Understanding Ratios | 理解比

📚 Understanding Ratios | 理解比

A ratio is a way of comparing two or more quantities of the same kind. It tells us how much of one thing there is compared to another. In real life, we use ratios when mixing paint, sharing money, reading maps, or even in sports statistics.

比（Ratio）是用于比较两个或两个以上同类数量之间关系的一种数学工具。它告诉我们某一种数量与另一种数量之间的相对大小。在生活中，我们调颜料、分钱、看地图、统计体育数据时都会用到比。

1. What Is a Ratio? | 什么是比？

A ratio shows a relationship between two quantities. For example, if a bag contains 4 red apples and 2 green apples, the ratio of red apples to green apples is 4 to 2. This means that for every 4 red apples, there are 2 green apples.

比表示两个数量之间的关系。例如，一个袋子里有4个红苹果和2个绿苹果，那么红苹果与绿苹果的比是4比2。这表示每有4个红苹果，就对应有2个绿苹果。

Key idea: The order of the quantities matters. Saying “red to green” is not the same as saying “green to red.”

关键：数量出现的顺序很重要。“红比绿”与“绿比红”意思完全不同。

A ratio compares parts to parts, or parts to a whole.
比既可以比较部分与部分，也可以比较部分与整体。

2. Ratio Notation | 比的表示方法

There are three common ways to write a ratio. Suppose a fish tank has 6 goldfish and 3 guppies. We can write the ratio of goldfish to guppies as:

假设一个鱼缸中有6条金鱼和3条孔雀鱼。金鱼与孔雀鱼的比可以写成以下三种形式：

6:3 or 6 to 3 or 6/3

The colon form “6:3” is the most common in mathematics. The fraction form “6/3” reminds us that a ratio is also a kind of division, although we usually leave it as a ratio rather than a decimal.

“6:3”这种带冒号的形式在数学中最常用。分数形式“6/3”提醒我们，比实际上也是一种除法，但我们通常保留为比的形式，而不是写成小数。

Read “6:3” as “six to three.”
“6:3” 读作“六比三”。
A ratio is usually simplified to its lowest terms, just like a fraction.
与分数类似，比通常需要化简为最简形式。

3. Equivalent Ratios | 等价比

Equivalent ratios are ratios that have the same value. For example, 2:3, 4:6, and 8:12 are all equivalent. This is similar to equivalent fractions.

等价比是指比值相等的比。例如，2:3、4:6和8:12都是等价比。这与等价分数非常相似。

To find an equivalent ratio, multiply or divide both terms of the ratio by the same non-zero number.

要得到等价比，可以将比的前项和后项同时乘以或除以同一个非零数。

2 × 2 : 3 × 2 = 4 : 6

If we divide both terms of 4:6 by 2, we return to 2:3. So simplifying a ratio means finding the smallest whole-number terms that keep the same relationship.

如果我们将4:6的前项和后项同时除以2，就能回到2:3。因此，化简比就是找到一组最小的整数项，使它们仍然保持原有的比例关系。

4. Simplifying Ratios | 化简比

To simplify a ratio, find the greatest common divisor (GCD) of all the terms and divide each term by it. For example, simplify 12:16.

化简比时，先求出各项的最大公约数（GCD），再将每一项都除以这个最大公约数。例如，化简12:16。

The GCD of 12 and 16 is 4. Dividing both terms by 4 gives 3:4. This is the simplest form.

12和16的最大公约数是4。两项同时除以4，得到3:4。这就是最简形式。

For ratios with three quantities, such as 6:8:10, we simplify all three terms using the same process. The GCD of 6, 8, and 10 is 2, so the simplified ratio is 3:4:5.

对于三项比，例如6:8:10，我们也用同样的方法处理所有项。6、8、10的最大公约数是2，因此化简后的比为3:4:5。

Always check that all terms are whole numbers after simplifying.
化简后一定要确认所有项都是整数。
If a ratio contains decimals, multiply every term by a power of 10 first to remove the decimal point.
如果比中含有小数，先将每一项乘以10的幂去掉小数点，然后再化简。

5. Ratio Tables | 比表

A ratio table is a helpful way to organise equivalent ratios. It can be used to solve problems involving proportional relationships.

比表是整理等价比的一种有效工具。它常用于解决与比例关系相关的问题。

Suppose a paint colour is made by mixing 4 parts blue paint with 3 parts yellow paint. The ratio of blue to yellow is 4:3. We can make a table of equivalent ratios.

