Mastering Ratio and Proportion | 掌握比和比例

📚 Mastering Ratio and Proportion | 掌握比和比例

Welcome to your essential guide on ratio and proportion – two of the most practical and powerful tools in Key Stage 3 mathematics. Whether you are scaling a recipe, sharing prize money fairly, or interpreting map scales, ratios and proportions help you compare quantities and see how they change together. In this article, you will learn what ratios and proportions mean, how to simplify them, how to divide amounts using ratios, and how to apply proportional reasoning to solve real‑world problems. Follow along step by step, and you will build a solid foundation for Cambridge KS3 maths and beyond.

欢迎来到比和比例的核心指南——它们是关键阶段3（KS3）数学中最实用、最有力的两个工具。无论你是在调整食谱分量、公平分配奖金，还是在解读地图比例尺，比和比例都能帮助你比较数量并观察它们如何共同变化。在本文中，你将学习比和比例的含义、如何化简比、如何按比例分配数量，以及如何运用比例推理解决实际问题。请逐步学习，你将打下扎实的剑桥KS3数学基础，并为以后的学习做好准备。

1. What Is a Ratio? | 什么是比？

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. We write ratios using a colon, for example, 3:2. This means that for every 3 of one item, there are 2 of the other. Ratios do not usually show the actual amounts, but the relative sizes.

比是一种比较两个或多个同类数量的方法。它告诉我们一个数量相对于另一个数量有多少。我们用冒号书写比，例如 3:2。这意味着每有3个某物品，就有2个另一物品。比通常不表示实际数量，只表示相对大小。

If a fruit bowl has 4 apples and 6 oranges, the ratio of apples to oranges is 4:6. We read this as ‘four to six’. The order is vital: apple ratio to oranges is different from oranges to apples (which would be 6:4).

如果一个果盘里有4个苹果和6个橙子，苹果与橙子的比是4:6。我们读作“四比六”。顺序至关重要：苹果与橙子的比不同于橙子与苹果的比（后者是6:4）。

Ratios can compare more than two quantities, such as 2:3:5 for red, green, and blue marbles. Each number in the ratio refers to the corresponding part in the same order.

比可以比较两个以上的数量，例如2:3:5表示红色、绿色和蓝色弹珠的比。比中的每个数字按相同顺序对应各个部分。

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2. Simplifying Ratios | 化简比

Just like fractions, ratios can often be simplified by dividing each part by the same whole number. This makes the ratio easier to understand without changing its meaning. To simplify, we find the highest common factor (HCF) of the numbers and divide all parts by it.

就像分数一样，比通常可以通过将每一部分除以相同的整数来化简。这样既不会改变比的意义，又能让它更易理解。化简时，我们找出各部分数字的最大公因数（HCF），并将所有部分除以它。

For example, simplify 8:12. The HCF of 8 and 12 is 4. Dividing both parts by 4 gives 8÷4 = 2 and 12÷4 = 3. So 8:12 simplifies to 2:3.

例如，化简 8:12。8和12的最大公因数是4。将两部分都除以4，得到8÷4=2，12÷4=3。因此 8:12 化简为 2:3。

If the ratio includes decimals, we can multiply by powers of 10 to obtain whole numbers first. For 0.5:2, multiply both parts by 2 (or 10 to get 5:20, then simplify to 1:4). The simplest whole-number form is always preferred.

如果比中包含小数，我们可以先乘以10的幂得到整数。例如 0.5:2，将两部分都乘以2（或乘以10得到5:20，再化简为1:4）。我们总是更喜欢使用最简单的整数比形式。

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3. Equivalent Ratios | 等比

Equivalent ratios are ratios that express the same relationship between quantities, just written with different numbers. They are created by multiplying or dividing each part of a ratio by the same non‑zero number, just like equivalent fractions.

等比是指用不同数字表达相同数量关系的比。它们是通过将比的每一部分乘以或除以同一个非零数字得到的，就像等值分数一样。

Starting from the ratio 1:3, we can multiply both parts by 2 to get 2:6, by 3 to get 3:9, and so on. All of these are equivalent because they reduce to the same simplest form, 1:3.

