Ratio and Proportion: Solving p195 Q1 Paint Mixing Problem | 比和比例：解决p195第1题颜料混合问题

📚 Ratio and Proportion: Solving p195 Q1 Paint Mixing Problem | 比和比例：解决p195第1题颜料混合问题

In this article, we will explore the concepts of ratio and proportion through a classic problem found on page 195, Question 1 of the Cambridge Checkpoint Mathematics Coursebook. The problem involves mixing paints in a given ratio, a practical application that builds a strong foundation for understanding ratios and direct proportion. We will break down the solution step by step and discuss the underlying principles to help you master this essential KS3 topic.

在这篇文章中，我们将通过剑桥数学 Checkpoint 教材第195页第1题探讨比和比例的概念。该题涉及按给定比例混合油漆，是一个很实际的应用，能为理解比和正比例打下坚实基础。我们将逐步分解解题过程，并讨论背后的原理，帮助你掌握 KS3 这一重要课题。

1. Understanding Ratios | 理解比

A ratio is a way of comparing two or more quantities. It tells us how much of one thing there is compared to another. For example, the ratio of red paint to blue paint could be 2 : 3, which means for every 2 parts of red paint, there are 3 parts of blue paint. The order is important: 2 : 3 is not the same as 3 : 2.

比是比较两个或多个数量的一种方式。它告诉我们一种事物相对于另一种事物有多少。例如，红色油漆与蓝色油漆的比可能是 2 : 3，这意味着每 2 份红色油漆对应 3 份蓝色油漆。顺序很重要：2 : 3 不同于 3 : 2。

Ratios can be simplified just like fractions. The ratio 4 : 6 is equivalent to 2 : 3 when both numbers are divided by 2. This is because ratios are based on multiplicative relationships, not total amounts.

比可以像分数一样进行简化。比例 4 : 6 等价于 2 : 3，只要把两个数同时除以 2。这是因为比基于乘法关系，而不是基于总数。

2. Simplifying Ratios | 简化比

To simplify a ratio, find the greatest common divisor (GCD) of all numbers involved and divide each by it. For instance, 12 : 18 simplifies to 2 : 3 by dividing by 6. Always express ratios in their simplest integer form unless otherwise stated.

要简化一个比，先找到所有数字的最大公约数，然后将每个数除以这个公约数。例如，12 : 18 同时除以 6 后简化为 2 : 3。除非另有要求，否则始终用最简整数比来表示。

Ratio problems often involve whole-number parts, but it’s possible to have fractions in ratios too. In KS3, we usually work with integer ratios, but you should be comfortable interpreting ratios like ½ : 1½, which can be multiplied by 2 to become 1 : 3.

比的问题通常涉及整数份数，但比中也可能出现分数。在 KS3 阶段，我们通常使用整数比，但你也应该能理解像 ½ : 1½ 这样的比，它可以同乘 2 变成 1 : 3。

3. Ratios and Parts | 比与份数

The concept of ‘parts’ is crucial. In the ratio 2 : 3, the total number of parts is 2 + 3 = 5 parts. This means if we had a mixture divided into 5 equal parts, 2 parts would be red and 3 parts would be blue. Understanding total parts helps when sharing amounts or finding unknown quantities.

“份数”这个概念非常关键。在 2 : 3 这个比中，总份数是 2 + 3 = 5 份。这意味着如果把一个混合物分成 5 等份，其中 2 份是红色，3 份是蓝色。理解总份数有助于按比例分配或求未知量。

For any ratio a : b, the total parts are a + b. For three quantities a : b : c, total parts = a + b + c. This is the foundation for solving many ratio problems.

对于任意的比 a : b，总份数为 a + b。对于三个量的比 a : b : c，总份数为 a + b + c。这是解决许多比的问题的基础。

4. The Paint Problem: p195 Q1 | 颜料问题：p195 第1题

Question 1 on page 195 reads: ‘A shade of purple paint is made by mixing red and blue paint in the ratio 2 : 3. A decorator has 12 litres of red paint. How much blue paint does he need to make the purple paint?’ This is a typical ratio problem where we are given one part quantity and need to find another using the ratio relationship.

第195页第1题是：”一种紫色油漆由红色和蓝色油漆按 2 : 3 的比例混合而成。装修工有 12 升红色油漆，他需要多少升蓝色油漆来调制这种紫色油漆？”这是一道典型的比的问题，已知一个份量的具体数值，需要利用比的关系求出另一个量。

At first glance, some students might think of adding or subtracting, but the key is to find the value of one ‘part’. Since red corresponds to the

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