假设一种颜色由4份蓝色颜料和3份黄色颜料混合而成。蓝色与黄色的比是4:3。我们可以列出等价比的表格。

Blue / 蓝	4	8	12	16
Yellow / 黄	3	6	9	12

If we need 20 liters of blue paint, we can use the table to find that the yellow paint needed is 15 liters, because both sides have been multiplied by 5.

如果我们需要20升蓝色颜料，可以通过表格找到所需黄色颜料为15升，因为两边都同时乘以了5。

6. Rates and Unit Rates | 速率与单位速率

A rate is a special kind of ratio that compares two quantities with different units. For example, driving 120 kilometers in 2 hours gives a rate of 120 km per 2 hours.

速率是一种特殊的比，它比较两个带有不同单位的数量。例如，2小时行驶120千米，可表示为每2小时120千米的速率。

A unit rate simplifies the comparison so that the second quantity is 1. For 120 km in 2 hours, the unit rate is 60 km per 1 hour, usually written as 60 km/h.

单位速率将后一项化为1，以便比较。例如，2小时120千米的单位速率是每小时60千米，通常写作60 km/h。

120 ÷ 2 : 2 ÷ 2 = 60 : 1

Unit rates are used in shopping to compare prices and in travel to compare speeds.

单位速率常用于购物时比较价格，也用于旅行时比较速度。

7. Real-World Applications | 实际应用

Ratios appear in many everyday situations. Using them correctly can help you interpret news, recipes, maps, and financial information.

比广泛出现在日常情境中。正确使用比，可以帮助你理解新闻、食谱、地图和财务信息。

Recipes often give ingredients as ratios, such as “2 cups flour to 1 cup sugar.”
食谱中的配料常以比表示，例如“2杯面粉比1杯糖”。
Map scales use ratios, such as 1:100,000, meaning 1 cm on the map represents 100,000 cm in real life.
地图比例尺使用比，例如1:100,000，表示地图上的1厘米代表实际中的100,000厘米。
Financial ratios, like profit-to-cost, help businesses make decisions.
财务比率，例如利润与成本之比，能帮助企业做出决策。

When solving ratio problems, always ask: “What units or items are being compared?”

解决比的问题时，一定要问：这里比较的是哪种单位或物体？

8. Common Misconceptions | 常见误区

Learners often make the same few mistakes when working with ratios. Understanding these pitfalls can help you avoid them.

学习比例时，同学们常常会犯一些相似的错误。理解这些陷阱可以帮助你避开它们。

Misconception 1: Swapping the order. 2:3 is not the same as 3:2.
误区一：颠倒顺序。2:3和3:2不是同一个比。
Misconception 2: Adding or subtracting the same number to both terms. This is incorrect; you must multiply or divide both terms by the same non-zero number.
误区二：给前后项加上或减去同一个数。这种做法是错的；必须同时乘以或除以同一个非零数。
Misconception 3: Treating a ratio as the same as a whole value. A ratio is a comparison, not an amount.
误区三：将比等同于一个总量。比是一种比较关系，而不是一个具体数值。

9. Practice Problems | 练习题目

Try these short questions to check your understanding of ratios.

尝试以下小问题来检验你对比的理解。

Write the ratio 45:60 in simplest form.
写出比45:60的最简形式。
A class has 8 boys and 12 girls. Find the ratio of boys to girls, and the ratio of girls to the whole class.
一个班有8名男生和12名女生。求男生与女生的比，以及女生与全班人数的比。
If 5 pens cost $3, find the cost of 20 pens using a rate or ratio method.
如果5支笔售价3美元，请用比率方法求出20支笔的价钱。

Answers: (1) 3:4; (2) 2:3 and 3:5; (3) $12.

答案如下：(1) 3:4；(2) 2:3和3:5；(3) 12美元。

10. Summary and Revision Tips | 总结与复习建议

A ratio compares quantities, uses careful order, and can be written with a colon, the word “to”, or a fraction. Equivalent ratios help you scale amounts up or down. Simplifying a ratio makes it easier to read and use.

比用于比较数量，需要注意顺序，可以用冒号、“比”字或分数形式表示。等价比帮助你按比例放大或缩小数量。化简比使它更容易阅读和使用。

For your revision, create your own ratio table and practise converting between the three notations. Then move on to word problems involving rates and unit rates.

复习时，你可以制作一个自己的比表，并练习在三种表示法之间转换。然后再练习包含速率与单位速率的应用题。

Ratio = ordered comparison → simplify → apply

比 = 有序比较 → 化简 → 应用

Published by TutorHao | 数学 Revision Series |

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Understanding Ratios | 理解比