从比 1:3 出发，将两部分都乘以2得到 2:6，乘以3得到 3:9，依此类推。所有这些都等价，因为它们化简后都得到相同的最简形式 1:3。

Recognising equivalent ratios is very useful when solving problems. If a recipe uses flour and sugar in the ratio 100:50, we can scale it down by dividing by 10 to get 10:5, then further to 2:1. This tells us that for every 2 cups of flour, we need 1 cup of sugar.

在解题时辨识等比十分有用。如果一份食谱中面粉和糖的比例是100:50，我们可以通过除以10将其缩小为10:5，再化简为2:1。这就告诉我们，每2杯面粉需要配1杯糖。

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4. Ratios in the Form 1:n | 写成 1:n 的形式

It is often helpful to write a ratio in the form 1:n, where one of the numbers becomes 1. This is especially common on maps and scale drawings, and it makes it easy to see the multiplier between quantities.

我们经常需要将比写成 1:n 的形式，其中某一个数字变成1。这在地图和比例图里尤其常见，也能让我们很容易看出数量之间的倍数关系。

To write a ratio a:b in the form 1:n, divide both parts by a (provided a ≠ 0). For example, to change 4:10 into the form 1:n, divide both parts by 4: we get 4÷4 = 1 and 10÷4 = 2.5. So 4:10 becomes 1:2.5.

要把比 a:b 写成 1:n 的形式，将两部分都除以 a（前提是 a 不为零）。例如，将 4:10 写成 1:n 的形式，两部分都除以4：4÷4=1，10÷4=2.5。所以 4:10 变为 1:2.5。

If the ratio includes separate units or measurements, we still use the same method. A map scale of 5 cm to 200 m can be written as 1 cm to 40 m after dividing both sides by 5. This tells us that every 1 cm on the map represents 40 m in real life.

如果比包含单位或测量值，我们仍使用相同方法。比例尺 5 cm 比 200 m，两边除以5后可以写成 1 cm 比 40 m。这告诉我们地图上每1厘米代表实际生活中的40米。

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5. Dividing Quantities in a Given Ratio | 按给定比例分配数量

One of the most common applications of ratios is sharing an amount into parts according to a given ratio. We first find the total number of parts by adding the numbers in the ratio. Then we calculate the value of one part, and multiply by each ratio number to get the individual shares.

比最常见的应用之一就是按照给定比例把一个数量分成若干份。我们首先把比里的数字加起来得到总份数。然后计算一份的价值，再分别乘以比中的每个数字，得到每一份的具体数额。

Divide £60 between two friends in the ratio 3:2. The total number of parts is 3 + 2 = 5. One part is worth £60 ÷ 5 = £12. The first friend receives 3 × £12 = £36, and the second receives 2 × £12 = £24. Always check that the shares add up to the original amount (£36 + £24 = £60).

将60英镑按照3:2分给两个朋友。总份数为3+2=5。一份的价值是60÷5=12英镑。第一个朋友得到3×12=36英镑，第二个得到2×12=24英镑。务必检查各份之和等于原始金额（36+24=60）。

When dividing more complex quantities involving three parts, the method is identical. If the ratio is 4:5:1, the total parts are 10. For a mass of 800 g, one part is 80 g, so the shares are 320 g, 400 g, and 80 g respectively.

当分配涉及三个部分的更复杂数量时，方法完全相同。如果比例是4:5:1，总份数为10。对于总质量800克，一份为80克，因此各部分分别为320克、400克和80克。

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6. What Is Proportion? | 什么是比例？

Proportion describes a relationship between two quantities where the ratio of one quantity to the other stays constant. There are two main types we study at KS3: direct proportion and inverse proportion (introduced briefly). In direct proportion, as one quantity increases, the other increases at the same rate.

比例描述两个数量之间的一种关系，其中一个数量与另一个数量的比值保持不变。在KS3阶段我们主要学习两种类型：正比例和反比例（简要介绍）。在正比例关系中，当一个数量增加时，另一个数量以相同的速率增加。

If apples cost 30p each, the total cost is directly proportional to the number of apples. You can write this relationship as total cost = 30p × number of apples. A table of values will show a constant ratio: 1 apple → 30p, 2 apples → 60p, and the ratio cost:apples remains 30:1 throughout.

