📚 Understanding Ratio and Proportion | 理解比率与比例
Ratio and proportion are fundamental concepts in mathematics that describe how quantities compare to each other, and how they scale up or down while keeping the same relationship. This revision guide covers writing and simplifying ratios, dividing amounts by ratio, understanding direct proportion, and solving real-world problems step by step — all aimed at building confidence for the KS3 Cambridge checkpoint.
比率与比例是数学中的基本概念，描述数量之间如何进行比较，以及在保持相同关系的前提下进行缩放。本复习指南涵盖比率的书写与化简、按比例分配量、理解正比例，并逐步解决实际问题——所有内容旨在为 KS3 Cambridge checkpoint 建立信心。
1. What Is a Ratio? | 什么是比率？
A ratio compares two or more quantities, showing how much of one thing there is compared to another. It can be written in the form a : b, using a colon, or as a fraction. The order in a ratio matters — the first number refers to the first quantity mentioned, the second number to the second, and so on.
比率用于比较两个或多个数量，表示一个事物相对于另一个事物有多少。比率可以用冒号形式 a : b 表示，或写成分数。比率中的顺序很重要——第一个数字对应首次提到的数量，第二个数字对应第二个，以此类推。
For example, if a class has 12 boys and 8 girls, the ratio of boys to girls is 12 : 8. If we wrote it as girls to boys, it would be 8 : 12. The ratio describes the relationship between the parts; it does not give the exact numbers unless a total is known.
例如，如果一个班级有 12 名男生和 8 名女生，男生与女生的比率为 12 : 8。如果我们写成女生与男生，则为 8 : 12。比率描述的是部分之间的关系；除非知道总数，否则它不给出确切数字。
2. Simplifying Ratios | 化简比率
To make a ratio easier to understand, we simplify it by dividing each part by the highest common factor (HCF). Every ratio can be reduced to its simplest integer form, just like simplifying a fraction.
为了使比率更容易理解，我们通过将每部分除以最大公因数（HCF）来化简。每个比率都可以约简为最简整数形式，就像化简分数一样。
Given 12 : 8, the HCF of 12 and 8 is 4. Dividing both numbers by 4 gives 3 : 2. This means that for every 3 boys, there are 2 girls. If the ratio contains a decimal or fraction, multiply all parts to clear them first. For example, 0.5 : 3 multiplied by 2 gives 1 : 6.
给定 12 : 8，12 和 8 的最大公因数是 4。将两数都除以 4 得到 3 : 2。这意味着每 3 名男生对应 2 名女生。如果比率包含小数或分数，先乘以一个数清除它们。例如，0.5 : 3 乘以 2 得到 1 : 6。
Ratios should always be written with whole numbers, in their simplest form, and without units — because a ratio compares parts of the same kind, the units cancel out.
比率应始终写成最简整数形式，且不带单位——因为比率比较的是同类部分，单位会被约掉。
3. Ratio as a Fraction | 用分数表示比率
A ratio a : b can be expressed as fractions of the total. The total number of parts is a + b. So the first part is a / (a + b) of the whole, and the second part is b / (a + b). Understanding this link is key for converting between ratios, fractions, and percentages.
比率 a : b 可以表示为占总体的分数。总份数为 a + b。因此第一部分占整体的 a / (a + b)，第二部分占 b / (a + b)。理解这一联系对于在比率、分数和百分比之间进行转换至关重要。
Example: A drink is made by mixing 1 part cordial to 4 parts water. The ratio of cordial to water is 1 : 4. The total parts = 1 + 4 = 5. So cordial is 1/5 of the drink, and water is 4/5.
示例：一种饮料由 1 份浓缩汁和 4 份水混合而成。浓缩汁与水的比率为 1 : 4。总份数 = 1 + 4 = 5。因此浓缩汁占饮料的 1/5，水占 4/5。
4. Sharing in a Given Ratio | 按给定比率分配
When sharing an amount in a ratio, first find the total number of parts by adding the ratio terms. Then divide the total amount by the total parts to get the value of one part. Multiply each ratio term by this unit value to find how much each person or category receives.
