Understanding Ratio and Proportion | 理解比率与比例

📚 Understanding Ratio and Proportion | 理解比率与比例

Ratio and proportion are fundamental concepts in KS3 mathematics, allowing us to compare quantities and see how they relate to one another. In this article, we will explore what ratios are, how to simplify them, how to share amounts using ratios, and how to work with direct proportion. By mastering these skills, you will be able to solve a wide range of real‑world problems, from mixing ingredients to reading maps. Let’s dive into the essential ideas behind ratio and proportion, ensuring you have a solid foundation for Cambridge Checkpoint and beyond.

比率和比例是 KS3 数学的基本概念，它们帮助我们比较数量并了解它们之间的关系。在本文中，我们将探讨什么是比率、如何简化比率、如何按比率分配量以及如何处理正比例。掌握这些技能后，你就能解决各种现实问题，从混合配料到阅读地图。让我们一起深入理解比率和比例背后的关键思想，为剑桥 Checkpoint 及更高级别的学习打下坚实基础。

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1. What Is a Ratio? | 什么是比率？

A ratio is a way of comparing two or more quantities. It tells us how much of one thing there is compared to another. We write a ratio using a colon, for example 2 : 3. This means that for every 2 parts of the first quantity, there are 3 parts of the second. The order in which we write the ratio is very important: 2 : 3 is not the same as 3 : 2.

比率是比较两个或多个数量的一种方式。它告诉我们一种东西相对于另一种有多少。我们用冒号书写比率，例如 2 : 3。这意味着每 2 份第一种数量，就有 3 份第二种数量。书写顺序非常重要：2 : 3 和 3 : 2 是不同的。

We can also use ratios to compare more than two quantities, such as 1 : 2 : 3. In a recipe, a ratio of 2 : 1 for flour to sugar means that if you use 200 g of flour, you need 100 g of sugar. Ratios do not tell us the actual amounts, only the relationship between them. For example, a ratio of 1 : 4 could represent 1 apple and 4 oranges, or 10 apples and 40 oranges.

我们也可以用比率比较两个以上的数量，比如 1 : 2 : 3。在一份食谱中，面粉和糖的比率为 2 : 1，意味着如果你用了 200 克面粉，就需要 100 克糖。比率并不告诉我们实际的数量，只告诉我们它们之间的关系。例如，1 : 4 可以代表 1 个苹果和 4 个橙子，也可以是 10 个苹果和 40 个橙子。

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2. Simplifying Ratios | 简化比率

Just like fractions, ratios can often be simplified by dividing every part by their highest common factor (HCF). For instance, the ratio 10 : 15 can be simplified by dividing both numbers by 5, giving 2 : 3. We always aim to give ratios in their simplest whole‑number form, where the numbers have no common factor other than 1.

就像分数一样，比率通常可以通过用最大公因数（HCF）除以每一项来简化。例如，比率 10 : 15 可以通过将两个数都除以 5 来简化，得到 2 : 3。我们总是力求用最简整数形式表示比率，即各数除了 1 之外没有其他公因数。

When a ratio contains decimals, we can multiply all parts by 10, 100, or 1000 to turn them into whole numbers before simplifying. For example, 1.5 : 2.5 can be multiplied by 10 to become 15 : 25, which then simplifies to 3 : 5. If the ratio includes mixed units, always convert to the same unit first. For instance, when comparing 3 cm to 5 mm, write both in mm: 30 mm : 5 mm, giving 6 : 1.

当比率中含有小数时，我们可以将所有项乘以 10、100 或 1000 将其化为整数再简化。例如，1.5 : 2.5 可以乘以 10 变成 15 : 25，然后简化为 3 : 5。如果比率涉及混合单位，务必先转换为同一单位。比如，比较 3 cm 和 5 mm 时，先都写成毫米：30 mm : 5 mm，得到 6 : 1。

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3. Equivalent Ratios | 等价比率

Equivalent ratios are produced by multiplying or dividing each part of a ratio by the same number. This is extremely useful when you need to scale a quantity up or down while keeping the same relationship. For example, 1 : 4 is equivalent to 2 : 8 (multiplying by 2) and to 0.5 : 2 (dividing by 2). Recognising equivalent ratios helps in solving proportion problems.

