Fractions, Decimals and Percentages for KS3 | KS3 分数、小数与百分比精通指南

📚 Fractions, Decimals and Percentages for KS3 | KS3 分数、小数与百分比精通指南

Mastering the link between fractions, decimals and percentages is a core skill in the Cambridge KS3 mathematics curriculum. Whether you are sharing a pizza, calculating discounts, or reading data from charts, these three forms of numbers appear everywhere. This guide will take you through each concept step by step, with clear explanations and plenty of examples, so you can confidently tackle any question that comes your way.

掌握分数、小数与百分比之间的联系是剑桥 KS3 数学课程的核心技能。无论是分披萨、计算折扣，还是阅读图表中的数据，这三种数字形式无处不在。本指南将带你逐步理解每一个概念，搭配清晰的说明和丰富的例题，让你能自信地应对任何考题。

1. Understanding Fractions | 理解分数

A fraction shows a part of a whole. It is written with a numerator (the top number) and a denominator (the bottom number), separated by a line. 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.

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，中间用一条分数线隔开。分母告诉你整体被分成了多少等份，分子则告诉你有多少份。

For example, in the fraction 3/4, the denominator 4 means the whole is cut into 4 equal slices, and the numerator 3 means we are looking at 3 of those slices. You can picture this as a pizza cut into 4 pieces; if you eat 3, you have eaten three-quarters of the pizza.

例如，在分数 3/4 中，分母 4 表示整体被切成 4 等份，分子 3 表示我们关注其中的 3 份。你可以想象一个被切成 4 块的披萨；如果你吃了 3 块，你就吃了这个披萨的四分之三。

Fractions can be proper (numerator smaller than denominator, e.g. 2/5), improper (numerator larger than or equal to denominator, e.g. 7/4), or mixed numbers (a whole number and a proper fraction, e.g. 1⅓). All types are used in real life and in exam questions.

分数可以是真分数（分子小于分母，如 2/5）、假分数（分子大于或等于分母，如 7/4）或带分数（一个整数和一个真分数，如 1⅓）。这些类型在现实生活和考试题目中都会用到。

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2. Equivalent Fractions | 等值分数

Equivalent fractions are different fractions that represent the same amount. You can find them by multiplying or dividing the numerator and denominator by the same non-zero number. The appearance changes, but the value stays identical.

等值分数是表示相同数量的不同分数。你可以通过将分子和分母同时乘或除以同一个非零数来找到它们。分数的样子变了，但数值不变。

For instance, 1/2, 2/4, 3/6, and 5/10 are all equivalent because multiplying the top and bottom of 1/2 by 2 gives 2/4, by 3 gives 3/6, and so on. In fact, if you draw a diagram, they all shade exactly half of the shape.

例如，1/2、2/4、3/6 和 5/10 都是等值分数，因为把 1/2 的分子和分母同时乘以 2 得到 2/4，乘以 3 得到 3/6，依此类推。事实上，如果你画图表示，它们都恰好涂满整个图形的一半。

Why is this useful? When adding or subtracting fractions, you often need to rewrite them with a common denominator. Being able to spot and create equivalent fractions makes this process much smoother. It also helps you simplify answers.

为什么这很有用？在进行分数加减法时，你通常需要把它们改写为具有公分母的分数。能够发现和构造等值分数会让这个过程顺畅许多，也有助于化简答案。

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3. Simplifying Fractions | 化简分数

Simplifying a fraction means reducing it to its smallest whole-number form. You do this by dividing both the numerator and the denominator by their highest common factor (HCF). When no number except 1 divides into both, the fraction is in its simplest form.

化简分数意味着把它约简到最小的整数形式。做法是用分子和分母的最大公因数（HCF）同时除以它们。当除了 1 以外没有其他整数能同时整除分子和分母时，这个分数就是最简形式。

Take the fraction 12/16. The highest common factor of 12 and 16 is 4. Dividing top and bottom by 4 gives 3/4. So 12/16 simplifies to 3/4. Always check whether you can cancel down further—sometimes you can do it in steps, e.g. 12/16 to 6/8 to 3/4.

以分数 12/16 为例。12 和 16 的最大公因数是 4。分子分母同时除以 4 得到 3/4。因此 12/16 化简为 3/4。总要检查是否还能继续约简——有时你可以分步进行，例如 12/16 到 6/8 再到 3/4。

Simplifying makes numbers easier to work with and is often required to get full marks in KS3 assessments. Always express your final answer as a fraction in its simplest form, unless the question specifies otherwise.