按比率分配一个量时，首先将比率各项相加得到总份数。然后用总量除以总份数，得到一份的值。再将比率每一项乘以这个单位值，得出每个人或类别应得多少。
Example: Share £64 between two friends in the ratio 3 : 5. Total parts = 3 + 5 = 8. One part = £64 ÷ 8 = £8. So the first friend gets 3 × £8 = £24, and the second gets 5 × £8 = £40.
示例：以 3 : 5 的比率与两位朋友分配 £64。总份数 = 3 + 5 = 8。一份 = £64 ÷ 8 = £8。因此第一位朋友得到 3 × £8 = £24，第二位得到 5 × £8 = £40。
Always check your answer by adding the two shares: £24 + £40 = £64, which matches the original total. This verification step is vital to catch arithmetic errors.
务必通过将各部分相加来检查答案：£24 + £40 = £64，与初始总量吻合。这一验证步骤对于发现计算错误至关重要。
5. Working with Three-Term Ratios | 处理三项比率
A ratio can have three or more parts, for instance when dividing an inheritance or mixing ingredients. The same principle applies: find the total parts by adding all terms, determine the value of one part, and multiply out.
比率可以有三项或更多项，例如在分配遗产或混合配料时。遵循相同原理：将所有项相加得出总份数，确定一份的值，然后相乘。
Example: A pet store has cats, dogs, and fish in the ratio 4 : 5 : 3. If there are 480 animals in total, how many of each? Total parts = 4 + 5 + 3 = 12. One part = 480 ÷ 12 = 40. Cats = 4 × 40 = 160, dogs = 5 × 40 = 200, fish = 3 × 40 = 120.
示例：一家宠物店中猫、狗和鱼的数量比为 4 : 5 : 3。如果总共有 480 只动物，各有多少？总份数 = 4 + 5 + 3 = 12。一份 = 480 ÷ 12 = 40。猫 = 4 × 40 = 160，狗 = 5 × 40 = 200，鱼 = 3 × 40 = 120。
Three-term ratios also simplify by dividing each term by their HCF. This makes problems more manageable and is often required in exam answers.
三项比率同样通过将每一项除以它们的最大公因数来化简。这使问题更易处理，且考试答案中常要求如此。
6. Map Scales and Ratio Notation | 地图比例尺与比率记法
Map scales are a special application of ratios. A scale like 1 : 25 000 means that 1 cm on the map represents 25 000 cm in real life. Since large numbers can be unwieldy, convert to more meaningful units — 25 000 cm = 250 m.
地图比例尺是比率的一种特殊应用。像 1 : 25 000 这样的比例尺表示地图上 1 厘米代表现实中的 25 000 厘米。由于大数字可能难以处理，可以转换成更有意义的单位——25 000 厘米 = 250 米。
To find actual distance from a map measurement, multiply the map length by the scale factor. For example, a 3.2 cm road on a 1 : 50 000 map represents 3.2 × 50 000 = 160 000 cm = 1 600 m = 1.6 km. Similarly, to find the map length, divide the real distance by the scale factor.
要根据地图测量求出实际距离，将地图长度乘以比例尺因子。例如，在 1 : 50 000 地图上一条 3.2 厘米的道路代表 3.2 × 50 000 = 160 000 厘米 = 1 600 米 = 1.6 千米。类似地，求地图长度时将实际距离除以比例尺因子。
Always ensure units are the same when using map scales. Converting everything to cm first, then to km or m, reduces mistakes.
使用地图比例尺时务必确保单位一致。先将所有量换算为厘米，再换算为千米或米，可减少错误。
7. Understanding Proportion | 理解比例
Proportion describes how one quantity changes in relation to another. If two quantities are in direct proportion, then when one is multiplied by a number, the other is multiplied by the same number. Their ratio stays constant.
