Mastering Ratios and Proportional Reasoning | 掌握比与比例推理

📚 Mastering Ratios and Proportional Reasoning | 掌握比与比例推理

Ratios appear everywhere in KS3 Cambridge mathematics, from sharing sweets to reading scale drawings. This revision guide works through the key ideas and skills you need for p177_2-style questions.

比在 KS3 Cambridge 数学中随处可见，从分配糖果到阅读比例尺图纸都会用到。本复习指南讲解 p177_2 类题目所需的关键概念与技巧。

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1. Understanding Ratio | 理解比

A ratio compares two or more quantities. It tells you how much of one thing there is compared with another. For example, in a fruit bowl with 3 apples and 2 pears, the ratio of apples to pears is written as 3 : 2.

比用于比较两个或多个数量。它告诉你一种事物相对于另一种事物的多少。例如，一个水果碗里有 3 个苹果和 2 个梨，苹果与梨的比写作 3 : 2。

The order of a ratio matters. The ratio 3 : 2 is not the same as 2 : 3. The first number links to the first item named.

比的顺序很重要。比 3 : 2 与 2 : 3 不同。第一个数字对应第一个被点名的项目。

Key notation:

关键记号：

apples : pears = 3 : 2

Ratios use a colon between the quantities. They do not have units because they compare amounts of the same kind.

比在数量之间使用冒号。因为比比较的是同类的量，所以不带单位。

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2. Simplifying Ratios | 化简比

Ratios can be simplified just like fractions. Divide every part of the ratio by the highest common factor, or HCF.

比可以像分数一样化简。用最大公因数（HCF）除以比的每一部分。

Example: Simplify 12 : 16. The HCF of 12 and 16 is 4, so divide both sides by 4.

示例：化简 12 : 16。12 和 16 的最大公因数是 4，所以两边都除以 4。

12 : 16 = (12 ÷ 4) : (16 ÷ 4) = 3 : 4

Always check whether the simplified ratio has whole numbers with no common factor other than 1.

始终检查化简后的比是否为只有公因数 1 的整数比。

If a ratio contains decimals, multiply all parts by a power of 10 first. For example, 0.3 : 0.8 becomes 3 : 8 after multiplying by 10.

如果比中含有小数，先将所有部分乘以 10 的幂。例如，0.3 : 0.8 乘以 10 后变为 3 : 8。

For a three-part ratio such as 18 : 24 : 30, divide all parts by the HCF 6 to get 3 : 4 : 5.

对于 18 : 24 : 30 这样的三部分比，将所有部分除以最大公因数 6，得到 3 : 4 : 5。

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3. Equivalent Ratios | 等价比

Equivalent ratios are formed by multiplying or dividing every part by the same non-zero number. This is useful when comparing ratios or scaling recipes.

等价比是通过将每一部分同时乘以或除以同一个非零数字得到的。这在比较比或调整食谱大小时非常有用。

For example, 1 : 3, 2 : 6 and 10 : 30 are equivalent ratios.

例如，1 : 3、2 : 6 和 10 : 30 都是等价比。

Multiply by 2: 1 : 3 = 2 : 6	乘以 2：1 : 3 = 2 : 6
Multiply by 10: 1 : 3 = 10 : 30	乘以 10：1 : 3 = 10 : 30

To compare ratios such as 5 : 8 and 3 : 5, write each as a fraction or scale to a common total. As decimals, 5/8 = 0.625 and 3/5 = 0.6, so 5 : 8 is larger.

要比较 5 : 8 和 3 : 5 这样的比，可以把每个都写成分数，或化成相同总量。化为小数，5/8 = 0.625，3/5 = 0.6，所以 5 : 8 更大。

In some questions you may need to find a missing term. If 2 : 5 = x : 20, the multiplier is 4, so x = 8.

有些题目可能需要求缺失项。如果 2 : 5 = x : 20，倍数关系为 4，因此 x = 8。

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4. Sharing in a Given Ratio | 按给定比例分配

To share an amount in a given ratio, first add the parts to find the total number of shares. Then divide the amount by the total shares, and multiply by each part.

按给定比分配时，先将各部分相加得到总份数。然后用总量除以总份数，再乘以每一部分。

Example: Share £60 between Anna and Ben in the ratio 2 : 3.

示例：按 2 : 3 的比在 Anna 和 Ben 之间分配 60 英镑。

Total parts = 2 + 3 = 5

One part = £60 ÷ 5 = £12

Anna gets 2 × £12 = £24, Ben gets 3 × £12 = £36

Check the answer by adding the shares: £24 + £36 = £60, which matches the original amount.

通过相加检查答案：24 + 36 = 60 英镑，与原始金额一致。

For three parts, the method is the same. Share £120 in the ratio 3 : 4 : 5. The total is 12 parts, so one part is £10. The shares are £30, £40 and £50.

对于三部分，方法相同。按 3 : 4 : 5 的比分配 120 英镑。总份数为 12，一份为 10 英镑。各部分为 30、40 和 50 英镑。

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5. Ratios and Fractions | 比与分数

A ratio can be converted into fractions once you know the total number of parts. The fraction of the whole belonging to each part is that part over the total.

一旦知道总份数，比就可以转化为分数。每一部分占总体的分数等于该部分除以总份数。

For the ratio 2 : 3, the total is 5, so the fractions are 2/5 and 3/5.

对于比 2 : 3，总份数为 5，因此分数为 2/5 和 3/5。

2 : 3 → 2/5 and 3/5

This helps in probability and data questions. If a bag has red and blue counters in the ratio 1 : 4, then 1/5 of the counters are red.

这对概率和数据题有帮助。如果袋中红蓝计数器的比为 1 : 4，那么红色计数器占 1/5。

Always use the total number of parts as the denominator, not just one side of the ratio.

始终用总份数作为分母，而不是比的一侧。

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6. Ratios and Percentages | 比与百分比

Ratios can be written as percentages by first finding the fractions and then multiplying by 100. This is common in survey and allocation questions.

比可以通过先求分数再乘以 100 来写成百分比。这在调查和分配问题中很常见。

Example: A school has boys and girls in the ratio 3 : 2. What percentage are boys?

示例：某学校男生和女生的比为 3 : 2。

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