化简能让数字更易于处理，而且通常在 KS3 测评中必须化简才能得到满分。除非题目另有要求，最终答案一定要用最简分数表达。

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4. Mixed Numbers and Improper Fractions | 带分数与假分数

A mixed number consists of a whole number and a proper fraction, like 2½. An improper fraction has a numerator that is greater than or equal to its denominator, such as 5/2. Both represent the same amount, and you need to be able to switch between the two forms freely.

带分数由一个整数和一个真分数组成，如 2½。假分数的分子大于或等于分母，例如 5/2。两者表示相同的数量，你需要能在这两种形式之间自由转换。

To convert a mixed number to an improper fraction: multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. For 2½, you do 2 × 2 + 1 = 5, so 5/2. To convert an improper fraction to a mixed number: divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator. For 7/3, 7 ÷ 3 = 2 remainder 1, so 2⅓.

将带分数转换为假分数：整数乘以分母，加上分子，结果放在原分母上。对于 2½，计算 2 × 2 + 1 = 5，因此得到 5/2。将假分数转换为带分数：用分子除以分母。商为整数部分，余数为新的分子。对于 7/3，7 ÷ 3 = 2 余 1，因此得到 2⅓。

This skill is essential when you start adding, subtracting, multiplying, or dividing mixed numbers. Most methods require you to convert mixed numbers into improper fractions first, perform the operation, and then convert back if necessary.

当你开始进行带分数的加、减、乘、除运算时，这项技能至关重要。大多数方法都要求先将带分数化为假分数，进行运算，必要时再转换回来。

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5. Adding and Subtracting Fractions | 分数的加减法

To add or subtract fractions, they need to have the same denominator. If the denominators are already the same, simply add or subtract the numerators and keep the denominator unchanged. Always simplify your answer.

进行分数的加减法时，它们需要有相同的分母。如果分母已经相同，直接将分子相加或相减，分母保持不变。最终答案务必要化简。

For example, 2/9 + 4/9 = (2+4)/9 = 6/9, which simplifies to 2/3. And 7/10 – 3/10 = 4/10 = 2/5. With like denominators, the shape of the pieces stays the same size, so we just count how many pieces we have.

例如，2/9 + 4/9 = (2+4)/9 = 6/9，化简为 2/3。而 7/10 – 3/10 = 4/10 = 2/5。当分母相同时，每一份的大小不变，因此只需要计算我们有多少份。

When denominators are different, you must find a common denominator—usually the lowest common multiple (LCM) of the two denominators. Rewrite each fraction as an equivalent fraction with that denominator, then add or subtract. For 1/4 + 1/6, the LCM of 4 and 6 is 12. 1/4 = 3/12 and 1/6 = 2/12, so 3/12 + 2/12 = 5/12.

当分母不同时，必须找到一个公分母——通常是两个分母的最小公倍数（LCM）。将每个分数改写为以该公分母为分母的等值分数，然后进行加减。对于 1/4 + 1/6，4 和 6 的最小公倍数是 12。1/4 = 3/12，1/6 = 2/12，因此 3/12 + 2/12 = 5/12。

If mixed numbers are involved, convert them to improper fractions first. Then follow the same process. Always remember to turn the final answer back into a mixed number and simplify if required.

如果涉及带分数，要先把它们化为假分数，然后遵循同样的步骤。务必记得将最终答案转回带分数，并根据需要进行化简。

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6. Multiplying Fractions | 分数的乘法

Multiplying fractions is often easier than adding them because you don’t need a common denominator. The rule is simple: multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator.

分数的乘法通常比加法更简单，因为你不需要公分母。规则很简单：分子相乘得到新的分子，分母相乘得到新的分母。

a/b × c/d = (a × c) / (b × d)

For example, 2/3 × 4/5 = (2 × 4) / (3 × 5) = 8/15. The fraction 8/15 is already in its simplest form. The key is to combine the numerators and the denominators in one step.

例如，2/3 × 4/5 = (2 × 4) / (3 × 5) = 8/15。分数 8/15 已经是最简形式。关键在于一步到位，将分子和分母分别结合相乘。

You can also simplify before multiplying by cancelling any common factor between a numerator and a denominator, even diagonally. This is called cross-cancelling. In 3/8 × 4/9, notice that 3 and 9 share a factor of 3, and 4 and 8 share a factor of 4. Cancel to get 1/2 × 1/3 = 1/6. Cross-cancelling keeps numbers smaller.

