📚 Ratio and Proportion: Simplifying and Sharing | 比率与比例：化简与分配
In KS3 Mathematics, ratio and proportion are essential tools for comparing quantities, sharing amounts fairly, and solving real-life problems such as recipes, scales, and money. This article reviews the key methods you need for Cambridge KS3, including simplifying ratios, sharing in a given ratio, converting ratios to fractions, and using the unitary method for proportion, with worked examples, exam tips and practice questions at the end.
在 KS3 数学中，比率和比例是比较数量、公平分配以及解决食谱、比例尺和金钱等实际问题的重要工具。本文复习 Cambridge KS3 所需的关键方法，包括化简比、按给定比例分配、将比率转换为分数，以及使用单位法解决比例问题，并在文末提供典型例题、考试技巧与练习题。
1. What is a Ratio? | 什么是比率？
A ratio is a way of comparing two or more quantities of the same type using division. We write it with a colon, for example 2 : 3, and we read it as 'two to three'. The order matters enormously: the ratio of boys to girls is not the same as the ratio of girls to boys. If a class contains 12 boys and 15 girls, the ratio of boys to girls is 12 : 15, which simplifies to 4 : 5, meaning that for every 4 boys there are 5 girls.
比率是用除法比较两个或多个同类量的方法。我们用一个冒号书写，例如 2 : 3，读作"二比三"。顺序极其重要：男生与女生的比并不等于女生与男生的比。如果一个班有 12 名男生和 15 名女生，男生与女生的比为 12 : 15，化简得 4 : 5，意思是每 4 名男生对应 5 名女生。
It is important to distinguish part-to-part ratios from part-to-whole ratios. A part-to-part ratio compares one part with another part; a part-to-whole ratio compares one part with the total. In the class above, the part-to-part ratio boys : girls is 4 : 5, while boys : total students is 12 : 27, which simplifies to 4 : 9.
区分"部分与部分"的比和"部分与整体"的比很重要。部分与部分的比是把一个部分与另一个部分进行比较；部分与整体的比是把一个部分与总量进行比较。在上述班级中，男生 : 女生的部分与部分之比是 4 : 5，而男生 : 全班人数是 12 : 27，化简为 4 : 9。
- A ratio can also be written as a fraction: 2 : 3 = 2/3 (as a part-to-part comparison).
- The quantities being compared must be in the same unit; otherwise convert first.
- Ratios can involve more than two parts, such as 2 : 3 : 5 for a three-way mixture.
- 比率也可以写成分数形式：2 : 3 = 2/3（作为部分与部分的比较）。
- 被比较的量必须单位相同，否则要先换算单位。
- 比率可以涉及两个以上的部分，例如三方混合物的 2 : 3 : 5。
2. Simplifying Ratios | 化简比
To simplify a ratio, divide every part by their highest common factor (HCF). This works exactly like simplifying fractions, except that the parts stay separate and are still joined by colons. A ratio is in its simplest form when the only common factor of all the parts is 1.
化简比时，把每一部分同时除以它们的最大公因数（HCF）。这与化简分数的做法完全一致，只是各部分保持独立，仍用冒号连接。当所有部分的公因数只有 1 时，比率就达到了最简形式。
Notice that 2 and 3 share no common factor other than 1, so the ratio is fully simplified.
示例：化简 24 : 36。24 与 36 的最大公因数是 12，两边同时除以 12，得 2 : 3。检验：24 ÷ 12 = 2，36 ÷ 12 = 3。注意 2 与 3 除 1 外没有公因数，因此该比已完全化简。
24 : 36 = (24 ÷ 12) : (36 ÷ 12) = 2 : 3
When a ratio contains decimals, first multiply every part by a suitable power of 10 to clear the decimals, then simplify. When it contains fractions, multiply every part by the lowest common multiple of the denominators.
当比率中含有小数时，先把各部分同乘一个合适的 10 的幂以去掉小数，再化简。当比率中含有分数时，把各部分同乘各分母的最小公倍数。
| Original ratio | Step | Simplest form |
| 24 : 36 | ÷ 12 | 2 : 3 |
| 45 : 60 : 90 | ÷ 15 | 3 : 4 : 6 |
| 0.5 : 0.25 | × 100, then ÷ 25 | 2 : 1 |
| 1/2 : 1/3 | × 6 | 3 : 2 |
Sometimes a ratio is written in the form 1 : n or n : 1. This is called a unit ratio and is very useful for comparison. To convert 8 : 12 into the form 1 : n, divide both parts by 8, giving 1 : 1.5. To convert it into n : 1, divide both parts by 12, giving 2/3 : 1.
