5% (15) + 1% (3) + 1% (3) = 51.
一种实用的非计算器方法是先求 10%、5% 和 1%，然后把它们组合起来。求 240 的 30% 时，先求 10% = 24，再乘以 3 得到 72。对于更复杂的数值，可以组合 10%、5% 和 1%。例如，300 的 17% = 10% (30) + 5% (15) + 1% (3) + 1% (3) = 51。
When the percentage is greater than 100%, the answer will be larger than the original amount. For example, 120% of 80 = 1.20 × 80 = 96.
当百分数大于 100% 时，答案会大于原始数量。例如，80 的 120% = 1.20 × 80 = 96。
7. Increasing and Decreasing by a Percentage | 按百分数增加和减少
An increase by P% means the new value is (100 + P)% of the original. A decrease by P% means the new value is (100 − P)% of the original. The decimal multiplier makes these calculations very efficient.
按 P% 增加意味着新值是原始值的 (100 + P)%。按 P% 减少意味着新值是原始值的 (100 − P)%。小数乘数可以让这些计算非常高效。
new value = original × (1 ± P/100)
For example, increase £80 by 15%. The multiplier is 1 + 15/100 = 1.15, so the new value is £80 × 1.15 = £92. To decrease £80 by 15%, the multiplier is 1 − 15/100 = 0.85, so the new value is £80 × 0.85 = £68.
例如，将 80 英镑增加 15%。乘数是 1 + 15/100 = 1.15，所以新值是 £80 × 1.15 = £92。将 80 英镑减少 15%，乘数是 1 − 15/100 = 0.85，所以新值是 £80 × 0.85 = £68。
Do not add or subtract the percentage amount in pounds directly without using the multiplier, especially for multi-step problems. The multiplier method is easier to apply and less likely to cause errors.
不要不使用乘数而直接加减百分比对应的金额，尤其是在多步问题中。乘数法更易于应用，也不容易出错。
8. Finding the Percentage Change | 求百分比变化
Percentage change compares the actual change to the original value, not to the new value. The original value is always the denominator. This is a common source of error, so read the question carefully.
百分比变化是将实际变化量与原始值进行比较，而不是与新值进行比较。原始值总是分母。这是一个常见的错误来源，所以请仔细阅读题目。
percentage change = (new value − original value) ÷ original value × 100%
For example, a price rises from £50 to £65. The actual change is £65 − £50 = £15. The original value is £50. So the percentage increase is 15 ÷ 50 × 100% = 30%.
例如，一个价格从 50 英镑上涨到 65 英镑。实际变化量是 £65 − £50 = £15。原始值是 £50。所以百分比增加量是 15 ÷ 50 × 100% = 30%。
numerator will be negative, giving a percentage decrease. For example, a price falls from £80 to £60. The change is −£20, so the percentage change is −20 ÷ 80 × 100% = −25%, a 25% decrease.
如果新值小于原始值，分子将为负数，得到百分比减少。例如，一个价格从 80 英镑下降到 60 英镑。变化量是 −£20，所以百分比变化是 −20 ÷ 80 × 100% = −25%，即减少 25%。
9. Reverse Percentages | 逆百分数
Reverse percentage problems give you a value after a percentage increase or decrease, and ask you to find the original value. To do this, divide by the decimal multiplier. Do not simply subtract or add the percentage to the final value.
逆百分数问题会给出按某个百分比增加或减少之后的值，并要求你求出原始值。要做到这一点，除以小数乘数即可。不要简单地从最终值中减去或加上百分比。
original value = final value ÷ (1 ± P/100)
For example, after a 20% increase, a price is £ he multiplier was 1 + 20/100 = 1.20. Divide the final value by this multiplier: original = £96 ÷ 1.20 = £80. So the price before the increase was £80.
例如，在增加 20% 之后，价格是 96 英镑。乘数是 1 + 20/100 = 1.20。用最终值除以这个乘数：原始值 = £96 ÷ 1.20 = £80。所以增加前的价格是 80 英镑。
Another example: after a 15% decrease, a value is £68. The multiplier was 1 − 15/100 = 0.85. The original value is £68 ÷ 0.85 = £ that the original value is the same in both examples, but different multipliers were used.
另一个例子：在减少 15% 之后，一个值是 68 英镑。乘数是 1 − 15
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