📚 Mastering Fractions, Decimals and Percentages | 精通分数、小数和百分数
Being able to move confidently between fractions, decimals and percentages is one of the most important skills you will develop in KS3 mathematics. Whether you are working out a discount in a shop, interpreting data in a science experiment, or solving problems in later topics such as ratio, proportion and probability, a solid understanding of these three forms is essential. This article will guide you through the key ideas, conversions and real‑life uses, giving you plenty of practice along the way.
能够在分数、小数和百分数之间自如转换，是你在 KS3 数学阶段需要掌握的最重要技能之一。无论是在商店里计算折扣，还是在科学实验中解读数据，亦或是在之后学习比、比例和概率时解决问题，牢固掌握这三种表示形式都至关重要。本文将带你梳理核心概念、转换方法以及生活中的应用，并在过程中提供大量的练习。
1. Understanding Fractions | 理解分数
A fraction represents a part of a whole. It is written with a numerator (top number) and a denominator (bottom number). The denominator tells you how many equal parts the whole is divided into, while the numerator tells you how many of those parts you have. For example, 3/4 means 3 parts out of 4 equal parts.
分数表示一个整体的一部分。它由分子（上面的数）和分母（下面的数）组成。分母告诉你整体被分成了多少等份，分子则告诉你拥有其中的多少份。例如，3/4 表示从 4 等份中取出 3 份。
Proper fractions have a numerator smaller than the denominator (e.g. 2/5). Improper fractions have a numerator larger than or equal to the denominator (e.g. 7/4). Mixed numbers combine a whole number and a proper fraction (e.g. 1 2/3). It is useful to be able to switch between improper fractions and mixed numbers during calculations.
真分数的分子比分母小（如 2/5）。假分数的分子大于或等于分母（如 7/4）。带分数则由一个整数和一个真分数组成（如 1 2/3）。在计算过程中，能够在假分数和带分数之间切换会非常有用。
2. Equivalent Fractions | 等值分数
Equivalent fractions look different but represent exactly the same amount. You can create an equivalent fraction by multiplying or dividing both the numerator and the denominator by the same non‑zero number. For example, 1/2 = 2/4 = 3/6 = 4/8. This is like slicing the same pizza into more pieces – the size of your portion stays the same.
等值分数虽然看起来不同，但表示的大小完全相同。你可以通过将分子和分母同时乘或除以同一个非零数来得到等值分数。例如，1/2 = 2/4 = 3/6 = 4/8。这就好比把同一张比萨切成更多块——你得到的份量保持不变。
To check whether two fractions are equivalent, cross‑multiply: if a/b = c/d then a × d = b × c. For 2/3 and 8/12, 2 × 12 = 24 and 3 × 8 = 24, so they are equivalent.
检查两个分数是否等值，可以用交叉相乘：如果 a/b = c/d，那么 a × d = b × c。对于 2/3 和 8/12，2 × 12 = 24，3 × 8 = 24，因此它们是等值分数。
3. Simplifying Fractions | 化简分数
Simplifying (or reducing) a fraction means writing it in its simplest form, where the numerator and denominator have no common factors other than 1. You do this by dividing both numbers by their highest common factor (HCF). For example, to simplify 12/16, the HCF of 12 and 16 is 4, so 12 ÷ 4 / 16 ÷ 4 = 3/4.
化简（或约分）一个分数，就是把它写成最简形式，也就是分子和分母除了 1 以外没有其他公因数。做法是用它们的最大公因数（HCF）同时去除分子和分母。例如，化简 12/16，12 和 16 的 HCF 是 4，所以 12÷4 / 16÷4 = 3/4。
Sometimes you may need to simplify in steps, dividing by small common factors several times until you cannot divide any further. Being able to recognise common factors quickly makes this process much faster.
有时你可能需要分步化简，用小的公因数多次去除，直到不能再分为止。能够快速识别公因数会大大加快这一过程。
4. Fractions and Decimals | 分数与小数
A decimal is another way of writing a fraction whose denominator is a power of ten. For instance, 0.3 means 3/10, and 0.07 means 7/100. The decimal point separates whole units from fractional parts. The place values after the point are tenths, hundredths, thousandths, and so on.
小数是分母为 10 的幂的分数的另一种写法。比如，0.3 表示 3/10，0.07 表示 7/100。小数点把整数部分和小数部分分开。小数点后面的数位分别是十分位、百分位、千分位等等。
Understanding this connection helps you see that terminating decimals (those that stop) come from fractions that in simplest form only have prime factors 2 and 5 in the denominator. For example, 3/8 = 0.375 because 8 = 2³, so the fraction turns into a terminating decimal.
