📚 Fractions, Decimals and Percentages | 分数、小数和百分数
In the Cambridge KS3 Mathematics curriculum, fractions, decimals and percentages are treated as three connected ways of describing the same proportion. Being able to switch between them quickly is an essential skill for Checkpoint tests and for later IGCSE work. This article explains the key methods and gives you exam-style examples to build confidence.
在剑桥 KS3 数学课程中，分数、小数和百分数被视为描述同一比例的三种相互关联的方式。能够快速在它们之间进行转换，是 Checkpoint 考试以及后续 IGCSE 学习中的一项基本技能。本文将解释关键方法，并给出考试风格的例题，帮助你建立信心。
1. Understanding Equivalent Forms | 理解等价形式
A fraction such as 1/2, a decimal such as 0.5 and a percentage such as 50% all represent the same part of a whole. The word 'percent' means 'out of 100', so 50% is literally 50 out of 100.
像 1/2 这样的分数、0.5 这样的小数和 50% 这样的百分数都表示整体的同一部分。'百分数' 一词的意思是 '每一百份中的若干份'，因此 50% 字面上就是 100 份中的 50 份。
When you see these three forms in a question, you should not think of them as separate topics. Instead, treat them as interchangeable labels for the same value. For example, 0.25, 1/4 and 25% all describe exactly the same proportion of a shape or quantity.
当你在题目中看到这三种形式时，不应把它们当作独立的知识点。相反，要把它们看作同一数值的可互换标签。例如，0.25、1/4 和 25% 都描述了一个图形或数量完全相同的比例。
A useful way to picture this is to draw a 100-square grid. If 25 small squares are shaded, then 25/100 of the grid is shaded, which can also be written as 0.25 or 25%. This visual link helps you check whether a conversion is reasonable.
一个有用的方法是在 100 格方格纸上画图。如果有 25 个小方格被涂色，那么整个方格的 25/100 被涂色，这也可以写成 0.25 或 25%。这种直观的联系能帮助你检查转换结果是否合理。
2. Converting Fractions to Decimals | 把分数转换为小数
To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/4 means 3 ÷ 4, which gives 0.75.
要把分数转换为小数，用分子除以分母。例如，3/4 表示 3 ÷ 4，结果是 0.75。
3 ÷ 4 = 0.75
division. For example, 1/8 = 1 ÷ 8 = 0.125 because 8 divides into 1000 exactly 125 times.
如果分母不是 10、100 或 1000 的因数，你可能需要进行长除法。例如，1/8 = 1 ÷ 8 = 0.125，因为 1000 正好有 125 个 8。
is often shown with a dot above the repeated digit, but in KS3 you can write the decimal to two or three decimal places if the question asks for a rounded answer.
有些分数会产生循环小数。例如，1/3 = 0.333...，2/3 = 0.666...。循环部分通常在重复数字上方加一个点来表示，但在 KS3 阶段，如果题目要求取近似值，你可以把小数写到两位或三位小数。
| Fraction | Decimal |
|---|---|
| 1/2 | 0.5 |
| 1/4 | 0.25 |
| 3/4 | 0.75 |
| 1/5 | 0.2 |
| 1/8 | 0.125 |
erator is greater than the denominator, you still divide: for example, 5/4 = 5 ÷ 4 = 1.25.
这些常见的转换应该熟记，这样你就能在应用题中立即使用它们。当分子大于分母时，仍然进行除法运算：例如，5/4 = 5 ÷ 4 = 1.25。
3. Converting Decimals to Percentages | 把小数转换为百分数
To convert a decimal to a percentage, multiply the decimal by 100 and add the % symbol. For example, 0.45 × 100 = 45%, and 0.07 × 100 = 7%.
要把小数转换为百分数，将小数乘以 100 并加上 % 符号。例如，0.45 × 100 = 45%，0.07 × 100 = 7%。
0.45 × 100 = 45%
Equivalently, you can move the decimal point two places to the right. This shortcut works because a percentage is just a fraction with denominator 100. So 0.625 becomes 62.5%, and 1.3 becomes 130%.
