📚 Mastering Ratios and Proportions | 掌握比例与比率
Ratios and proportions are among the most frequently tested topics in the Edexcel IGCSE Mathematics examination. They appear not only in the Number and Algebra sections but also underpin many problems in Geometry, Statistics, and Practical Mathematics. In this article, we will break down the key skills you need, from simplifying ratios to solving complex word problems involving direct and inverse proportion.
比率与比例是 Edexcel IGCSE 数学考试中考查频率最高的知识点之一。它们不仅出现在数与代数部分，更贯穿几何、统计以及实际应用问题。本文将从化简比率到解决涉及正比例与反比例的复杂文字题，逐一拆解你所需的核心技能。
1. Understanding Ratios | 理解比率
A ratio compares two or more quantities of the same kind. For example, if a fruit basket contains 4 apples and 3 oranges, we say the ratio of apples to oranges is 4:3. The order of the numbers matters: 4:3 is not the same as 3:4. Ratios have no units because the units cancel out during comparison.
比率是比较两个或多个同类型数量的工具。例如，如果一个水果篮中有 4 个苹果和 3 个橙子，我们说苹果与橙子的比率为 4:3。数字的顺序非常重要：4:3 与 3:4 完全不同。比率没有单位，因为在比较过程中单位相互抵消了。
Ratios can be written in three equivalent ways: using a colon (4:3), as a fraction (4⁄3), or in words ("four to three"). In the IGCSE exam, the colon form is most common, but you must be comfortable converting between all three.
比率可以用三种等价方式书写：使用冒号（4:3）、分数形式（4⁄3）或用文字表述（"四比三"）。在 IGCSE 考试中，冒号形式最为常见，但你必须熟练地在三种形式之间进行转换。
2. Simplifying Ratios | 化简比率
To simplify a ratio, divide every part by the same number until no common factor remains. For instance, the ratio 12:8 simplifies to 3:2 by dividing both sides by 4. Ratios involving decimals or fractions require an extra step: first multiply all parts by a power of 10 or the LCM of the denominators to make them integers, then simplify.
化简比率时，需要将每一项同时除以同一个数，直到没有公因数为止。例如，12:8 可以通过同时除以 4 化简为 3:2。当比率涉及小数或分数时，需要多一步处理：先将各项乘以 10 的幂或分母的最小公倍数，使其变为整数，然后再化简。
Example: Simplify the ratio 0.4 : 0.6. Multiply both parts by 10 to obtain 4 : 6, then divide by 2 to get 2 : 3. Example: Simplify 1⁄2 : 3⁄4. Multiply both parts by 4 (the LCM of 2 and 4) to get 2 : 3.
例如：化简比率 0.4 : 0.6。先将两项同时乘以 10，得到 4 : 6，再同时除以 2，得到 2 : 3。又如：化简 1⁄2 : 3⁄4。先将两项同时乘以 4（2 和 4 的最小公倍数），得到 2 : 3。
Simplification rule: divide every term by the HCF | 化简法则：每一项都除以最大公因数
3. Dividing a Quantity in a Given Ratio | 按给定比率分配数量
Suppose we need to share £ e: add the parts (3 + 5 = 8), divide the total by this sum to find the value of one part (£80 ÷ 8 = £10), then multiply each part of the ratio by this value.
假设我们需要将 80 英镑按 3:5 的比率分给 Alice 和 Bob。方法很简单：先将各部分相加（3 + 5 = 8），用总量除以这个和得到每一份的价值（80 ÷ 8 = 10 英镑），然后再用这个值乘以比率的每一项。
Alice receives 3 × £10 = £30, and Bob receives 5 × £10 = £50. Always check: the two amounts should add back to the original total (£30 + £50 = £80). This checking step is quick and can save you from careless errors.
Alice 得到 3 × 10 = 30 英镑，Bob 得到 5 × 10 = 50 英镑。务必验证：两个数额相加应等于原来的总数（30 + 50 = 80 英镑）。这一步验证非常快速，能帮你避免粗心错误。
herefore the three lengths are 30 cm, 120 cm, and 90 cm. Notice that the middle piece is four times the smallest, and the largest two pieces combined equal 7 parts of the total.
