Sharing in a Ratio: Cambridge KS3 Maths | 按比例分配：剑桥 KS3 数学

📚 Sharing in a Ratio: Cambridge KS3 Maths | 按比例分配：剑桥 KS3 数学

Sharing quantities in a given ratio is a core skill in the Cambridge KS3 mathematics curriculum. It appears regularly in written exercises, including the practice set on page 119 question 2, and is essential for later work on proportion, percentages and real-life financial problems.

按给定比例分配数量是剑桥 KS3 数学课程的核心技能。它经常出现在书面练习中，包括第119页第2题的练习，并且对后续的比例、百分数和实际财务问题都至关重要。

1. What is a Ratio? | 什么是比？

A ratio compares two or more quantities in the same unit. We write ratios using a colon, for example 3 : 2. This means for every 3 parts of the first quantity, there are 2 parts of the second quantity. The total number of equal parts is found by adding the numbers in the ratio.

比用来比较两个或两个以上相同单位的量。我们用冒号写比，例如 3 : 2。这表示第一种量每占 3 份，第二种量就占 2 份。总份数等于比中各项数字之和。

For the ratio 3 : 5, there are 3 + 5 = 8 equal parts in total. Understanding this idea is the foundation for all sharing problems.

对于比 3 : 5，总共有 3 + 5 = 8 个相等的份。理解这个概念是解决所有分配问题的基础。

3 : 5 means total parts = 8; 3 : 5 表示总份数 = 8
A ratio has no units; 比没有单位
The order of the ratio matters; 比的顺序很重要

2. Reading a Ratio Question | 阅读比例题

When a question says “Share £120 between A and B in the ratio 3 : 5”, it is asking you to split the whole amount into equal parts according to the ratio. Identify the total amount and the ratio terms carefully before you start.

当题目说“按 3 : 5 的比例在 A 和 B 之间分配 120 英镑”时，是要求你按照该比例将总量分成相等的份。开始之前请仔细识别总量和比中的各项。

Useful steps for reading the question:

阅读题目的有用步骤：

Find the total amount to be shared; 找到要分配的总量
Find the ratio and check its order; 找到比并检查其顺序
Add the ratio terms to get total parts; 将比中各项相加得到总份数

3. The Bar Model Method | 条形模型法

A bar model is a visual way to represent sharing in a ratio. Draw one bar, divide it into equal sections matching the total parts, then label the known total. Each section represents one part.

条形模型是一种按比例分配的可视化方法。画一个条形，把它分成与总份数相等的若干段，然后标出已知总量。每一段代表一份。

For £120 shared in the ratio 3 : 5, draw a bar with 8 equal sections. The first 3 sections represent A, and the remaining 5 sections represent B. The whole bar equals £120.

对于按 3 : 5 分配 120 英镑，画一个分成 8 个等份的条形。前 3 段代表 A，后 5 段代表 B。整个条形等于 120 英镑。

Total parts = 3 + 5 = 8 | One part shown by one section | 总份数 = 3 + 5 = 8 | 一份用一段表示

4. The Unit-Method | 单位法

The unit-method uses three steps: find the total number of parts, divide the given amount by the total parts to find the value of one part, then multiply by each ratio term.

单位法使用三个步骤：求出总份数，用已知数量除以总份数得到一份的值，再分别乘以比中的各项。

This method works for money, lengths, masses, volumes and any other quantity that can be divided into equal parts.

该方法适用于金钱、长度、质量、体积以及任何可以分成相等份的量。

One part = Total amount ÷ Total parts | 一份 = 总量 ÷ 总份数

Step 1: Find total parts; 步骤1：求总份数
Step 2: Divide total amount by total parts; 步骤2：总量除以总份数
Step 3: Multiply one part by each ratio term; 步骤3：一份乘以比中各项

5. Worked Example: Sharing Money | 示例：分钱问题

Work through this example: Share £120 between A and B in the ratio 3 : 5.