如果每个苹果售价30便士，那么总费用与苹果的数量成正比例。你可以将这个关系写成：总费用 = 30便士 × 苹果数量。数值表会显示恒定的比值：1个苹果 → 30便士，2个苹果 → 60便士，费用与苹果数量的比始终保持为30:1。

We also recognise direct proportion when one quantity is a constant multiple of another, i.e., y = kx, where k is the constant of proportionality. In the apple example, the constant k = 30p per apple.

我们还可以通过一个数量是另一个数量的常数倍来识别正比例，即 y = kx，其中k是比例常数。在苹果的例子中，常数k = 每苹果30便士。

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7. Direct Proportion and the Unitary Method | 正比例与单位法

The unitary method is a powerful technique for solving proportion problems. First you find the value of one unit (the ‘per’ amount), and then you scale up to any quantity you need. This method works perfectly for direct proportion.

单位法是解决比例问题的一种强大技巧。你先求出一个单位的价值（“每”多少），然后按需要扩大到任意数量。这个方法对正比例问题非常有效。

A car travels 210 km on 15 litres of petrol. How far will it travel on 22 litres? First find the distance per litre: 210 ÷ 15 = 14 km per litre. Then multiply by 22: 14 × 22 = 308 km. The car covers 308 km.

一辆汽车用15升汽油行驶210公里。用22升汽油可以行驶多远？首先求出每升行驶的距离：210 ÷ 15 = 每升14公里。再乘以22：14 × 22 = 308公里。这辆车可以行驶308公里。

We can also use a table approach: write the known pair (15 litres, 210 km) and the unknown pair (22 litres, ? km). Because the ratio of litres to distance is constant, the distance must be 22 × (210/15) = 308 km.

我们也可以使用表格法：写下已知配对（15升，210公里）和未知配对（22升，?公里）。因为升数与距离的比值恒定，距离必定是 22 × (210/15) = 308公里。

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8. Inverse Proportion (Introduction) | 反比例（入门）

In inverse proportion, as one quantity increases the other decreases in such a way that their product remains constant. We touch on this lightly at KS3 to recognise the pattern, without heavy formula work.

在反比例关系中，当一个量增大时，另一个量会减小，且二者的乘积保持不变。我们在KS3阶段会初步接触这种模式，不涉及繁重的公式推导。

Imagine a fixed amount of work. If 2 workers take 6 hours to complete a job, how long would 3 workers take? The total work is 2 × 6 = 12 worker‑hours. For 3 workers, the time t must satisfy 3 × t = 12, so t = 4 hours. As the number of workers goes up, the time goes down, and the product is constant.

设想一个固定的工作量。如果2个工人需要6小时完成一项工作，那么3个工人需要多长时间？总工作量为 2 × 6 = 12 人时。对于3个工人，时间t必须满足 3 × t = 12，所以 t = 4小时。随着工人数量增加，所需时间减少，且乘积保持不变。

This is different from direct proportion, where doubling one quantity doubles the other. In inverse proportion, doubling one quantity halves the other.

这与正比例不同，正比例中一个量翻倍时另一个量也翻倍。而在反比例中，一个量翻倍会使另一个量减半。

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9. Ratio and Proportion in Word Problems | 文字题中的比和比例

Cambridge KS3 exams love to embed ratio and proportion in everyday contexts. You might encounter problems about mixing paint, sharing money, converting recipes, or interpreting scale diagrams. The key is to extract the ratio, decide what type of relationship it is, and then use the appropriate method.

剑桥KS3考试喜欢把比和比例融入日常情境中。你可能会遇到混合油漆、分配金钱、转换食谱或解读比例图等问题。解题的关键在于提取出比，判断它是哪种关系，然后选用合适的方法。

Example: In a school, the ratio of students wearing glasses to those not wearing glasses is 2:7. If there are 540 students in total, how many wear glasses? The total parts = 2 + 7 = 9. One part = 540 ÷ 9 = 60. So the number wearing glasses = 2 × 60 = 120 students.

例题：在一所学校里，戴眼镜与不戴眼镜的学生比为2:7。如果共有540名学生，有多少人戴眼镜？总份数 = 2+7=9。一份 = 540÷9=60。因此戴眼镜的人数 = 2×60=120名学生。

Always underline the key numbers and the ratio, identify the total, and check that the answer makes sense in the original situation. Practicing different word problems builds confidence.