你还可以在相乘之前约简，即找出任意分子和分母（哪怕是对角线上的）的公因数进行约分，这叫做交叉约分。在 3/8 × 4/9 中，注意 3 和 9 有公因数 3，4 和 8 有公因数 4。约分后得到 1/2 × 1/3 = 1/6。交叉约分能让数字保持小一些。

For mixed numbers, convert to improper fractions first. So 1½ × 2⅔ becomes 3/2 × 8/3, which you can simplify before multiplying: 3/2 × 8/3 = (1 × 4)/(1 × 1) = 4. The answer is 4.

对于带分数，先化为假分数。因此 1½ × 2⅔ 变成 3/2 × 8/3，可在相乘前约简：3/2 × 8/3 = (1 × 4)/(1 × 1) = 4。答案是 4。

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7. Dividing Fractions | 分数的除法

To divide by a fraction, you multiply by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator. So the division problem a/b ÷ c/d becomes a/b × d/c.

除以一个分数，相当于乘上它的倒数。一个分数的倒数就是将它的分子和分母互换位置。因此除法算式 a/b ÷ c/d 变成 a/b × d/c。

a/b ÷ c/d = a/b × d/c

For example, 3/5 ÷ 2/7 = 3/5 × 7/2 = (3 × 7) / (5 × 2) = 21/10. You can leave this as an improper fraction or convert it to the mixed number 2⅒. Always check if the final fraction can be simplified.

例如，3/5 ÷ 2/7 = 3/5 × 7/2 = (3 × 7) / (5 × 2) = 21/10。你可以保留这个假分数，或将它化为带分数 2⅒。总要检查最终分数是否可以化简。

If you are dividing whole numbers by fractions, write the whole number as a fraction with denominator 1. So 4 ÷ 2/3 becomes 4/1 × 3/2 = 12/2 = 6. This technique works every time and is a favourite in KS3 tests.

如果你用整数除以分数，就把整数写成分母为 1 的分数。因此 4 ÷ 2/3 变成 4/1 × 3/2 = 12/2 = 6。这个方法屡试不爽，也是 KS3 考试中的常见技巧。

When mixed numbers appear, convert them to improper fractions first. 1½ ÷ 3/4 = 3/2 ÷ 3/4 = 3/2 × 4/3 = 12/6 = 2. With regular practice, dividing fractions becomes as straightforward as multiplying.

当涉及带分数时，先将它们化为假分数。1½ ÷ 3/4 = 3/2 ÷ 3/4 = 3/2 × 4/3 = 12/6 = 2。经过经常练习，分数除法会变得和乘法一样容易。

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8. Converting Fractions to Decimals | 分数转换为小数

A fraction can be turned into a decimal by dividing the numerator by the denominator. You can use short division or long division, and the result might be a terminating decimal or a recurring decimal. This link is fundamental because percentages are based on decimals.

分数可以通过用分子除以分母转换为小数。你可以使用短除法或长除法，结果可能是有限小数，也可能是循环小数。这一联系非常关键，因为百分比正是建立在十进制的基础上的。

For example, 3/8 means 3 ÷ 8 = 0.375. Since the division ends after three decimal places, 0.375 is a terminating decimal. For 1/3, dividing 1 by 3 gives 0.3333… with the 3 repeating forever; we write this as 0.3̄ (with a dot or line over the 3).

例如，3/8 表示 3 ÷ 8 = 0.375。因为除法在小数点后三位就结束了，0.375 就是一个有限小数。对于 1/3，1 除以 3 得到 0.3333…，3 永远循环；我们写为 0.3̄（在 3 上加一个点或横线）。

It helps to memorise some common fractions and their decimal equivalents: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, 1/10 = 0.1, and 1/8 = 0.125. Knowing these can speed up your work and build number sense.

记住一些常见分数及其对应的小数会很有帮助：1/2 = 0.5，1/4 = 0.25，3/4 = 0.75，1/5 = 0.2，1/10 = 0.1 以及 1/8 = 0.125。了解这些能提高你的运算速度并培养数感。

If a fraction has a denominator that can be changed into a power of 10 (10, 100, 1000, …) using equivalent fractions, you can instantly write the decimal. For 7/20, multiply top and bottom by 5 to get 35/100 = 0.35. This trick is excellent for mental maths.