现在考虑一个三项比率，例如将 240 cm 的绳子按 1:4:3 分配。总份数为 1 + 4 + 3 = 8。每一份等于 240 ÷ 8 = 30 cm。因此三段长度分别为 30 cm、120 cm 和 90 cm。注意，中间一段是最小段的四倍，而最大两段合计占总数的 7 份。
4. Ratios as Fractions and Percentages | 比率与分数及百分数的互化
or the ratio 1:4:3, the first quantity is 1⁄8 of the total, the second is 4⁄8 = 1⁄2 (or 50%), and the third is 3⁄8 (or 37.5%). Converting ratios to percentages is a very common exam requirement.
在比率 a:b 中，第一项数量占总量的分数为 a⁄(a+b)。对于比率 1:4:3，第一项占总量的 1⁄8，第二项占 4⁄8 = 1⁄2（即 50%），第三项占 3⁄8（即 37.5%）。将比率转换为百分数是考试中的常见要求。
ol has students in the ratio of boys to girls of 7:9. What percentage of students are boys? Total parts = 16. Boys fraction = 7⁄16 = 0.4375 = 43.75% the second, e.g., boys:girls = 7:9 means the number of boys is 7⁄9 of the number of girls.
例如：某学校男生与女生的人数比率为 7:9。男生占全校学生的百分比是多少？总份数 = 16。男生占比 = 7⁄16 = 0.4375 = 43.75%。若要将比率表示为某一量相对于另一量的分数，只需将第一项除以第二项，例如，男生:女生 = 7:9 意味着男生人数是女生人数的 7⁄9。
5. Equivalent Ratios and the Unitary Method | 等价比率与单位法
Two ratios are equivalent if one can be obtained from the other by multiplying or dividing both parts by the same non-zero number. The ratios 2:3, 4:6, and 10:15 are all equivalent. The unitary method involves finding the value of one unit first, then scaling up or down. This technique is invaluable for solving proportion word problems.
如果一个比率可以通过将两项同时乘以或除以同一个非零数得到另一个比率，则这两个比率是等价的。比率 2:3、4:6 和 10:15 都是等价的。单位法是指先求出"一份"所对应的值，然后再按比例放大或缩小。这一技巧在解决比例应用题时极为重要。
ple: A recipe for 6 people requires 240 g of flour. How much flour is needed for 15 people? One person needs 240 ÷ 6 = 40 g. For 15 people: 40 × 15 = 600 g. This "find one part first" strategy works whenever the relationship is directly proportional.
例如：一份供 6 人食用的食谱需要 240 克面粉。15 人需要多少面粉？每人需要 240 ÷ 6 = 40 克。15 人需要：40 × 15 = 600 克。只要两个量成正比关系，"先求一份"的策略就始终适用。
Unitary method: value of one unit → multiply by required number | 单位法：求一份的值 → 乘以所需数量
6. Direct Proportion | 正比例
t of proportionality. On a graph, direct proportion appears as a straight line passing through the origin.
如果两个量按相同的比率同时增加或减少，则称它们成正比例。一个量加倍，另一个量也随之加倍。一般关系式为 y = kx，其中 k 是比例常数。在图像上，正比例表现为一条通过原点（0,0）的直线。
2 gives y = 48. Always write the constant with its correct units; in this case k has units "y-units per x-unit", which here simplifies to 4.
例如：y 与 x 成正比例。当 x = 5 时，y = 20。求当 x = 12 时 y 的值。先求 k：k = y⁄x = 20⁄5 = 4。所以 y = 4x。代入 x = 12，得到 y = 48。写比例常数时务必带上正确的单位；本例中 k 的单位是"y 单位每 x 单位"，即化简为 4。
Exam tip: IGCSE questions often describe direct proportion in words, such as "the cost of apples is directly proportional to their weight." If 3 kg of apples cost £ find the cost of 8 kg. Here k = 7.50 ÷ 3 = 2.50 (pounds per kg), so 8 kg costs 2.50 × 8 = £20.00.
考试提示：IGCSE 题目常用文字描述正比例，例如"苹果的总价与其重量成正比"。若 3 kg 苹果售价 7.50 英镑，求 8 kg 的价格。这里 k = 7.50 ÷ 3 = 2.50（英镑每公斤），所以 8 kg 的价格为 2.50 × 8 = 20.00 英镑。
7. Inverse Proportion | 反比例
taken to complete a job is inversely proportional to the number of workers: more workers mean less time.
