📚 Mastering Ratio and Proportion for Cambridge KS3 Maths | 掌握剑桥KS3数学中的比与比例
Ratio and proportion are central to the Cambridge KS3 maths curriculum. They appear in number problems, algebra, scale drawing, geometry and everyday comparisons, and they form the foundation of the work you will later meet at IGCSE. This article explains the key methods step by step so that you can simplify ratios, share amounts, link ratios to fractions and percentages, and solve both direct and inverse proportion questions accurately.
比和正比例、反比例是剑桥 KS3 数学课程的核心内容。它们出现在数值问题、代数、比例尺、几何以及日常比较中，也是你日后学习 IGCSE 的基础。本文将一步步讲解关键方法，帮助你准确化简比、按比分配数量、把比与分数和百分数联系起来，并解答正比例与反比例问题。
1. What Is a Ratio? | 什么是比？
A ratio compares two or more quantities of the same type in a fixed order. The ratio 3 : 2 means that for every 3 parts of the first quantity there are 2 parts of the second quantity. The colon is read as the word to, so 3 : 2 is read as three to two.
比用于按固定顺序比较两个或多个同类的数量。3 : 2 表示第一项每有 3 份，第二项就有 2 份。冒号读作"比"，因此 3 : 2 读作"三比二"。
Order matters. 3 : 2 is not the same as 2 : 3, because swapping the terms changes the meaning completely. If a recipe mixes flour and sugar in the ratio 3 : 2, the flour is the larger amount. If the ratio were written the other way round, the recipe would need more sugar than flour.
顺序很重要。3 : 2 与 2 : 3 并不相同，因为交换两项会完全改变含义。如果一份食谱中面粉与糖的比是 3 : 2，那么面粉的量更大；如果反过来写，就意味着糖比面粉多。
A ratio describes relative size, not the actual total. A class with 12 boys and 8 girls has the same ratio as a class with 30 boys and 20 girls, because both simplify to 3 : 2. The second class is simply a scaled-up version of the first, so the balance between boys and girls is identical.
比描述的是相对大小，而不是实际总量。一个有 12 名男生和 8 名女生的班级，与一个有 30 名男生和 20 名女生的班级具有相同的比，因为两者都化简为 3 : 2。第二个班级只是第一个班级按比例放大的版本，男女生之间的比例完全一致。
- Ratio = a relative comparison, not a total | 比 = 相对比较，而不是总量
- Term or part = one value inside the ratio | 项 = 比中的一个数值
- Order matters: a : b ≠ b : a | 顺序很重要：a : b ≠ b : a
- Both quantities must be in the same unit | 两个量必须使用相同单位
A ratio can also be written in fraction form, because a : b can be written as a/b. This is useful later, when ratios are used in algebra and in proportion equations.
比也可以写成分数形式，因为 a : b 可以写成 a/b。这在后面把比用于代数和比例方程时非常有用。
2. Simplifying Ratios | 化简比
To simplify a ratio, divide every part by the highest common factor, or HCF. A ratio is unchanged when all parts are multiplied or divided by the same non-zero number, so simplifying keeps the meaning exactly the same while making the numbers easier to read and compare.
化简比时，将比中的每一项除以它们的最大公因数，简称 HCF。由于比的所有项同时乘或除以同一个不为零的数时比值不变，所以化简不改变含义，只让数字更容易阅读和比较。
For example, 12 and 8 have HCF 4, so both parts are divided by 4. The ratio keeps the same value but is now written in its simplest form.
例如，12 和 8 的最大公因数是 4，因此两项都除以 4。比的值没有改变，但现在写成最简形式。
12 : 8 = (12 ÷ 4) : (8 ÷ 4) = 3 : 2
dividing both parts by any common factor and repeating the process also works, but it takes longer.
再看一例：24 : 36 的最大公因数是 12，所以 24 : 36 = 2 : 3。如果一时看不出最大公因数，也可以用任意公因数先除，再重复这个过程，只是会慢一些。
If the ratio includes decimals, multiply all parts by the same power of 10 to obtain integers first. If it includes fractions, multiply all parts by the lowest common multiple of the denominators.
如果比中包含小数，先给所有项同时乘以 10 的幂，转化为整数比。如果比中包含分数，就给所有项同时乘以各分母的最小公倍数。
1.2 : 0.8 × 10 = 12 : 8 = 3 : 2
½ : ⅓ × 6 = 3 : 2
Always check the units before simplifying. A ratio must compare quantities measured in the same unit, so 50p : £2 must first be changed to 50 : 200, which simplifies to 1 : 4.