经常划出关键数字和比，确定总数，并检查答案在原始情境中是否合理。练习不同的文字题可以积累信心。

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10. Using Ratio Tables | 使用比数表

Ratio tables offer a visual way to handle equivalent ratios and proportions. You place one quantity in the top row and the corresponding other quantity in the bottom row, then use multiplication or division to move between columns.

比数表提供了一种可视化的方式来处理等比和比例。你把一个量放在表格的上行，对应的另一个量放在下行，然后通过乘法或除法在列之间移动。

For the car that uses 5 litres per 40 km, build a table: start with the pair (40 km, 5 L). To find distance for 12 L, first find distance for 1 L (40 ÷ 5 = 8 km), then multiply by 12 to get 96 km. The table grows column by column, reinforcing the constant multiplier between kilometres and litres.

对于那辆每5升油行驶40公里的汽车，可以建一个表：从(40公里, 5升)这一对开始。要计算12升油能行驶的距离，先求1升油的距离（40÷5=8公里），再乘以12得到96公里。表格逐列扩展，强化了公里与升之间恒定的倍数关系。

Ratio tables also help with scaling down. If a mixture uses 200 g flour and 120 g butter (ratio 200:120 = 5:3), we can put these in a table and divide both quantities by the same numbers to reach a desired total amount or a 1:n form.

比数表也有助于缩小比例。如果混合物用200克面粉和120克黄油（比例为200:120=5:3），我们可以将它们放入表格，并让两个数量同时除以相同的数，从而得到所需的总量或1:n形式。

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11. Common Misconceptions to Avoid | 要避免的常见误区

Order matters: The ratio 1:3 is not the same as 3:1. Reversing the order completely changes the meaning. Always read the question carefully to know which quantity is being compared to which.

顺序很重要： 比 1:3 与 3:1 不同。颠倒顺序会完全改变含义。务必仔细读题，弄清哪个量与哪个量进行比较。

Units must be the same: When writing ratios, ensure both quantities are expressed in the same unit. A ratio of 1 m to 50 cm should be converted to 100 cm : 50 cm, which simplifies to 2:1, not 1:50.

单位必须一致： 书写比时，确保两个量使用相同单位。1米比50厘米应当转换为100厘米:50厘米，化简为2:1，而不是1:50。

Do not always add the ratio parts: If a ratio is 2:3 and you are told the larger quantity is 75, you cannot simply add 2+3=5 to find the smaller. Instead, find the value of one part: 75 ÷ 3 = 25, then the smaller part is 2 × 25 = 50.

不要总想着把比的各项相加： 如果比是2:3，且已知较大的量为75，你不能简单地把2+3=5来求较小的量。正确的做法是，先求出一份的值：75÷3=25，然后较小的部分为2×25=50。

Proportional or not? Not every relationship is a direct proportion. Check whether the ratio between quantities remains constant. If the first hour of parking costs £2 and the next costs £1, the total cost is not proportional to time because the ratio changes.

是否成比例？ 并非所有关系都是正比例。检查两个量之间的比值是否保持不变。如果停车第一小时收费2英镑，之后每小时1英镑，总费用与时间不成比例，因为比值在变化。

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12. Summary and Key Points | 总结与要点

Ratio compares quantities, proportion describes the relationship when that comparison stays constant. Remember to simplify ratios, use the total number of parts for sharing, and convert to 1:n form when helpful. For proportion, identify direct or inverse type, then apply the unitary method or constant multiplier.

比用于比较数量，比例则描述当这种比较保持不变时的关系。记住要化简比，利用总份数来进行分配，并在有用时转换成1:n的形式。对于比例问题，要辨别是正比例还是反比例，然后运用单位法或恒定倍数来解题。

Master these topics by practising a wide range of exercises: sharing money, scale drawings, recipe scaling, and speed‑time‑distance problems. The more you work with ratios and proportions, the easier it becomes to spot patterns and apply the correct strategy. Keep a sharp eye on units and the order of quantities – success is in the detail.

通过大量练习来掌握这些主题：分配金钱、比例图、食谱缩放以及速度‑时间‑距离问题。你对比和比例的运用越熟练，就越容易发现规律并采用正确的策略。请密切关注单位和数量的顺序——成功就在于细节之中。

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