📚 Fractions, Decimals and Percentages | 分数、小数和百分比
At KS3 level, Cambridge mathematics expects you to convert fluently between fractions, decimals and percentages, and to order them in real-life and abstract problems.
在 KS3 阶段，剑桥数学要求你能够在分数、小数和百分比之间熟练转换，并能在实际问题和抽象问题中对它们进行排序。
This skill is essential for later topics such as probability, ratio, proportion and interpreting statistical data.
这项技能对于概率、比、比例和数据解读等后续主题至关重要。
1. Understanding the Three Forms | 理解三种表示形式
A fraction, a decimal and a percentage are three different ways of representing the same proportion of a whole.
分数、小数和百分比是表示同一个整体比例的三种不同方式。
The fraction 1/4, the decimal 0.25 and the percentage 25% all describe the same shaded part of a grid.
分数 1/4、小数 0.25 和百分比 25% 都描述了网格中相同的阴影部分。
Fractions are useful for exact division, decimals are useful for measurement, and percentages are useful for comparing amounts quickly.
分数适用于精确除法，小数适用于测量，百分比适用于快速比较数量。
2. Converting Fractions to Decimals | 分数转换为小数
To convert a fraction to a decimal, divide the numerator by the denominator.
要将分数转换为小数，用分子除以分母。
For example, 3/8 means 3 ÷ 8, which gives 0.375.
例如，3/8 表示 3 ÷ 8，结果是 0.375。
If the fraction has a denominator of 10, 100 or 1000, you can write it directly as a decimal by moving the decimal point.
如果分数的分母是 10、100 或 1000，你可以直接写成小数，只需移动小数点。
| Fraction | Decimal |
|---|---|
| 7/10 | 0.7 |
| 23/100 | 0.23 |
| 9/1000 | 0.009 |
When the denominator is a power of ten, the number of zeros in the denominator tells you how many decimal places are needed.
当分母是 10 的幂时，分母中零的个数告诉你需要多少位小数。
For fractions with other denominators, you may need to use short division or recognise equivalent fractions.
对于分母不是 10 的幂的分数，你可能需要使用短除法，或者识别等值分数。
3. Converting Decimals to Fractions | 小数转换为分数
To convert a terminating decimal to a fraction, write the decimal as a fraction with denominator 10, 100, 1000 or another power of ten, then simplify.
要将有限小数转换为分数，先将小数写成分母为 10、100、1000 或其他 10 的幂的分数，然后化简。
For instance, 0.65 = 65/100 = 13/20 after dividing the numerator and denominator by 5.
例如，0.65 = 65/100 = 13/20，即分子和分母同除以 5 之后的结果。
For recurring decimals, a different method is needed, but at KS3 the focus is usually on terminating decimals.
对于循环小数，需要使用不同的方法，但在 KS3 阶段通常重点是有限小数。
0.8 = 8/10 = 4/5
Always check whether the fraction can be simplified by finding common factors.
一定要通过寻找公因数来检查分数是否可以化简。
A quick way is to count the decimal places first: one decimal place means tenths, two decimal places mean hundredths, and three decimal places mean thousandths.
一个快速的方法是先数小数位数：一位小数表示十分位，两位小数表示百分位，三位小数表示千分位。
4. Converting Fractions and Decimals to Percentages | 分数和小数转换为百分比
To convert a decimal to a percentage, multiply by 100 and write the % symbol.
要将小数转换为百分比，乘以 100 并写上 % 符号。
For example, 0.42 = 0.42 × 100% = 42%.
例如，0.42 = 0.42 × 100% = 42%。
To convert a fraction to a percentage, first change the fraction to a decimal by division, then multiply by 100.
要将分数转换为百分比，先用除法把分数化为小数，然后乘以 100。
So 2/5 = 2 ÷ 5 = 0.4 = 40%.
因此 2/5 = 2 ÷ 5 = 0.4 = 40%。
Alternatively, you can make an equivalent fraction with denominator 100.
或者，你也可以先化成分母为 100 的等值分数。
3/4 = 75/100 = 75%
This method is often quicker when the denominator divides exactly into 100.
当分母能整除 100 时，这种方法通常更快。
5. Converting Percentages to Fractions and Decimals | 百分比转换为分数和小数
To convert a percentage to a decimal, divide by 100.
要将百分比转换为小数，除以 100。
For example, 17% = 17 ÷ 100 = 0.17.
例如，17% = 17 ÷ 100 = 0.17。
To convert a percentage to a fraction, write the percentage over 100 and simplify.
要将百分比转换为分数，将百分比写在 100 上方，然后化简。
Thus, 45% = 45/100 = 9/20.
因此，45% = 45/100 = 9/20。
When the percentage includes a decimal, such as 7.5%, write it as 7.5/100 and multiply the numerator and denominator by 10 to get 75/1000 = 3/40.
当百分比包含小数时，例如 7.5%，写成 7.5/100，再将分子和分母同乘以 10，得到 75/1000 = 3/40。
If the percentage is greater than 100, the fraction will be an improper fraction or a mixed number.
如果百分比大于 100，那么得到的分数将是假分数或带分数。
6. Ordering Fractions, Decimals and Percentages | 分数、小数和百分比的排序
To order a mixture, convert all values to the same form, usually decimals or percentages, and then compare.
要对混合数字排序，先将所有值转换为同一种形式，通常是小数或百分比，然后再比较。
For example, order 0.3, 1/2, 40% and 2/5 from smallest to largest.
例如，将 0.3、1/2、40% 和 2/5 从小到大排序。
| Value | As a decimal |
|---|---|
| 0.3 | 0.3 |
| 1/2 | 0.5 |
| 40% | 0.4 |
| 2/5 | 0.4 |
The order is 0.3, 0.4, 0.4, 0.5, so the original order is 0.3, 40% = 2/5, 1/2.
排序为 0.3、0.4、0.4、0.5，因此原始顺序为 0.3、40% = 2/5、1/2。
Watch out for recurring decimals and values that are very close to each other.
注意循环小数和彼此非常接近的值。
Converting everything to decimals is usually the safest approach in an exam.
在考试中，将所有值都转换为小数通常是最稳妥的方法。
7. Common Equivalent Values to Memorise | 需要记住的常见等价值
Some equivalent forms appear so often that you should learn them by heart.
有些等值形式出现得十分频繁，你应该牢记它们。
- 1/10 = 0.1 = 10%
- 1/8 = 0.125 = 12.5%
- 1/4 = 0.25 = 25%
- 1/3 = 0.333… = 33⅓%
- 1/2 = 0.5 = 50%
- 2/3 = 0.666… = 66⅔%
- 3/4 = 0.75 = 75%
- 1 = 1.0 = 100%
Knowing these values speeds up ordering and percentage calculations in mental maths.
掌握这些值可以加快排序和百分比计算的心算速度。
You should also know that 0.125 is one eighth and 0.375 is three eighths, since these appear in many Cambridge questions.
你还应知道 0.125 是八分之一，0.375 是八分之三，因为它们在许多剑桥题目中都会出现。
8. Using a Number Line | 使用数轴
A number line helps you see
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