化简前一定要检查单位。比必须比较用相同单位表示的数量，因此 50p : £2 要先化为 50 : 200，再化简为 1 : 4。
3. Equivalent Ratios | 等价比
Equivalent ratios are produced by multiplying or dividing every part of a ratio by the same number. Starting from 3 : 2 and multiplying by 2, 3 and 10 gives 6 : 4, 9 : 6 and 30 : 20. All of these describe the same relationship, and all of them simplify back to 3 : 2.
把比的每一项同时乘或除以同一个数，就得到等价比。从 3 : 2 出发，分别乘以 2、3 和 10，得到 6 : 4、9 : 6 和 30 : 20。它们描述的都是同一种关系，而且都能化简回 3 : 2。
Since 3 : 4 = 21 : 28 and 5 : 7 = 20 : 28, the ratio 3 : 4 is slightly larger than 5 : 7.
等价比可以用来比较两个看起来差别很大的比。要比较 3 : 4 和 5 : 7，可以把它们都改写为第二项相同的比。由于 3 : 4 = 21 : 28，5 : 7 = 20 : 28，可见 3 : 4 略大于 5 : 7。
g x = 25. Cross-multiplication gives the same result: 5 × 40 = 8 × x, so x = 200 ÷ 8 = 25.
等价比也可用于求比中缺失的项。如果 5 : 8 = x : 40，第二项被乘以了 5，那么第一项也必须乘以 5，得到 x = 25。交叉相乘得到同样的结果：5 × 40 = 8 × x，所以 x = 200 ÷ 8 = 25。
5 : 8 = 25 : 40
- Multiply every part by the same number to scale up | 各项同乘一个数即放大
- Divide every part by the same number to scale down | 各项同除一个数即缩小
- Simplify only when the question asks for simplest form | 只有在题目要求时才必须化为最简
4. Sharing an Amount in a Given Ratio | 按比分配数量
Sharing an amount in a given ratio is one of the most common KS3 questions. There are three steps: add the parts of the ratio to find the total number of parts, divide the total amount by that number to find the value of one part, then multiply to find each share.
按比分配数量是 KS3 中最常见的题型之一。解题分三步：先把比的各项相加，求出总份数；再用总量除以总份数，求出一份的值；最后分别相乘，求出每一份的数量。
Suppose £ ween two people in the ratio 2 : 3. The total number of parts is 2 + 3 = 5, so one part is £60 ÷ 5 = £12. The first person receives 2 × £12 = £24 and the second receives 3 × £12 = £36. As a check, £24 + £36 = £60, which matches the original total.
假设把 60 英镑按 2 : 3 分给两个人。总份数是 2 + 3 = 5，所以一份是 60 ÷ 5 = 12 英镑。第一个人得到 2 × 12 = 24 英镑，第二个人得到 3 × 12 = 36 英镑。检验一下：24 + 36 = 60 英镑，与原总量一致。
share = total × (part ÷ total number of parts)
| Ratio | 比 | Total parts | 总份数 | One part | 一份 | Shares | 各份 |
|---|---|---|---|
| 2 : 3 (£60) | 5 | £12 | £24 and £36 |
| 5 : 3 (240 g) | 8 | 30 g | 150 g and 90 g |
| 1 : 4 (250 ml) | 5 | 50 ml | 50 ml and 200 ml |
Always add the shares back together at the end of the question. This single check catches most arithmetic slips and is worth a mark in many mark schemes.
每道题做完后，都要把各份相加检验一次。这个简单的检查能发现大多数计算失误，在不少评分标准中也能拿到方法分。
5. Three-Part Ratios and Word Problems | 三项比与应用题
the total amount is 400 g, one part is 40 g and the three shares are 80 g, 120 g and 200 g.
像 2 : 3 : 5 这样的三项比，处理方法完全相同。总份数是 2 + 3 + 5 = 10，因此如果总量是 400 克，一份就是 40 克，三份分别是 80 克、120 克和 200 克。
e of 3 and 4 is 12, so 2 : 3 becomes 8 : 12 and 4 : 5 becomes 12 : 15. The combined ratio is a : b : c = 8 : 12 : 15.
当两个比含有同一项时，可以把它们合并。已知 a : b = 2 : 3，b : c = 4 : 5，先要让 b 项相同。3 和 4 的最小公倍数是 12，所以 2 : 3 变成 8 : 12，4 : 5 变成 12 : 15。合并后的比是 a : b : c = 8 : 12 : 15。
2 : 3 = 8 : 12 and 4 : 5 = 12 : 15 → a : b : c = 8 : 12 : 15
hares instead of the total. If two quantities are in the ratio 7 : 4 and the difference is £36, then the difference in parts is 7 − 4 = 3, so one part is £36 ÷
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