如果两个量中一个增大而另一个减小，且它们的乘积保持不变，则称它们成反比例。一般关系式为 y = k⁄x。例如，完成一项工作所需的时间与工人人数成反比例：工人越多，所需时间越少。
Example: It takes 6 painters 8 days to paint a house. How long would it take 4 painters? ime and number of painters are inversely proportional, the product is constant: 6 × 8 = 48 "painter-days" ters: time = 48 ÷ 4 = 12 days. Note that this assumes each painter works at the same rate.
例如：6 名油漆工需要 8 天粉刷一栋房子。4 名油漆工需要多少天？因为时间与油漆工人数成反比例，乘积保持不变：6 × 8 = 48（人·天）。4 名油漆工所需时间 = 48 ÷ 4 = 12 天。注意，这个结论假设每位油漆工的工作效率相同。
Do not confuse inverse and direct proportion. A quick check: ask yourself "if one quantity doubles, what happens to the other?" If it also doubles, it is direct; if it halves, it is inverse. This simple mental test can prevent many exam errors.
切勿混淆反比例与正比例。快速检验方法：问自己"如果一个量加倍，另一个量会怎样变化？"如果另一个量也随之加倍，则为正比例；如果减半，则为反比例。这个简单的思考测试可以避免许多考试错误。
8. Ratio and Maps: Scale Drawings | 比率与地图：比例尺作图
A map scale expresses the ratio between a distance on a drawing and the corresponding real distance. A scale of 1:50000 means that 1 cm on the map represents 50000 cm (500 m) on the ground. To convert, multiply the map distance by the scale factor, or divide the real distance by it.
比例尺表示图纸上的距离与实际距离之间的比率。比例尺 1:50000 意味着地图上的 1 cm 代表实际地面上的 50000 cm（即 500 m）。换算时，用图上距离乘以比例因子，或用实际距离除以比例因子。
features is 1.5 km and the scale is 1:50000, the map distance is 150000 cm ÷ 50000 = 3 cm.
例如：在一张比例尺为 1:20000 的地图上，两个城镇相距 3.5 cm。实际距离为 3.5 × 20000 = 70000 cm = 700 m。反过来，如果两个地物之间的实际距离为 1.5 km，而比例尺为 1:50000，则图上距离为 150000 cm ÷ 50000 = 3 cm。
ite IGCSE extension question. For a scale of 1:200, the area scale factor is 200² = 40000.
面积和体积的比例因子与长度比例因子之间存在特殊关系。若长度比例因子为 k，则面积比例因子为 k²，体积比例因子为 k³。这是 IGCSE 常见的拓展题型。对于 1:200 的比例尺，面积比例因子为 200² = 40000。
9. Proportional Reasoning with Money and Best Buys | 货币比例推理与最优购买
Best buy problems require comparing prices of different package sizes to determine which offers greater value. The standard approach is to calculate the unit price (price per gram, per litre, or per item) for each option and compare the figures. The lower unit price represents the better value.
"最优购买"问题要求比较不同包装规格的价格，以判断哪个更划算。标准做法是计算每个选项的单位价格（每克、每升或每件的价格），然后进行比较。单位价格更低者性价比更高。
Example: A 500 g jar of coffee costs £6.50, and a 250 g jar costs £3.30. Which is the better buy? rice of the small jar: 330 p ÷ 250 g = 1.32 p per gram. The large jar is the better value.
例如：一罐 500 g 的咖啡售价 6.50 英镑，一罐 250 g 的咖啡售价 3.30 英镑。哪个更划算？大罐的单位价格：650 便士 ÷ 500 g = 每克 1.3 便士。小罐的单位价格：330 便士 ÷ 250 g = 每克 1.32 便士。大罐更划算。
In currency conversion problems, the exchange rate is a direct proportion relationship. If £1 = €1.15, then to convert £200 to euros, multiply: 200 × 1.15 = € by 1.15. Remember that banks apply a commission, which slightly alters the effective rate.
在货币兑换问题中，汇率就是一种正比例关系。如果 £1 = €1.15，那么将 200 英镑兑换成欧元时，乘以汇率：200 × 1.15 = €230。将欧元兑换回英镑时，除以 1.15。记住，银行会收取手续费，这会使实际汇率略有变化。
10. Changing Ratios: Adding or Removing Quantities | 比率变化：增加或减少数量
Some problems ask how a ratio changes when a quantity is added or removed. These are two-step problems. First, use the original ratio and the known total to find the quantities involved. Second, apply the change and write the new ratio, simplifying it if required.