等价比率是通过将比率的每一项乘以或除以同一个数得到的。当需要按比例放大或缩小一个量而又保持关系不变时，这非常有用。例如，1 : 4 等价于 2 : 8（乘以 2）和 0.5 : 2（除以 2）。识别等价比率有助于解决比例问题。

You can check whether two ratios are equivalent by writing them as fractions and seeing if they simplify to the same value. The ratio 3 : 5 is equivalent to 30 : 50 because both simplify to the fraction 3/5. Being comfortable with equivalent ratios will make sharing amounts and finding missing values much easier.

你可以通过把两个比率写成分数并看它们是否化简为相同的值来检验它们是否等价。3 : 5 等价于 30 : 50，因为两者都化简为 3/5。熟练运用等价比率会使分配量和求未知值变得简单得多。

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4. Writing Ratios as Fractions | 将比率写成分数

A ratio between two quantities can be expressed as a fraction to show what part of the whole one quantity represents. If the ratio of boys to girls in a class is 4 : 5, the total number of parts is 4 + 5 = 9. The fraction of the class that is boys is 4/9, and girls is 5/9. It is crucial to remember that the ratio parts are not the actual numbers of students, but shares of the whole.

两个数量之间的比率可以表示为分数，以显示其中一个数量占总体的一部分。如果一个班级男生和女生的比率是 4 : 5，总份数为 4 + 5 = 9。男生占全班的比例是 4/9，女生是 5/9。必须牢记，比率中的份数并不是实际的学生人数，而是整体的份额。

When the ratio involves three parts, such as A : B : C = 2 : 3 : 4, the total parts = 2+3+4 = 9. The fraction for A is 2/9, for B is 3/9 (or 1/3), and for C is 4/9. Being able to switch between ratios and fractions is essential when calculating percentages or working with mixing problems.

当比率涉及三项时，例如 A : B : C = 2 : 3 : 4，总份数 = 2+3+4 = 9。A 的分数是 2/9，B 是 3/9（即 1/3），C 是 4/9。能够在比率和分数之间转换对于计算百分数或处理混合问题至关重要。

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5. Sharing in a Given Ratio | 按给定比率分配

Sharing an amount in a given ratio means dividing it into parts according to the ratio numbers. To do this, first find the total number of parts by adding the numbers in the ratio. Then, divide the total amount by the total number of parts to find the value of one part. Finally, multiply the value of one part by each ratio number.

按给定比率分配一笔钱意味着根据比率数字将其分成若干份。做法是：首先将比率中的数字相加求出总份数。然后，将总金额除以总份数，得到每一份的值。最后，用每一份的值乘以比率中的每个数字。

For example, share £40 in the ratio 3 : 5. Total parts = 3+5 = 8. One part = £40 ÷ 8 = £5. The first share is 3 parts = 3 × £5 = £15, and the second share is 5 × £5 = £25. Always check your answer: £15 + £25 = £40. This method works for any number of parts and any type of quantity, such as lengths, weights, or numbers of people.

例如，将 40 英镑按 3 : 5 的比例分配。总份数 = 3+5 = 8。一份 = 40 ÷ 8 = 5 英镑。第一份是 3 份 = 3 × 5 = 15 英镑，第二份是 5 × 5 = 25 英镑。一定记得检查答案：15 + 25 = 40 英镑。此方法适用于任何份数和任何类型的量，如长度、重量或人数。

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6. Finding Missing Values Using Ratios | 使用比率求未知值

When you know that two ratios are equal, you can find an unknown quantity by using the unitary method or by setting up equivalent fractions. Suppose the ratio of pens to pencils is 2 : 7, and you have 14 pens. To find how many pencils, divide 14 by 2 to find the multiplier: 14 ÷ 2 = 7. Then multiply 7 by the pencil ratio number: 7 × 7 = 49. So there are 49 pencils.

当你知道两个比率相等时，可以通过单位法或建立等价分数来求未知量。假设钢笔与铅笔的比率是 2 : 7，你有 14 支钢笔。要求铅笔的数量，用 14 除以 2 求得倍数：14 ÷ 2 = 7。然后将铅笔的比率数字乘以 7：7 × 7 = 49。所以有 49 支铅笔。

You can also use a table of equivalent ratios to visualise the scaling. For the ratio 3 : 8, if 3 corresponds to 15, then ask: what did I multiply 3 by to get 15? The multiplier is 5. Apply the same multiplier to 8, giving 40. This technique is a powerful tool for map scales, recipes, and similar real‑life applications.