有时比率会写成 1 : n 或 n : 1 的形式，这称为单位比，对比较非常有用。要把 8 : 12 化为 1 : n 的形式，两边同除以 8，得 1 : 1.5；要化为 n : 1 的形式，两边同除以 12，得 2/3 : 1。
3. Equivalent Ratios, Ratio Tables and Unit Ratios | 等值比、比率表与单位比
Two ratios are equivalent if one can be obtained from the other by multiplying or dividing all the parts by the same non-zero number. So 2 : 3, 4 : 6, 6 : 9 and 20 : 30 are all equivalent. Equivalent ratios describe exactly the same relationship between quantities, just measured in different-sized units.
如果可以通过把所有部分同乘或同除一个非零数由其中一个比率得到另一个比率，那么这两个比率就是等值比。因此 2 : 3、4 : 6、6 : 9 与 20 : 30 都是等值比。等值比描述的数量关系完全相同，只是所用的度量单位大小不同。
A ratio table is a neat way of organising equivalent ratios and spotting the scaling factor. It is especially helpful in recipe and mixture problems where you repeatedly scale up or down.
比率表是整理等值比、发现缩放因子的好方法。在需要反复按倍数放大或缩小的食谱和混合问题中尤其有用。
| Flour (g) | 200 | 400 | 600 | 100 |
| Butter (g) | 50 | 100 | 150 | 25 |
nit ratios let you compare two different ratios directly. Which is larger, 7 : 9 or 8 : 11? reater than 0.727, the ratio 7 : 9 represents a larger first quantity relative to the second.
单位比可以让你直接比较两个不同的比率。7 : 9 与 8 : 11 哪个更大？把它们都写成 n : 1 的形式：7 : 9 = 0.778 : 1，8 : 11 = 0.727 : 1。由于 0.778 大于 0.727，所以比率 7 : 9 相对于第二个量来说第一个量更大。
th 72 : 99, and again 7 : 9 wins. Both methods are accepted in exams; choose whichever you find faster and less error-prone.
另一种方法是用等值比把第二个数变成相同。因为 9 × 11 = 99，我们比较 77 : 99 与 72 : 99，结果同样是 7 : 9 更大。两种方法在考试中都被接受，选择你觉得更快、更不易出错的那一种即可。
4. Ratios with Different Units | 不同单位的比率
A ratio only makes sense when the quantities being compared are measured in the same unit. If they are not, convert one of them first, and remember that the unit itself is not written in the final answer.
只有当被比较的量用同一单位度量时，比率才有意义。如果单位不同，先换算其中一个，并记住最终的答案中不写单位。
2 hours to 120 minutes, giving 45 : 120. The HCF of 45 and 120 is 15, so 45 : 120 = 3 : 8.
示例：把 45 分钟与 2 小时的比写成最简形式。先把 2 小时换算为 120 分钟，得 45 : 120。45 与 120 的最大公因数是 15，所以 45 : 120 = 3 : 8。
45 minutes : 2 hours = 45 : 120 = 3 : 8
Example with money: write £1.50 to 90p as a ratio. Convert pounds to pence: £1.50 = 150p, so the ratio is 150 : 90 = 5 : 3 after dividing by 30.
关于金钱的示例：把 1.50 英镑与 90 便士写成比率。把英镑换算为便士：1.50 英镑 = 150 便士，所以比率为 150 : 90，除以 30 后得 5 : 3。
- Length: convert km ↔ m ↔ cm (× 1000, × 100).
- Mass: convert kg ↔ g (× 1000).
- Time: convert hours ↔ minutes ↔ seconds (× 60).
- Money: convert pounds ↔ pence (× 100).
- 长度：千米 ↔ 米 ↔ 厘米（× 1000、× 100）。
- 质量：千克 ↔ 克（× 1000）。
- 时间：小时 ↔ 分钟 ↔ 秒（× 60）。
- 金钱：英镑 ↔ 便士（× 100）。
s 3 : 500. This is wrong, because the units differ. Convert first: 3000 g : 500 g = 6 : 1.