有些题目会询问当增加或减少某个数量后比率如何变化。这类问题分两步解决。首先，利用原比率和已知总量求出各数量。其次，实施变化后写出新比率，并按需化简。
he ratio 5:2. There is 21 kg in total. How much cement must be added to make the ratio 2:1? w part = 15 ÷ 2 = 7.5 kg, meaning cement should be 7.5 kg. Add 7.5 − 6 = 1.5 kg of cement.
例如：某混合物中沙子和水泥的比率为 5:2，总重 21 kg。需要加入多少水泥才能使比率变为 2:1？原有量：总份数 = 7，每份 = 21 ÷ 7 = 3 kg。沙子 = 15 kg，水泥 = 6 kg。我们希望沙子:水泥 = 2:1。沙子仍为 15 kg，所以新的一份 = 15 ÷ 2 = 7.5 kg，即水泥应为 7.5 kg。需要加入 7.5 − 6 = 1.5 kg 水泥。
Always keep the unchanged quantity as your fixed reference point | 始终以不变的数量作为固定参考点
11. Exam-Style Word Problems | 考试风格应用题
Edexcel IGCSE questions frequently combine ratios with algebraic equations. For example: "The angles of a quadrilateral are in the ratio 2:3:4:6. Find the largest angle." x + 3x + 4x + 6x = 360°. Therefore 15x = 360°, so x = 24°. The largest angle is 6x = 144°.
Edexcel IGCSE 考试经常将比率与代数方程结合。例如："一个四边形的四个角之比为 2:3:4:6。求最大的角。"因为四边形的内角和为 360°，可以列方程 2x + 3x + 4x + 6x = 360°。因此 15x = 360°，解得 x = 24°。最大的角为 6x = 144°。
Another classic type involves ratios expressed in terms of variables: "If a:b = 3:4 and b:c = 6:5, find a:b:c." 3 to get a:b = 9:12, and multiply the second by 2 to get b:c = 12:10. Now a:b:c = 9:12:10.
另一类经典题型涉及用变量表示的比率："若 a:b = 3:4 且 b:c = 6:5，求 a:b:c。"为合并这些比率，需使 b 的值一致。将第一个比率乘以 3，得到 a:b = 9:12；将第二个比率乘以 2，得到 b:c = 12:10。因此 a:b:c = 9:12:10。
Always read ratio problems carefully to identify the known information and the target. Underline whether you are being asked for a quantity, a fraction, a percentage, or a simplified ratio. Organise your working with clear steps; even if the final answer is wrong, method marks are available.
在做比率应用题时务必仔细阅读，判断已知信息与所求目标。在试卷上标注出题目要求的是数量、分数、百分数还是化简后的比率。解题过程要条理清晰、步骤分明；即使最终答案有误，方法分仍然可以获得。
12. Common Mistakes and Revision Checklist | 常见错误与复习清单
The most common mistakes in ratio questions include: writing the ratio in the wrong order; forgetting to simplify fully; using the wrong operation (multiplying instead of dividing) in the unitary method; and confusing direct with inverse proportion. Another frequent slip is when converting between units: ensure you use the correct multiplier for cm to m (divide by 100) and for m to km (divide by 1000).
比率题最常见的错误包括：比率顺序写反；忘记将比率完全化简；在单位法中使用错误的运算（应为除却用了乘）；混淆正比例与反比例。另一个常见疏忽是单位换算：将 cm 换算为 m 应除以 100，将 m 换算为 km 应除以 1000，务必正确使用换算倍数。
Use this checklist when revising: (1) Can I simplify any ratio, including those with fractions or decimals? (2) Can I divide a quantity in a two-part or three-part ratio? (3) Can I solve direct and inverse proportion word problems with the constant of proportionality? (4) Can I apply scale factors to lengths, areas, and volumes? (5) Can I combine multiple ratios into one?
复习时请使用以下清单：（1）我能否化简任何比率，包括含分数或小数的比率？（2）我能否将数量按两项或三项比率分配？（3）我能否利用比例常数解决正比例与反比例应用题？（4）我能否将比例因子应用于长度、面积和体积？（5）我能否将多个比率合并为一个连比？
Practice consistently with past papers and timed conditions. