📚 Ratio, Proportion and Scale Drawings | 比、比例与比例图
Ratio and proportion are central tools in the Cambridge Lower Secondary and KS3 mathematics course. They let you compare quantities, share amounts fairly, solve everyday problems and read maps or scale drawings with confidence. This revision guide covers the key definitions, methods, worked examples, common errors and exam tips you need.
比和比例是剑桥初中与 KS3 数学课程中的核心工具。它们帮助你比较数量、公平分配、解决日常问题，并自信地读懂地图或比例图。本复习指南涵盖关键定义、方法、典型例题、常见错误与考试技巧。
1. Understanding Ratios | 理解比
A ratio is a way of comparing two or more quantities of the same kind using a colon. If a fruit bowl has 4 apples and 6 bananas, the ratio of apples to bananas is written as 4 : 6. We read this as "4 to 6", meaning that for every 4 apples there are 6 bananas.
比是一种用冒号比较两个或多个同类数量的方式。如果一个水果碗里有 4 个苹果和 6 根香蕉，苹果与香蕉的比写作 4 : 6。读作"4 比 6"，表示每 4 个苹果对应 6 根香蕉。
The order of the numbers must match the order of the words. So 4 : 6 is apples to bananas, while 6 : 4 is bananas to apples. Reversing the order changes the meaning completely, even though both ratios describe the same bowl of fruit.
数字的顺序必须与文字的顺序一致。所以 4 : 6 是苹果与香蕉的比，而 6 : 4 是香蕉与苹果的比。颠倒顺序会完全改变含义，尽管两个比描述的是同一碗水果。
Equivalent ratios describe the same relationship. Multiplying or dividing every part of a ratio by the same non-zero number produces an equivalent ratio. For example, 4 : 6, 2 : 3, 8 : 12 and 20 : 30 are all equivalent.
等价比描述相同的关系。将比的每一项同时乘或除以同一个非零数，会得到等价比。例如 4 : 6、2 : 3、8 : 12 和 20 : 30 都是等价比。
4 : 6 = 2 : 3 = 8 : 12 = 20 : 30
Ratios are dimensionless: they compare quantities of the same type, so no unit is attached to the final answer. A ratio can compare one part with another part, or one part with the whole. However, a part-to-part ratio is not the same as a fraction of the total unless you work out the total number of parts first.
比是无量纲的：它比较的是同类型的数量，所以最终答案不带单位。比可以比较部分与部分，也可以比较部分与整体。但是，部分与部分之比并不等同于总量中的分数，除非你先算出总份数。
2. Simplifying Ratios | 化简比
To simplify a ratio, divide every part by their highest common factor (HCF). For example, 12 : 18 simplifies by dividing both numbers by 6, giving 2 : 3. The simplified ratio has no common factor other than 1, and it is the clearest way to present a ratio in an exam.
化简比时，把每一项都除以它们的最大公因数 (HCF)。例如，12 : 18 两边同时除以 6，得到 2 : 3。化简后的比除 1 以外没有其他公因数，这是考试中呈现比的最清晰方式。
If the ratio contains decimals, multiply every part by a suitable power of 10 to make all terms whole numbers, then divide by the HCF. For instance, 0.5 : 0.25 becomes 50 : 25, which simplifies to 2 : 1.
如果比中含有小数，先把每一项乘以适当的 10 的幂，使所有项都变成整数，再除以 HCF。例如 0.5 : 0.25 变为 50 : 25，化简得 2 : 1。
If the ratio contains fractions, multiply every part by the lowest common multiple (LCM) of the denominators. For example, 1½ : 2¼ can be written as 3/2 : 9/4; multiplying both parts by 4 gives 6 : 9, which simplifies to 2 : 3.
如果比中含有分数，先把每一项乘以所有分母的最小公倍数 (LCM)。例如 1½ : 2¼ 可以写成 3/2 : 9/4；两边同乘 4 得 6 : 9，化简为 2 : 3。
Step 1: check that all quantities are in the same unit.
第一步：确认所有数量使用同一单位。
Step 2: remove decimals or fractions so that every term is a whole number.
第二步：去掉小数或分数，使每一项都是整数。
Step 3: divide every term by the HCF to finish.