同样地，你可以把小数点向右移动两位。这个快捷方法之所以有效，是因为百分数就是分母为 100 的分数。因此 0.625 变为 62.5%，1.3 变为 130%。
Be careful with zeros: 0.08 is not 8% until you move the point two places. It becomes 8%, but 0.8 becomes 80%. A quick way to check is to ask whether the decimal is less than, equal to or greater than 1. Decimals less than 1 give percentages less than 100%; decimals greater than 1 give percentages greater than 100%.
要注意零的位置：0.08 只有在把小数点移动两位之后才是 8%。0.8 会变成 80%。一个快速的检查方法是：判断小数是小于、等于还是大于 1。小于 1 的小数得出的百分数小于 100%；大于 1 的小数得出的百分数大于 100%。
4. Converting Percentages to Fractions | 把百分数转换为分数
age to a fraction, write the percentage over 100 and simplify if possible. For example, 25% = 25/100 = 1/4 after dividing the numerator and denominator by 25.
要把百分数转换为分数，把百分数写在 100 之上，并尽可能化简。例如，25% = 25/100，将分子和分母同时除以 25 后得到 1/4。
25% = 25/100 = 1/4
- 50% = 50/100 = 1/2
- 20% = 20/100 = 1/5
- 75% = 75/100 = 3/4
- 10% = 10/100 = 1/10
- 5% = 5/100 = 1/20
If the percentage includes a decimal, such as 12.5%, first write it as 12.5/100, then multiply numerator and denominator by 10 to get 125/1000, which simplifies to 1/8.
如果百分数包含小数，例如 12.5%，先写成 12.5/100，然后将分子和分母同时乘以 10，得到 125/1000，化简后为 1/8。
You can also convert a percentage greater than 100% to an improper fraction or mixed number. For example, 150% = 1 1/2. This is useful in percentage increase problems where an amount more than doubles.
你也可以将大于 100% 的百分数转换为假分数或带分数。例如，150% = 150/100 = 3/2 = 1 1/2。这在数量增加超过一倍的百分数增加问题中很有用。
5. Ordering Fractions, Decimals and Percentages | 比较分数、小数和百分数的大小
When a question asks you to order a mixture such as 2/5, 0.35 and 30%, convert all of them into the same form first. Decimals are often the easiest common form.
当题目要求你比较 2/5、0.35 和 30% 这样混合形式的数时，先把它们都转换成同一种形式。小数通常是最容易使用的共同形式。
Convert 2/5 to 0.4 and 30% to 0.30. Then the order from smallest to largest is clear: 0.30 < 0.35 < 0.40.
将 2/5 转换为 0.4，将 30% 转换为 0.30。然后从小到大的顺序就很清楚了：0.30 < 0.35 < 0.40。
30% < 0.35 < 2/5
3/5 = 9/15, so 2/3 > 3/5. Choosing decimals is often quicker when a calculator is allowed.
直接比较分数时，可以使用公分母。例如，要比较 2/3 和 3/5，把两者都化为十五分之几：2/3 = 10/15，3/5 = 9/15，所以 2/3 > 3/5。在允许使用计算器时，选择小数通常更快。
Always line up the decimal points and compare place values carefully. Adding extra zeros, such as writing 0.35 as 0.350 and 0.4 as 0.400, can help you avoid mistakes.
始终把小数点对齐，并仔细比较数位。补上额外的零，例如把 0.35 写成 0.350，把 0.4 写成 0.400，可以帮助你避免错误。
6. Finding a Percentage of a Quantity | 求一个数量的百分之几
centage as a fraction with denominator 100, then multiply by the quantity. For example, 15% of 60 is (15/100) × 60 = 9.
要求一个数量的百分之几，把百分数写成分母为 100 的分数，然后乘以这个数量。例如，60 的 15% 是 (15/100) × 60 = 9。
15% of 60 = (15/100) × 60 = 9
You can also use a decimal multiplier. Since 15% ecomes 0.15 × 60 = 9. This method is especially useful in IGCSE-style percentage problems.
你也可以使用小数乘数。因为 15% = 0.15，所以计算变为 0.15 × 60 = 9。这种方法在 IGCSE 风格的百分数问题中特别有用。
Without a calculator, you can use a simple breakdown. For example, to find 17% of 80, you could find 10% = 8, 5% = 4, 1% = 0.8 and 1% again = 0.8, then add 8 + 4 + 0.8 + 0.8 =
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