Mastering Ratio and Proportion for KS3 Cambridge Maths | KS3剑桥数学比与比例精讲

📚 Mastering Ratio and Proportion for KS3 Cambridge Maths | KS3剑桥数学比与比例精讲

Ratio and proportion are essential tools in KS3 mathematics, helping us compare quantities and solve real-world problems such as sharing costs, scaling recipes and converting units. In this article, we will build your confidence with ratio notation, simplification, division in a given ratio, direct proportion and the unitary method. Each step is explained in English and Chinese to support bilingual learners.

比与比例是 KS3 数学的重要工具，帮助我们比较数量并解决现实问题，例如分摊费用、调整食谱和单位换算。本文将帮助你掌握比的表示、化简、按比分配、正比例和单位法。每个步骤都用中英双语解释，方便双语学习者理解。

1. Understanding Ratio Notation | 理解比的表示

A ratio compares two or more quantities, written as a : b. For example, if a class has 12 boys and 8 girls, the ratio of boys to girls is 12 : 8. The order is important: the ratio of girls to boys would be 8 : 12.

比用于比较两个或多个数量，写作 a : b。例如，一个班有 12 名男生和 8 名女生，男生与女生的比是 12 : 8。顺序很重要：女生与男生的比则为 8 : 12。

Ratios do not tell us the actual amounts, only the relationship. The ratio 12 : 8 can represent 12 boys and 8 girls, but also 24 boys and 16 girls, or 3 boys and 2 girls after simplification.

比并不告诉我们实际数量，只表示关系。比 12 : 8 可以代表 12 名男生和 8 名女生，也可以代表 24 名男生和 16 名女生，或化简后的 3 名男生和 2 名女生。

2. Simplifying Ratios | 化简比

To simplify a ratio, divide all parts by their greatest common factor. For example, 12 : 8 simplifies to 3 : 2 because both parts can be divided by 4. Always check that the simplified ratio uses whole numbers with no common factor other than 1.

化简比时，将比的每一项除以它们的最大公因数。例如，12 : 8 化简为 3 : 2，因为两项都能除以 4。要始终确保化简后的比使用整数，且除 1 外没有其他公因数。

If a ratio contains decimals or fractions, first multiply every part by the same number to make whole numbers. For example, 0.5 : 1.5 becomes 5 : 15 after multiplying by 10, then simplifies to 1 : 3.

如果比中含有小数或分数，先将每一项乘以相同的数，化为整数。例如，0.5 : 1.5 乘以 10 后变为 5 : 15，再化简为 1 : 3。

3. Equivalent Ratios | 等价比

Equivalent ratios have the same relationship, just like equivalent fractions. You can find an equivalent ratio by multiplying or dividing every part by the same non-zero number. For example, 3 : 2 is equivalent to 6 : 4, 9 : 6 and 30 : 20.

等价比具有相同的关系，就像等值分数一样。你可以将比的每一项乘以或除以同一个非零数来得到等价比。例如，3 : 2 与 6 : 4、9 : 6 和 30 : 20 等价。

To decide if two ratios are equivalent, simplify both to their lowest terms. If they match, they are equivalent. This is useful when solving problems involving recipes or scale factors.

要判断两个比是否等价，可以把它们都化为最简形式。如果相同，则等价。这在解决食谱或比例因子问题时很有用。

4. Dividing in a Given Ratio | 按比分配

When sharing a total amount in a given ratio, first add the parts of the ratio to find the total number of shares. For example, to split £50 in the ratio 3 : 2, total parts = 3 + 2 = 5. One share is £50 ÷ 5 = £10, so the amounts are 3 × £10 = £30 and 2 × £10 = £20.

当按给定比分配总量时，先将比的各项相加，得到总份数。例如，按 3 : 2 分 50 英镑，总份数 = 3 + 2 = 5。一份是 50 ÷ 5 = 10 英镑，因此分配金额为 3 × 10 = 30 英镑和 2 × 10 = 20 英镑。

Always check that the parts add up to the original total. In the example above, £30 + £20 = £50, which confirms the division is correct.

务必检查各部分之和是否等于原来的总量。上面的例子中，30 + 20 = 50 英镑，验证了分配正确。

5. Finding Missing Values | 求未知值

If you know one part of a ratio and the total or another part, you can use a multiplier. For example, if the ratio of red to blue counters is 2 : 5 and there are 12 red counters, the multiplier is 12 ÷ 2 = 6. Therefore blue counters = 5 × 6 = 30.

如果知道比中的一部分以及总数或另一部分，可以使用乘数。例如，红球与蓝球的比是 2 : 5，红球有 12 个，乘数是 12 ÷ 2 = 6。因此蓝球有 5 × 6 = 30 个。

You can also set up a table to see the relationship clearly. Each row shows an equivalent ratio until the required value appears.

你也可以用表格清晰地展示关系。每一行表示一个等价比，直到出现所需的值。