第三步：每一项除以最大公因数，完成化简。
3. Ratios with Different Units and Three Terms | 不同单位与三项比
When quantities use different units, first convert them to the same unit before writing or simplifying the ratio. A ratio of 2 m to 50 cm should be written as 200 cm : 50 cm, then simplified to 4 : 1. Never leave a ratio with mixed units, because the comparison would be meaningless.
当数量使用不同单位时，先换算成同一单位，再写比或化简。2 m 与 50 cm 的比应先写成 200 cm : 50 cm，再化简为 4 : 1。绝不要让比中混用不同单位，因为那样的比较没有意义。
Ratios can also have three or more terms, such as 1 : 2 : 3 for a concrete mix or 2 : 3 : 5 for a fruit blend. Every term is compared with every other term, and the same simplification rules apply to all parts.
比也可以有三项或更多项，例如混凝土配比的 1 : 2 : 3，或水果混合比的 2 : 3 : 5。每一项都与其他项相比较，化简规则同样适用于所有项。
m b the same in both. Multiply the first ratio by 4 and the second by 3 to get a : b = 8 : 12 and b : c = 12 : 15, so a : b : c = 8 : 12 : 15.
有时两个单独的比共享同一个量。如果 a : b = 2 : 3 且 b : c = 4 : 5，可以通过让公共项 b 在两个比中相同，把它们合并成三项比。第一个比乘 4、第二个比乘 3，得到 a : b = 8 : 12 和 b : c = 12 : 15，因此 a : b : c = 8 : 12 : 15。
a : b : c = 8 : 12 : 15
4. Ratio and Fraction: What Is the Difference? | 比与分数的区别
A ratio compares quantities, while a fraction can represent a part of a whole. In the fruit bowl with 4 apples and 6 bananas, the ratio of apples to bananas is 4 : 6, or 2 : 3 when simplified. The fraction of the fruit that is apples is 4/10, not 4/6, because there are 10 pieces of fruit in total.
比用于比较数量，而分数可以表示整体中的一部分。在有 4 个苹果和 6 根香蕉的水果碗中，苹果与香蕉的比是 4 : 6，化简为 2 : 3。苹果占全部水果的分数是 4/10，而不是 4/6，因为水果总共有 10 个。
are 2 + 3 = 5 shares in total. The first ingredient is 2/5 of the mixture and the second is 3/5.
要把部分与部分之比转化为占整体的分数，先把各项相加求出总份数。如果混合物按 2 : 3 配制，总份数为 2 + 3 = 5。第一种成分占混合物的 2/5，第二种占 3/5。
for a fraction, a percentage or a decimal rather than a ratio. For example, 3/5 = 0.6 = 60%, so the second ingredient makes up 60% of the mixture.
当题目要求的是分数、百分数或小数而不是比时，这种转换非常关键。例如 3/5 = 0.6 = 60%，所以第二种成分占混合物的 60%。
5. Sharing an Amount in a Given Ratio | 按比分配
Sharing in a ratio is one of the most common exam question types. The method is: add the parts to find the total number of shares, divide the total amount by the number of shares to find the value of one share, then multiply by each part of the ratio.
按比分配是最常见的考试题型之一。方法是：先把各项相加求出总份数，用总量除以总份数得到一份的值，再分别乘以比中的每一项。
For example, share £60 in the ratio 2 : 3. The total number of shares is 2 + 3 = 5. One share is £60 ÷ 5 = £12. The two parts are 2 × £12 = £24 and 3 × £12 = £36.
例如，把 60 英镑按 2 : 3 分配。总份数为 2 + 3 = 5。一份是 60 ÷ 5 = 12 英镑。两部分分别是 2 × 12 = 24 英镑和 3 × 12 = 36 英镑。
£60 ÷ 5 = £12, then 2 × £12 = £24 and 3 × £12 = £36
e original total, and their ratio should simplify to the ratio given in the question. Here £24 + £36 = £60 and 24 : 36 = 2 : 3, so the answer is correct.
务必检查答案：两个金额相加应等于原来的总量，且它们的比化简后应等于题目给出的比。这里 24 + 36 = 60 英镑，且 24 : 36 = 2 : 3，所以答案正确。
io 2 : 3, the first share is 2/5 of the total and the second is 3/5 of the total. So 2/5 × £60 = £24 and 3/5 × £60 = £36. Both methods give the same result.