你也可以用等价比率表来直观地展示缩放过程。对于比率 3 : 8，如果 3 对应 15，那么问自己：3 乘以什么数得到 15？乘数是 5。将 8 乘以同一个数，得到 40。这一技巧是解决地图比例尺、食谱及类似现实应用的强大工具。

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7. Direct Proportion | 正比例

Two quantities are in direct proportion if they increase or decrease at the same rate. This means that when one quantity is multiplied by a number, the other is multiplied by the same number. The ratio between the two quantities remains constant. If 5 apples cost £2.00, then 10 apples cost £4.00; the cost is directly proportional to the number of apples.

如果两个量以相同的速率增加或减少，它们就是成正比例的。这意味着当一个量乘以一个数时，另一个量也乘以同一个数。两个量之间的比率保持不变。如果 5 个苹果售价 2.00 英镑，那么 10 个苹果售价 4.00 英镑；价格与苹果的数量成正比。

In problems on direct proportion, you can always find the value of 1 unit first – this is the unitary method. For example, if 4 identical books weigh 2.8 kg, the weight of one book is 2.8 ÷ 4 = 0.7 kg. Then the weight of 11 such books is 11 × 0.7 = 7.7 kg. The key thing to check is whether doubling one quantity doubles the other; if it does, it is likely a direct proportion.

在正比例问题中，你总是可以先求出 1 个单位的值——这就是单位法。例如，如果 4 本相同的书重 2.8 千克，那么一本书的重量是 2.8 ÷ 4 = 0.7 千克。那么 11 本这样的书重 11 × 0.7 = 7.7 千克。需要检查的关键是两倍变化是否导致另一个量也翻倍；如果是，则很可能是正比例。

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8. Using the Unitary Method | 使用单位法

The unitary method is a step‑by‑step approach to solving proportion problems by first finding the value of 1 unit. It is particularly helpful when the numbers are not immediately obvious multiples. For example, if 7 tickets cost £26.25, find the cost of 12 tickets. Cost of 1 ticket = £26.25 ÷ 7 = £3.75. Then 12 tickets cost 12 × £3.75 = £45.00.

单位法是通过先求出 1 个单位的值来逐步解决比例问题的方法。当数字不是显而易见的倍数时，它特别有用。例如，如果 7 张票花费 26.25 英镑，求 12 张票的价格。1 张票的价格 = 26.25 ÷ 7 = 3.75 英镑。那么 12 张票的价格为 12 × 3.75 = 45.00 英镑。

This method can also be used in reverse: if you know the value of several units, you can find the value of just one. Always label your units clearly and check that your answer is sensible. In exams, showing the unitary step demonstrates your reasoning and often earns method marks even if the final answer has a small arithmetic error.

这个方法也可以反过来用：如果你知道若干个单位的总值，可以求出单个单位的值。务必清楚地标明单位并检查答案是否合理。在考试中，展示单位步骤可以体现你的推理过程，即使最终答案有小的算术错误也通常能拿到方法分。

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9. Scale Drawings and Map Scales | 比例图与地图比例尺

Maps and scale drawings use ratios to represent real‑life distances. A map scale might be given as a ratio, such as 1 : 50 000, which means 1 cm on the map represents 50 000 cm (or 0.5 km) in real life. If two towns are 4.2 cm apart on the map, the real distance is 4.2 × 50 000 = 210 000 cm = 2.1 km.

地图和比例图使用比率来表示现实生活中的距离。地图比例尺可以用比率表示，例如 1 : 50 000，这意味着地图上 1 厘米代表实际中的 50 000 厘米（即 0.5 公里）。如果地图上两个城镇相距 4.2 厘米，那么实际距离为 4.2 × 50 000 = 210 000 厘米 = 2.1 公里。

When working with scale drawings, always convert all measurements to the same unit before doing calculations. A plan of a house with scale 1 : 100 means 1 cm on the plan stands for 100 cm (1 m) in reality. If a room measures 4.5 cm by 3.2 cm on the plan, its real dimensions are 4.5 m by 3.2 m. Understanding scales is a direct application of ratio and proportion.

处理比例图时，务必在计算前将所有测量值转换为相同的单位。一栋房子的平面图比例为 1 : 100，意味着图上 1 厘米代表现实中的 100 厘米（1 米）。如果一间房间在图上量得 4.5 厘米乘 3.2 厘米，其实际尺寸就是 4.5 米乘 3.2 米。理解比例尺是比率和比例的直接应用。

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10. Common Misconceptions | 常见误区

One common mistake is confusing the parts of a ratio with the total. A student might see the ratio 2 : 3 and think ‘oh, the total is 5, so 2 is 2/5 and 3 is 3/5’, which is actually correct for fractions of the whole – but sometimes they mistakenly think 2 and 3 are the actual quantities. Remember, the ratio tells you the relationship, not the amounts. Another error is adding the same number to both sides of a ratio, for example changing 2 : 3 to 3 : 4 by adding 1; ratios are not preserved by addition, only by multiplication or division.