完成这个示例：按 3 : 5 的比例在 A 和 B 之间分配 120 英镑。

Total parts = 3 + 5 = 8; 总份数 = 3 + 5 = 8
One part = £120 ÷ 8 = £15; 一份 = 120 ÷ 8 = 15 英镑
A = 3 × £15 = £45; A = 3 × 15 = 45 英镑
B = 5 × £15 = £75; B = 5 × 15 = 75 英镑
Check: £45 + £75 = £120; 检查：45 + 75 = 120 英镑

The answer is £45 for A and £75 for B. Always check that the amounts add back to the original total.

答案是 A 得 45 英镑，B 得 75 英镑。一定要检查各数量相加是否等于原总量。

6. Using Algebra to Share in a Ratio | 用代数按比例分配

You can also set up an equation. Let one part be x. Then the amounts are 3x and 5x, and the sum equals the total.

你也可以设方程。设一份为 x。那么数量就是 3x 和 5x，它们的和等于总量。

3x + 5x = 120 → 8x = 120 → x = 15

Then A = 3 × 15 = 45 and B = 5 × 15 = 75. This algebraic approach links ratio sharing to solving linear equations.

那么 A = 3 × 15 = 45，B = 5 × 15 = 75。这种代数方法把按比例分配与解一元一次方程联系起来。

7. Ratio and Fractions | 比与分数

A ratio can be converted into fractions of the whole. In the ratio 3 : 5, the first part receives 3/8 of the total and the second receives 5/8 of the total.

比可以转化为整体的分数。在 3 : 5 中，第一部分得到总量的 3/8，第二部分得到总量的 5/8。

A = (3/8) × £120 = £45 | B = (5/8) × £120 = £75

This is useful when a question asks for one share as a fraction of the whole, or when comparing two related ratio problems.

当题目要求求某一份占整体的几分之几，或者比较两个相关的比例问题时，这种方法很有用。

8. Common Mistakes | 常见错误

Students often forget to add the ratio terms before dividing. Another mistake is using the ratio term as the actual amount. Always calculate the value of one part first.

学生常常在除法之前忘记将比中各项相加。另一个错误是把比中的项直接当作实际数量。一定要先计算一份的值。

Mistake: dividing by 3 instead of by total parts 8; 错误：除以 3 而不是总份数 8
Mistake: leaving the answer as 3 : 5 without finding amounts; 错误：只写出 3 : 5 而没有求出实际数量
Mistake: mixing units, such as pounds and pence; 错误：单位混用，例如英镑和便士混用
Mistake: swapping the order of shares; 错误：调换份额顺序

Careful reading and a clear layout will help you avoid these errors.

认真读题和清晰的书写布局能帮助你避免这些错误。

9. Exam-Style Practice | 考试型练习

Try these quick questions to check your understanding:

尝试以下快速练习来检查理解：

Share £200 in the ratio 3 : 7. 按 3 : 7 分配 200 英镑。
Divide 360 ml in the ratio 1 : 5. 按 1 : 5 分配 360 毫升。
The ratio of boys to girls in a class is 4 : 5. If there are 27 students in total, how many boys are there? 班级男女比例为 4 : 5。如果共有 27 名学生，男生有多少人？

For the last question, total parts = 4 + 5 = 9, one part = 27 ÷ 9 = 3, so boys = 4 × 3 = 12.

最后一题中，总份数 = 4 + 5 = 9，一份 = 27 ÷ 9 = 3，所以男生 = 4 × 3 = 12。

10. Key Takeaways | 核心要点

To share in a ratio, always find the total number of parts, divide to find one part, then multiply. Use a bar model if you need a visual check. Verify that the amounts add back to the original total.

按比例分配时，总是先求总份数，除以总量得到一份，再乘以各项。如果需要直观检查，可以用条形图。验证各数量相加等于原总量。

This skill is tested regularly at KS3 and forms the basis for harder proportion questions later in Cambridge IGCSE mathematics.

这门技能在 KS3 阶段经常考查，也为剑桥 IGCSE 数学后续更复杂的比例问题打下基础。

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Sharing in a Ratio: Cambridge KS3 Maths | 按比例分配：剑桥 KS3 数学