比例描述了一个量相对于另一个量如何变化。如果两个量成正比例，那么当一个量乘以一个数时，另一个量也乘以相同的数。它们的比率保持不变。
For instance, if 4 apples cost £1.20, then cost and number of apples are in direct proportion. To find the cost of 10 apples, first find the cost per apple: £1.20 ÷ 4 = £0.30. Then multiply: 10 × £0.30 = £3.00. This is the unitary method — finding the value of one unit first.
例如，如果 4 个苹果售价 £1.20，那么总价和苹果数量成正比例。求 10 个苹果的价格，先找出每个苹果的价格：£1.20 ÷ 4 = £0.30。然后相乘：10 × £0.30 = £3.00。这是单份法——首先求出一个单位的值。
Direct proportion relationships can be written as y = k x, where k is the constant of proportionality. For the apples, k = 0.30, so y = 0.30x.
正比例关系可写为 y = k x，其中 k 是比例常数。对于苹果，k = 0.30，所以 y = 0.30x。
8. Inverse Proportion | 反比例
In inverse proportion, as one quantity increases, the other decreases so that their product remains constant. This often appears in contexts like the number of workers and time taken to complete a job, or speed and time for a fixed distance.
在反比例中，当一个量增加时，另一个量减少，使它们的乘积保持不变。这种关系常出现于工人数量与完成工作所需时间、或固定距离下的速度与时间等情境中。
For example, if 6 workers can build a wall in 4 days, then the total work is 6 × 4 = 24 worker-days. If we have 8 workers, the time needed is 24 ÷ 8 = 3 days. Note that more workers mean fewer days — that is the key property of inverse proportion.
例如，如果 6 名工人可在 4 天内砌好一堵墙，则总工作量为 6 × 4 = 24 个“工人-天”。如果我们有 8 名工人，所需时间为 24 ÷ 8 = 3 天。注意工人越多，天数越少——这是反比例的关键特性。
Problems often involve doubling/halving situations. Approach them by first calculating the total “work” or “distance” and then adapting to the new condition.
题目常涉及翻倍或减半的情形。处理时可先计算总“工作量”或“路程”，再根据新条件进行调整。
9. Comparing Ratios Using 1 : n and n : 1 | 将比率化为 1 : n 或 n : 1 进行比较
When comparing ratios, it is often useful to rewrite them in the form 1 : n or n : 1. This makes it easy to see which mixture or offer gives more of one component. To convert a : b to 1 : n, divide both sides by a. To get n : 1, divide both sides by b.
在比较比率时，通常将它们重写为 1 : n 或 n : 1 的形式会有帮助。这样可以轻松看出哪种混合物或优惠给出更多某种成分。要将 a : b 转换为 1 : n，两边都除以 a。要得到 n : 1，两边都除以 b。
Example: A pancake recipe uses 200 g flour to 4 eggs (ratio 200 : 4). Divide by 4 to get 50 : 1, meaning 50 grams of flour per egg. Another recipe uses 250 g flour to 5 eggs (250 : 5), which simplifies to 50 : 1 as well — so the flour-to-egg ratio is the same.
示例：一个煎饼食谱使用 200 克面粉和 4 个鸡蛋（比率 200 : 4）。除以 4 得到 50 : 1，表示每个鸡蛋对应 50 克面粉。另一个食谱用 250 克面粉和 5 个鸡蛋（250 : 5），同样化简为 50 : 1——因此面粉与鸡蛋的比例相同。
When comparing prices, writing 1 : n can reveal unit pricing. For example, 3 batteries for £1.20 and 5 batteries for £1.80: first ratio as 1 : 0.40 (cost per battery £0.40), second as 1 : 0.36 (£0.36 per battery), so the second deal is better.