一个常见的误区是混淆了比率中的份额和总数。学生看到比率 2 : 3 可能会想“哦，总数是 5，所以 2 是 2/5，3 是 3/5”——这在表示整体的分数时实际上是对的——但有时他们会误以为 2 和 3 就是实际数量。记住，比率告诉你的只是关系，而不是实际数量。另一个错误是给比率的每一边加上相同的数，例如通过加 1 把 2 : 3 变成 3 : 4；比率不能通过加法保持不变，只有乘或除才能得到等价比率。

Students also sometimes write ratios in the wrong order. When a problem says ‘the ratio of A to B’, the first number corresponds to A and the second to B. Mixing up the order can lead to completely wrong answers. Always read the question carefully and double‑check which quantity comes first. Practising with real‑life examples helps avoid these pitfalls.

同学们有时也会把比率的顺序写反。当题目说“A 与 B 的比率”时，第一个数字对应 A，第二个对应 B。弄错顺序可能导致完全错误的答案。要仔细读题，反复确认哪一个量在前。通过生活中的例子来练习有助于避免这些陷阱。

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11. Ratio and Proportion in Problem Solving | 应用题中的比率和比例

In exams, ratio problems are often presented as word problems. A typical example: ‘The ratio of adults to children on a bus is 3 : 2. There are 30 adults. How many children are there?’ The multiplier is 30 ÷ 3 = 10, so children = 2 × 10 = 20. Another type: ‘Divide 180 in the ratio 1 : 2 : 3.’ Total parts = 6, one part = 30, so the shares are 30, 60, and 90.

考试中，比率问题通常以文字题的形式出现。一个典型的例子：“一辆公共汽车上成年人与儿童的比率是 3 : 2。有 30 名成年人。儿童有多少人？”乘数是 30 ÷ 3 = 10，所以儿童 = 2 × 10 = 20。另一类：“将 180 按 1 : 2 : 3 分配。”总份数 = 6，一份 = 30，因此各份为 30、60 和 90。

To solve more complex problems, draw a simple table with ratio and actual values. For instance, if the ratio of red to blue to green marbles is 5 : 3 : 2 and there are 80 marbles in total, add the ratio numbers to get 10 parts. One part = 80 ÷ 10 = 8 marbles. Then red = 5 × 8 = 40, blue = 3 × 8 = 24, and green = 2 × 8 = 16. A table helps you organise your thinking and reduce errors.

要解决更复杂的问题，可以画一个简单的表格，列出比率和实际值。例如，如果红、蓝、绿弹珠的比率是 5 : 3 : 2，总共有 80 颗弹珠，把比率数字相加得到 10 份。一份 = 80 ÷ 10 = 8 颗。那么红色 = 5 × 8 = 40，蓝色 = 3 × 8 = 24，绿色 = 2 × 8 = 16。通过表格可以帮助整理思路并减少错误。

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12. Summary and Key Points | 总结与要点

Let’s recap the most important ideas. A ratio compares quantities part‑to‑part. It can be simplified by dividing by the HCF. Equivalent ratios are formed by multiplying or dividing all parts by the same number. To share an amount in a given ratio, divide the total by the total number of parts, then multiply by each ratio number. Direct proportion means the two quantities always stay in the same ratio, and the unitary method (find 1 first) is a reliable strategy.

我们来回顾一下最重要的概念。比率对数量进行部分与部分的比较。可以通过除以最大公因数来简化。等价比率是将所有项乘以或除以同一个数得到的。要按给定比率分配一个量，用总数除以总份数，然后乘以比率的每个数字。正比例意味着两个量始终保持相同的比率，单位法（先求 1 个单位）是一种可靠的策略。

Always write the ratio in the correct order, and check that your simplified ratio has no common factor. When using map scales, convert units carefully. Practice is essential to become confident with ratio and proportion. The skills you learn here will be used again in topics like similar shapes, percentages, speed, and density in later years, so building a strong foundation now is very important.

务必按正确的顺序书写比率，并检查简化后的比率是否没有公因数。使用地图比例尺时，要仔细换算单位。练习对于自信地掌握比率和比例至关重要。你在这里学到的技能将在未来的相似形、百分比、速度和密度等课题中再次使用，因此现在打下扎实的基础非常重要。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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