理解这种联系有助于你看出，有限小数（能够写完的小数）来自那些化简后分母只含有质因数 2 和 5 的分数。例如，3/8 = 0.375，因为 8 = 2³，所以这个分数能化成有限小数。
5. Converting Fractions to Decimals | 分数转小数
To convert a fraction to a decimal, divide the numerator by the denominator. You can use short division or a calculator. For example, 5/8 = 5 ÷ 8 = 0.625. With a mixed number, convert the whole number part just as it is and divide the fraction part, then add them: 2 3/4 = 2 + (3 ÷ 4) = 2.75.
把分数转换为小数，用分子除以分母即可。你可以使用短除法或计算器。例如，5/8 = 5 ÷ 8 = 0.625。对于带分数，整数部分保持不变，只对分数部分做除法，然后把两部分相加：2 3/4 = 2 + (3 ÷ 4) = 2.75。
If the division does not stop and the digits repeat, you get a recurring decimal. For instance, 1/3 = 0.333…, written as 0.3̅. Similarly, 5/11 = 0.4545…, written as 0.4̅5̅. Cambridge KS3 expects you to recognise and use standard notation for recurring decimals.
如果除法不能除尽，数字开始循环，你就会得到循环小数。例如，1/3 = 0.333…，记作 0.3̅。同样，5/11 = 0.4545…，记作 0.4̅5̅。剑桥 KS3 课程要求你认识并使用循环小数的标准记法。
6. Converting Decimals to Fractions | 小数转分数
Write the decimal as a fraction over its place value. For a terminating decimal, 0.6 = 6/10 (simplify to 3/5), 0.75 = 75/100 (simplify to 3/4), and 0.125 = 125/1000 (simplify to 1/8). Always simplify the fraction if possible.
根据小数的数位把它写成分数。对于有限小数，0.6 = 6/10（化简为 3/5），0.75 = 75/100（化简为 3/4），0.125 = 125/1000（化简为 1/8）。如果可以，最后一定要化简分数。
For a recurring decimal, a slightly different method is used. For a simple recurring decimal like 0.7̅ (meaning 0.777…), let x = 0.777…, then 10x = 7.777… Subtract: 10x – x = 7.777… – 0.777… → 9x = 7 → x = 7/9. So 0.7̅ = 7/9.
对于循环小数，则需要使用一种略微不同的方法。以简单的循环小数 0.7̅（即 0.777…）为例，设 x = 0.777…，则 10x = 7.777…。相减：10x – x = 7.777… – 0.777… → 9x = 7 → x = 7/9。因此 0.7̅ = 7/9。
| Decimal | 小数 | Fraction | 分数 |
|---|---|
| 0.5 | 1/2 |
| 0.25 | 1/4 |
| 0.2 | 1/5 |
| 0.1̅ | 1/9 |
| 0.3̅ | 1/3 |
7. Percentages | 百分数
A percentage is a fraction with a denominator of 100. The word ‘per cent’ means ‘out of 100’. So 45% means 45/100, or 0.45 as a decimal. Percentages are especially useful when comparing proportions or describing changes, because they always refer to the same whole of 100.
百分数就是分母为 100 的分数。“百分之”的意思就是“每一百份中的”。因此 45% 表示 45/100，也就是小数 0.45。在比较比例或描述变化时，百分数特别有用，因为参照的总量总是 100。
To find a percentage of an amount, multiply the amount by the percentage as a decimal. For example, 20% of 60 is 0.20 × 60 = 12. You can also use fraction equivalents: 20% = 1/5, so 60 ÷ 5 = 12.
求一个数的百分之几，用这个数乘上百分数对应的小数。例如，60 的 20% 是 0.20 × 60 = 12。你也可以使用等值分数：20% = 1/5，所以 60 ÷ 5 = 12。
8. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分数的相互转换
The three forms are interchangeable. The table below summarises the conversions you need to be able to do fluently at KS3. Being able to recall common equivalents (such as 1/4 = 0.25 = 25%) without having to calculate each time saves time in exams.
这三种形式可以相互转换。下表总结了你在 KS3 阶段需要熟练掌握的转换方法。能够不假思索地回想起常见的等值关系（如 1/4 = 0.25 = 25%），可以节省考试中的时间。
| From | 从 | To | 到 | Method | 方法 |
|---|---|---|
| Fraction | Decimal | Divide numerator by denominator |
| Decimal | Fraction | Write over 10, 100, 1000 etc. and simplify |
| Decimal | Percentage | Multiply by 100 |
| Percentage | Decimal | Divide by 100 |
| Percentage | Fraction | Write as fraction over 100 and simplify |
| Fraction | Percentage | Convert to decimal first, then multiply by 100 |
9. Comparing and Ordering | 比较与排序
To compare fractions, decimals and percentages, you should convert them all to the same form. Often, converting to decimals is the easiest, because you can simply compare the digits. For example, to order 3/5, 58% and 0.61 from smallest to largest, write 3/5 = 0.6, 58% = 0.58, and compare with 0.61. So the order is 0.58 (58%), 0.6 (3/5), 0.61.
要比较分数、小数和百分数，你应该把它们全部转化成同一种形式。通常，转换成小数是最简单的，因为可以直接比较数字的大小。例如，将 3/5、58% 和 0.61 从小到大排列，先写出 3/5 = 0.6，58% = 0.58，再与 0.61 比较。因此顺序是 0.58（58%），0.6（3/5），0.61。
With fractions that have different denominators, you can either find a common denominator or convert to decimals. For 2/3 and 3/4, a common denominator is 12: 2/3 = 8/12, 3/4 = 9/12, so 3/4 is larger. As decimals, 2/3 ≈ 0.667 and 3/4 = 0.75, confirming the same result.
对于分母不同的分数，你可以通分或者转换成小数进行比较。对于 2/3 和 3/4，公分母是 12：2/3 = 8/12，3/4 = 9/12，所以 3/4 更大。用小数比较，2/3 ≈ 0.667，3/4 = 0.75，同样得出 3/4 更大的结果。
10. Real‑life Applications | 实际应用
Fractions, decimals and percentages appear in everyday situations. In shopping, a 25% discount on a £40 item means you save £10, so you pay £30. Here, 25% = 0.25, and 0.25 × 40 = 10. In cooking recipes, if a recipe for 4 people needs 3/4 cup of flour, for 2 people you would halve it to 3/8 cup.
分数、小数和百分数在日常生活中随处可见。购物时，一件 £40 的商品打 25% 的折扣，就意味着你省了 £10，所以最终付款 £30。这里 25% = 0.25，而 0.25 × 40 = 10。在烹饪中，如果一个 4 人份的食谱需要 3/4 杯面粉，那么 2 人份时就该减半为 3/8 杯。
In statistics and probability, a chance of 0.3 means the same as 30% or 3/10. When reading charts or data tables, you often need to switch between these forms to make sense of the numbers. Mastery of this topic therefore directly supports your work in science, geography and everyday numeracy.
在统计和概率中，0.3 的几率等同于 30% 或 3/10。在阅读图表或数据表时，你经常需要在这些形式之间切换，以便更好地理解数字。因此，掌握本课题直接有助于你在科学、地理以及日常计算中的学习。
11. Practice Problems | 练习题
Test your understanding with these questions. Try to complete them without a calculator first.
用这些问题来检测你的理解。先试着不借助计算器完成。
- Convert 7/8 to a decimal and then to a percentage.
- Convert 0.6̅ to a fraction in its simplest form.
- Write 0.04 as a fraction and as a percentage.
- Order 2/5, 0.45 and 43% from largest to smallest.
- Find 15% of 200.
- A drink is made from 2/5 orange juice and the rest lemonade. What percentage of the drink is lemonade?
- 将 7/8 转换为小数，再转换为百分数。
- 将 0.6̅ 化为最简分数。
- 将 0.04 写成分数和百分数。
- 将 2/5、0.45 和 43% 从大到小排列。
- 求 200 的 15%。
- 一款饮料由 2/5 的橙汁和其余的柠檬水组成。柠檬水占饮料的百分之几？
Check your answers: 7/8 = 0.875 = 87.5%; 0.6̅ = 2/3; 0.04 = 1/25 = 4%; Largest to smallest: 0.45, 43% (0.43), 2/5 (0.4); 15% of 200 = 30; Lemonade is 3/5 = 60%.
核对你的答案：7/8 = 0.875 = 87.5%；0.6̅ = 2/3；0.04 = 1/25 = 4%；从大到小：0.45，43%（0.43），2/5（0.4）；200 的 15% = 30；柠檬水占 3/5 = 60%。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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