如果一个分数的分母可以利用等值分数化为 10 的乘方（10，100，1000……），你就能立刻写出对应的小数。对于 7/20，将分子分母同时乘 5 得到 35/100 = 0.35。这个技巧对心算非常有帮助。

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9. Converting Decimals to Percentages | 小数转换为百分比

Percent means ‘out of 100’. To turn a decimal into a percentage, multiply it by 100. This is the same as moving the decimal point two places to the right. The reverse is equally important: to change a percentage to a decimal, divide by 100.

百分数表示“每一百中”。要将小数转换为百分比，将它乘以 100。这相当于把小数点向右移动两位。反过来也同样重要：要将百分比转换为小数，就除以 100。

For instance, 0.47 becomes 0.47 × 100 = 47%. A decimal like 0.6 is 0.6 × 100 = 60%. If the decimal has more than two places, like 0.375, it becomes 37.5%, which is perfectly valid. Percentages don’t have to be whole numbers.

例如，0.47 变为 0.47 × 100 = 47%。像 0.6 这样的小数变为 0.6 × 100 = 60%。如果小数位数超过两位，比如 0.375，它可以写成 37.5%，这完全有效。百分比不一定是整数。

What about recurring decimals? For 0.3333… (or 1/3 as a decimal), we often write 33.3̄% or 33⅓% to show the exact value. In KS3 you are usually expected to give a decimal percentage like 33.3% (to one decimal place) or keep it as a fraction form like 33⅓%, depending on the question.

那么循环小数呢？对于 0.3333…（或 1/3 的小数形式），我们通常写作 33.3̄% 或 33⅓% 来表示精确的值。在 KS3 中，一般要求给出像 33.3%（保留一位小数）这样的小数百分数，或根据题意保留成分数形式，如 33⅓%。

This conversion allows you to compare numbers easily. Knowing that 0.45 is 45% and 0.4 is 40% makes it clear that 0.45 is larger. The ability to move freely between decimals and percentages is tested frequently, especially in data and probability questions.

这种转换使你能轻松比较数值。知道 0.45 是 45%，0.4 是 40%，就能清楚地看出 0.45 更大。在数据和概率题目中，经常考到小数与百分比之间来回转换的能力。

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10. Converting Percentages Back to Fractions | 百分比转换为分数

To convert a percentage to a fraction, write the percentage as a number over 100, and then simplify if possible. This returns you to the most basic form of the number. For example, 25% is 25/100, which simplifies to 1/4.

要将百分比转换为分数，先把百分数写成一个分母为 100 的分数，然后尽可能地化简。这样你就回到了这个数最基本的形式。例如，25% 是 25/100，可化简为 1/4。

If the percentage includes a decimal, like 12.5%, write it as 12.5/100, then multiply top and bottom by 10 to clear the decimal: 125/1000. Simplify by dividing by 125 to get 1/8. So 12.5% is exactly 1/8. This method works for any decimal percentage.

如果百分数包含小数，比如 12.5%，把它写成 12.5/100，然后分子分母同乘 10 以消除小数：125/1000。通过除以 125 化简得到 1/8。因此 12.5% 恰好等于 1/8。这个方法适用于任何带有小数的百分数。

Here is a table of common percentage-fraction equivalents every KS3 student should know:

下面是一个每位 KS3 学生都应熟记的常见百分数与分数的等值表：

Percentage	Fraction	百 分 数	分 数
50%	1/2	50%	1/2
25%	1/4	25%	1/4
75%	3/4	75%	3/4
20%	1/5	20%	1/5
10%	1/10	10%	1/10
1%	1/100	1%	1/100

Being confident with these conversions helps when solving problems involving discounts, interest, probability, and interpreting data from graphs. It ties the whole topic together.

对这些转换充满信心，有助于解决涉及折扣、利息、概率和解读图表数据的问题。它将整个专题串联起来。

The complete cycle—from fraction to decimal to percentage and back—is a powerful tool in KS3 mathematics. If you can smoothly navigate between these three forms, you have mastered one of the most practical and frequently assessed skills in the curriculum.

从分数到小数到百分比再回到分数的完整循环，是 KS3 数学中的一个强大工具。如果你能在这三种形式之间自如地穿梭，你就掌握了课程中最实用、最常考核的技能之一。

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