Mastering Ratio and Proportion | 掌握比与比例

📚 Mastering Ratio and Proportion | 掌握比与比例

Ratio and proportion appear throughout KS3 mathematics, from sharing money and mixing ingredients to reading scale drawings and maps. This article explains the key ideas, methods and exam techniques you need for Cambridge Lower Secondary Mathematics, including the type of question often found around Exercise 2 on page 147.

比与比例贯穿 KS3 数学的各个部分，从分钱、混合配料到阅读比例图和地图都会用到。本文讲解剑桥初中数学中你需要掌握的核心概念、常用方法和考试技巧，包括第 147 页练习 2 中常见的题型。

1. What Is a Ratio? | 什么是比？

A ratio compares two or more quantities of the same kind. For example, if a class has 12 boys and 18 girls, the ratio of boys to girls is written as 12 : 18.

比用来比较两个或两个以上同类量。例如，一个班有 12 名男生和 18 名女生，那么男生与女生的比写作 12 : 18。

The order of a ratio matters. The ratio 2 : 3 is not the same as 3 : 2, because the first number is linked to the first quantity named.

比的顺序很重要。2 : 3 与 3 : 2 不同，因为第一个数对应第一个提到的量。

You can only compare quantities in a ratio if they use the same unit. For example, 50 cm to 1 m must first be written as 50 cm : 100 cm.

只有单位相同的量才能用比进行比较。例如，50 cm 比 1 m 必须先写作 50 cm : 100 cm。

2. Simplifying Ratios | 化简比

To simplify a ratio, divide every part by the highest common factor (HCF). For 12 : 18, the HCF of 12 and 18 is 6, so 12 : 18 simplifies to 2 : 3.

化简比时，要把比的每一项都除以最大公约数（HCF）。对于 12 : 18，12 和 18 的最大公约数是 6，因此 12 : 18 化简为 2 : 3。

If a ratio contains fractions or decimals, multiply all parts by the same number to make integers first. For example, ¼ : ½ becomes 1 : 2 after multiplying both sides by 4.

如果比中含有分数或小数，可以先将所有项乘以同一个数化为整数。例如，¼ : ½ 两边同时乘以 4 后变为 1 : 2。

The simplified form must contain only integers with no common factor except 1.

最简比只能含有整数，且除了 1 以外没有其他公因数。

3. Equivalent Ratios | 等价比

Equivalent ratios are produced by multiplying or dividing every part by the same non-zero number. For instance, 2 : 3 is equivalent to 4 : 6 and 10 : 15.

等价比是将比的每一项同时乘或除以同一个非零数得到的。例如，2 : 3 等价于 4 : 6 和 10 : 15。

To compare ratios, it is often useful to write them with the same total number of parts or with one side equal to 1.

为了比较不同的比，通常需要将它们写成总份数相同或其中一项为 1 的形式。

Example: Are 3 : 5 and 9 : 15 equivalent? Multiply both sides of 3 : 5 by 3 to get 9 : 15, so yes.

例子：3 : 5 和 9 : 15 是否等价？将 3 : 5 两边同乘 3 得到 9 : 15，所以等价。

4. Dividing a Quantity in a Given Ratio | 按给定比分配

To share a quantity in a given ratio, first find the total number of parts by adding

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