考试中常见的陷阱是把 3 千克 : 500 克直接写成 3 : 500。这是错误的，因为单位不同。应先换算：3000 克 : 500 克 = 6 : 1。
5. Sharing in a Given Ratio | 按给定比例分配
Sharing an amount in a given ratio is one of the most heavily tested KS3 skills. The method has three clear steps: add the parts of the ratio to find the total number of parts, divide the total amount by the number of parts to find the value of one part, then multiply to find each share.
按给定比例分配一个总量，是 KS3 考试中最常考的技能之一。方法有三个清晰的步骤：先把比率各部分相加得到总份数，再用总量除以总份数求出一份的值，最后相乘求出每一份。
Example: share £240 in the ratio 3 : 5. Total parts = 3 + 5 = 8. One part = £240 ÷ 8 = £30. So the shares are 3 × £30 = £90 and 5 × £30 = £150. Always check: £90 + £150 = £240, which matches the original amount.
示例：把 240 英镑按 3 : 5 分配。总份数 = 3 + 5 = 8。一份 = 240 ÷ 8 = 30 英镑。因此两份分别是 3 × 30 = 90 英镑与 5 × 30 = 150 英镑。务必检验：90 + 150 = 240 英镑，与原总量一致。
Total parts = 3 + 5 = 8 → One part = £240 ÷ 8 = £30 → £90 : £150
so one part is 500 ÷ 10 = 50 sweets. The shares are 100, 150 and 250 sweets respectively.
同样的方法可推广到三部分之比。把 500 颗糖按 2 : 3 : 5 分配，总份数为 2 + 3 + 5 = 10，所以一份是 500 ÷ 10 = 50 颗糖。三份分别是 100 颗、150 颗和 250 颗。
Sometimes the question does not give the total but gives one share or the difference between shares. In those cases, work out how many parts that piece of information represents, then find one part.
有时题目不给总量，而是给出其中一份或两份之差。这种情况下，先判断这条信息对应多少份，再求出一份的值。
- Given one share: if a share of £90 corresponds to 3 parts, one part is £90 ÷ 3 = £30.
- Given the difference: if the difference between the shares is 2 parts and equals £60, one part is £30.
- Given the total: divide the total by the total number of parts.
- 已知其中一份：若 90 英镑对应 3 份，则一份为 90 ÷ 3 = 30 英镑。
- 已知差额：若两份之差为 2 份且等于 60 英镑，则一份为 30 英镑。
- 已知总量：用总量除以总份数。
6. Sharing Problems in Real Contexts | 实际情境中的分配问题
Ratio questions in exams are almost always wrapped in a real-life context. Recognising the structure behind the story is the key skill: once you strip away the recipe, the paint or the prize money, the mathematics is the same three-step sharing method.
考试中的比率题几乎总是包裹在实际情境中。识别情境背后的结构才是关键技能：一旦剥去食谱、油漆或奖金的"外衣"，数学上都是同样的三步分配法。
our and sugar in the ratio 5 : 3. If a baker uses 750 g of flour, how much sugar is needed? One part = 750 ÷ 5 = 150 g, so sugar = 3 × 150 = 450 g.
食谱示例：某个蛋糕配方中面粉与糖的比为 5 : 3。如果烘焙师用了 750 克面粉，需要多少糖？一份 = 750 ÷ 5 = 150 克，所以糖 = 3 × 150 = 450 克。
Money example: Ali, Ben and Cara share a prize in the ratio 2 : 3 : 7. If Cara receives £140 more than Ali, how much was the total prize? Cara has 7 parts and Ali has 2 parts, so the difference of 5 parts equals £140, giving one part = £28. The total is (2 + 3 + 7) = 12 parts = 12 × £28 = £336.
金钱示例：Ali、Ben 和 Cara 按 2 : 3 : 7 分配奖金。若 Cara 比 Ali 多得 140 英镑，奖金总额是多少？Cara 占 7 份，Ali 占 2 份，差的 5 份等于 140 英镑，所以一份 = 28 英镑。总量为 (2 + 3 + 7) = 12 份 = 12 × 28 = 336 英镑。
s, so 3 parts = 12, meaning one part = 4. Red beads = 4 × 4 = 16 and blue beads = 7 × 4 = 28, giving 44 beads in total.