你也可以使用分数法。对于比 2 : 3，第一份是总量的 2/5，第二份是总量的 3/5。所以 2/5 × 60 = 24 英镑，3/5 × 60 = 36 英镑。两种方法结果相同。
6. Proportion and the Unitary Method | 比例与单位法
oportion problems usually ask you to find a missing value when three of the four quantities are known.
比例是表示两个比相等的陈述，例如 3 : 4 = 15 : 20。写成分数形式是 3/4 = 15/20。比例问题通常要求你在四个量中已知三个的情况下求出缺失的值。
red value. If 3 : 4 = x : 20, then 20 ÷ 4 = 5, so one unit is 5. Therefore x = 3 × 5 = 15.
单位法是最稳妥的方法：先求一个单位的值，再乘出所需的值。如果 3 : 4 = x : 20，那么 20 ÷ 4 = 5，所以一个单位是 5。因此 x = 3 × 5 = 15。
For example, if 5 pens cost £3.50, find the cost of 8 pens. One pen costs £3.50 ÷ 5 = £0.70, so 8 pens cost 8 × £0.70 = £ ing out the working in clear steps earns method marks even if you make an arithmetic slip.
例如，如果 5 支笔售价 3.50 英镑，求 8 支笔的价格。一支笔为 3.50 ÷ 5 = 0.70 英镑，所以 8 支笔为 8 × 0.70 = 5.60 英镑。把过程分步写清楚，即使算错也能拿到方法分。
he same answer, but you must still understand the underlying proportion and check that your final value is reasonable.
交叉相乘是一种捷径：如果 a/b = c/d，那么 a × d = b × c。它给出同样的答案，但你仍要理解背后的比例关系，并检查最终结果是否合理。
7. Direct and Inverse Proportion | 正比例与反比例
then y = kx for some constant k, and the ratio y : x stays the same. Doubling x doubles y.
在正比例中，两个量按相同的比一起增大或减小。如果 y 与 x 成正比，那么 y = kx，其中 k 为常数，且比 y : x 保持不变。x 加倍，y 也加倍。
then y = k/x, and doubling x halves y. A classic example is the number of workers and the time taken to complete a job.
在反比例中，一个量增大而另一个量减小，使它们的乘积保持不变。如果 y 与 x 成反比，那么 y = k/x，x 加倍则 y 减半。经典例子是工人数量与完成一项工作所需的时间。
| Type | Relationship | Example |
| Direct | y = kx | More hours, more pay |
| Inverse | y = k/x | More workers, less time |
Always identify which type of proportion the question describes before calculating. Mixing up direct and inverse proportion is one of the most frequent errors in this topic.
计算前一定要判断题目描述的是哪种比例。混淆正比例与反比例是本主题最常见的错误之一。
8. Proportion in Recipes and Everyday Life | 食谱与生活中的比例
6/4 = 300 g. The scaling factor is 6 ÷ 4 = 1.5, so every ingredient is multiplied by 1.5.
食谱本质上是伪装的比例问题。如果一份 4 人份的食谱需要 200 克面粉，那么 6 人份需要 200 × 6/4 = 300 克。缩放因子是 6 ÷ 4 = 1.5，所以每种配料都乘以 1.5。
Best-buy questions compare prices using unit rates. If 500 g of rice costs £2.40, the cost per 100 g is £2.40 ÷ 5 = £0.48. If 750 g costs £3.30, the cost per 100 g is £3.30 ÷ 7.5 = £0.44. The second packet is better value.
最划算问题用单价来比较。如果 500 克大米售价 2.40 英镑，每 100 克价格为 2.40 ÷ 5 = 0.48 英镑。如果 750 克售价 3.30 英镑，每 100 克为 3.30 ÷ 7.5 = 0.44 英镑。第二包更划算。
£2.40 ÷ 5 = £0.48 per 100 g; £3.30 ÷ 7.5 = £0.44 per 100 g
Other everyday examples include mixing paint, diluting juice, currency exchange, speed and density. In every case, set up the ratio carefully, keep the units consistent and check that your final answer is sensible.
其他日常例子包括调配颜料、稀释果汁、货币兑换、速度与密度。无论哪种情况，都要小心列出比，保持单位一致，并检查最终答案是否合理。
9. Scale Drawings and Map Scales | 比例图与地图比例尺
A scale drawing represents a real object using a ratio called a scale. A scale of 1 : 50 means that 1 cm on the drawing represents 50 cm in real life. Map scales work in exactly the same way.
比例图用一个称为比例尺的比来表示真实物体。比例尺 1 : 50 表示图上 1 厘米代表实际 50 厘米。地图比例尺的原理完全相同。
presents 250 m on the ground. The scale is dimensionless, but the units you use must be consistent.
在比例尺为 1 : 25 000 的地图上，图上 1 厘米代表实际 25 000 厘米。由于 25 000 厘米 = 250 米，所以图上 1 厘米代表地面 250 米。比例尺是无量纲的，但你使用的单位必须一致。
cm long on a map with scale 1 : 50 000, the real length is 6 × 50 000 = 300 000 cm = 3 km.
求实际长度时，把图上的长度乘以比例尺的倍数。如果一条路在比例尺 1 : 50 000 的地图上是 6 厘米，实际长度为 6 × 50 000 = 300 000 厘米 = 3 千米。
6 cm × 50 000 = 300 000 cm = 3 km
To find a drawing length, divide the real length by the scale factor, converting units first. Remember that for areas the scale factor is squared: a scale of 1 : 100 gives an area ratio of 1 : 10 000.
求图上长度时，把实际长度除以比例尺的倍数，并先换算单位。记住，对于面积，比例尺的倍数是平方关系：比例尺 1 : 100 对应的面积比为 1 : 10 000。
10. Common Errors to Avoid | 常见错误与避免方法
Many marks are lost through small slips rather than a lack of understanding. Knowing the typical mistakes helps you avoid them in timed conditions.
很多分数不是丢在不会做，而是丢在小失误上。了解典型错误有助于你在限时考试中避免它们。
Reversing the order of a ratio. Always match the numbers to the words in the question.
把比的顺序写反。始终让数字与题目中的文字顺序对应。
Forgetting to convert units. Write both quantities in the same unit before simplifying.
忘记换算单位。化简前先把两个数量写成同一单位。
Not simplifying fully. Divide by the HCF, not just any common factor.
没有化到最简。要除以最大公因数，而不只是任意公因数。
Confusing a part-to-part ratio with a fraction of the whole. Add the parts first to find the total.
把部分与部分之比与占整体的分数混淆。先相加求出总份数。
Forgetting to add the parts when sharing an amount. The total number of shares is essential.
按比分配时忘记把各项相加。总份数是关键。
Mixing up direct and inverse proportion. Ask whether both quantities rise together or one falls as the other rises.
混淆正比例与反比例。先问自己：两个量是一起增大，还是一个增大时另一个减小。
11. Exam Technique and Checking | 考试技巧与检验
Show every step of your working, especially in ratio-sharing and proportion questions. Even if the final answer is wrong, clear method steps can earn valuable marks.
把每一步过程都写出来，尤其是在按比分配和比例问题中。即使最终答案错误，清晰的方法步骤也能拿到宝贵的分数。
Check your answer in three ways: does it add back to the original total; does the ratio simplify to the one given; and is the size of the answer reasonable in context?
用三种方式检查答案：相加是否回到原来的总量；化简后的比是否等于题目给出的比；答案的大小在情境中是否合理。
version clearly, for example 300 000 cm = 3 km, so the examiner can follow your reasoning.
全程保留单位。在比例图题目中，把换算写清楚，例如 300 000 厘米 = 3 千米，这样阅卷老师可以跟上你的推理。
Finally, practise interpreting word problems. Underline the quantities, identify what is being compared and decide whether the relationship is a ratio, a proportion or a scale before you start calculating.
最后，多练习解读应用题。划出数量，识别比较的对象，在开始计算前判断它是比、比例还是比例尺。
12. Quick Practice and Answers | 快速练习与答案
Try these questions without a calculator first, then check your answers below.
先尝试不用计算器完成下列题目，然后对照下面的答案。
1. Simplify 24 : 36.
1. 化简 24 : 36。
2. Write 1.5 m : 60 cm as a simplified ratio.
2. 把 1.5 米 : 60 厘米写成最简比。
3. Share £84 in the ratio 3 : 4.
3. 把 84 英镑按 3 : 4 分配。
4. If 4 notebooks cost £6, how much do 7 notebooks cost?
4. 如果 4 本笔记本售价 6 英镑，7 本多少钱？
5. A map has scale 1 : 20 000. Two towns are 9 cm apart on the map. Find the real
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