在比较价格时，写成 1 : n 可揭示单价。例如，3 节电池 £1.20 和 5 节电池 £1.80：第一个比率化为 1 : 0.40（每节电池 £0.40），第二个为 1 : 0.36（每节 £0.36），因此第二个交易更划算。
10. Ratio and Proportion in Recipes | 食谱中的比率与比例
Recipes are perfect real-life examples of ratio and proportion. If a recipe is for 4 people and you need to serve 10, you must scale all ingredients by the same factor. Scaling factor = desired servings ÷ original servings = 10 ÷ 4 = 2.5.
食谱是比率与比例在生活中绝佳的例子。如果一份食谱供 4 人份量，而你需要供 10 人食用，你必须将所有配料乘以相同的倍率。倍率 = 所需份数 ÷ 原份数 = 10 ÷ 4 = 2.5。
For instance, a cake recipe requiring 200 g flour for 8 slices. To make 20 slices, multiply 200 g by (20/8) = 2.5, giving 500 g flour. Always multiply every ingredient by the same factor to maintain taste and texture.
例如，一个蛋糕食谱需要 200 克面粉制作 8 片。要制作 20 片，将 200 克乘以 (20/8) = 2.5，得到 500 克面粉。务必每个配料都乘以相同的因子，以维持味道和质地。
This is direct proportion in action. The ratio of ingredients remains constant even as the total amount changes — it is the essence of scaling.
这正是正比例的应用。即使总量改变，配料之间的比率依然不变——这就是缩放的实质。
11. Common Mistakes to Avoid | 常见错误与避免方法
One frequent error is mixing up the order in a ratio. “The ratio of boys to girls” is not the same as “girls to boys.” Always write the terms in the exact order given in the problem.
一个常见错误是混淆比率中的顺序。“男生与女生的比率”与“女生与男生”不同。务必按题目给定的准确顺序写出各项。
Another mistake is forgetting to simplify ratios fully. A ratio like 6 : 10 must be reduced to 3 : 5. Leaving it unsimplified can lead to wrong answers in sharing problems because the total parts would be incorrect.
另一个错误是忘记将比率彻底化简。像 6 : 10 的比率必须化简为 3 : 5。不化简可能导致分配问题中总份数出错，从而得到错误答案。
When scaling recipes or maps, confusing units is a pitfall. Always convert measurements to the same unit before doing calculations. And finally, check your answer by seeing if the parts add up to the original total, or if the proportion holds.
在缩放食谱或地图时，混淆单位是一项陷阱。计算前务必将度量换算为相同单位。最后，通过检查各部分之和是否等于原始总量，或比例关系是否成立，来验证你的答案。
12. Key Skills for the Checkpoint Exam | 关键考试技能
For success in the KS3 Cambridge maths exam on ratio and proportion, you must be able to simplify a ratio given in any form (including decimals and fractions), divide a quantity in a given ratio, solve direct proportion problems using the unitary method, interpret map scales, and apply ratio reasoning to word problems.
要在 KS3 Cambridge 数学考试中比率与比例部分取得成功，你必须能够：化简任意形式的比率（包括小数和分数），按给定比率分配一个量，使用单份法解决正比例问题，解读地图比例尺，并将比率推理应用于文字题。
Practice mixed problems: combine ratio with fractions, percentages, and algebra. For example, a question might say “the ratio of red to blue marbles is 3 : 2, and 60% are red — find something.” The key is staying calm, identifying what is given, and working step by step.
练习混合题型：将比率与分数、百分率和代数结合起来。例如，一道题可能会说“红色弹珠与蓝色弹珠的比率为 3 : 2，且 60% 是红色——求某某。”关键在于保持冷静，识别已知条件，并一步步推进。
Displaying answers clearly, with working shown, earns method marks even if the final number is wrong. Remember the golden rule: a ratio has no units and should be in simplest integers.
清晰地展示答案，并写出解题步骤，即使最终数字错误，也能得到过程分。记住黄金法则：比率不带单位，且应为最简整数。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导