年龄或比较示例：袋中红珠与蓝珠之比为 4 : 7，蓝珠比红珠多 12 颗。差为 7 − 4 = 3 份，所以 3 份 = 12，即一份 = 4。红珠 = 4 × 4 = 16 颗，蓝珠 = 7 × 4 = 28 颗，总共 44 颗。
Mixture example: paint is mixed from red and white in the ratio 3 : 8. To make 33 litres of paint, the total number of parts is 11, one part is 3 litres, so 9 litres of red and 24 litres of white are required.
混合示例：油漆由红色与白色按 3 : 8 混合。要配制 33 升油漆，总份数为 11，一份为 3 升，因此需要 9 升红色和 24 升白色。
7. Converting Ratios to Fractions | 将比率转换为分数
Converting a ratio into fractions is essential when a question asks what fraction of the whole each part represents. For a two-part ratio a : b, the whole is a + b parts, so the first part is a/(a + b) of the whole and the second part is b/(a + b).
当题目问每一部分占整体的几分之几时，把比率转换为分数就非常必要。对于两部分之比 a : b，整体是 a + b 份，因此第一部分占整体的 a/(a + b)，第二部分占 b/(a + b)。
a : b → first part = a ÷ (a + b) and second part = b ÷ (a + b)
ol has 1,080 students, the number of boys is (4/9) × 1080 = 480 and the number of girls is (5/9) × 1080 = 600.
示例：某校男生与女生之比为 4 : 5。那么男生占学生总数的 4/9，女生占 5/9。若全校有 1080 名学生，则男生人数为 (4/9) × 1080 = 480 人，女生人数为 (5/9) × 1080 = 600 人。
getarian, then 3/7 are vegetarian and 4/7 are not, so vegetarian : non-vegetarian = 3 : 4.
反向转换同样重要：分数也可以化为比率。如果某群体中 3/7 是素食者，那么 3/7 是素食者、4/7 不是，所以素食者 : 非素食者 = 3 : 4。
s. Since a ratio of 3 : 5 means the first part is 3/8 of the whole, and 3/8 = 0.375 = 37.5%, we can say that 37.5% of the total belongs to the first share.
百分数与这两个概念天然相连。由于 3 : 5 意味着第一部分占整体的 3/8，而 3/8 = 0.375 = 37.5%，因此可以说总量的 37.5% 属于第一份。
- Ratio to fraction: use the total number of parts as the denominator.
- Fraction to ratio: keep the numerator as the first part and use (denominator − numerator) as the second part.
- Ratio to percentage: convert to a fraction first, then multiply by 100.
- 比率转分数：以总份数作分母。
- 分数转比率：分子作为第一部分，用（分母 − 分子）作为第二部分。
- 比率转百分数：先化成分数，再乘以 100。
8. Ratio, Scale and Similar Shapes | 比率、比例尺与相似形
A scale drawing or map uses a ratio to link a distance on the drawing to the real distance. A scale of 1 : 25,000 means that 1 cm on the map represents 25,000 cm in real life, which is 250 m or 0.25 km.
比例尺图纸或地图用比率把图上的距离与真实距离联系起来。比例尺 1 : 25,000 表示图上 1 厘米代表现实中的 25,000 厘米，即 250 米或 0.25 千米。
direction, a real distance of 7.5 km would appear as 750,000 ÷ 50,000 = 15 cm on the map.
示例：在比例尺为 1 : 50,000 的地图上，两镇相距 6 厘米。实际距离为 6 × 50,000 = 300,000 厘米 = 3,000 米 = 3 千米。反向计算时，7.5 千米的实际距离在图上应画 750,000 ÷ 50,000 = 15 厘米。
e is 1.5 times the matching length in the smaller one. The ratio of the perimeters is also 2 : 3, but the ratio of the areas is 2² : 3² = 4 : 9.
相似图形遵循同样的逻辑。如果两个三角形相似且边长比为 2 : 3，那么大三角形中每一条对应边都是小三角形中对应边的 1.5 倍。周长比同样是 2 : 3，但面积比为 2² : 3² = 4 : 9
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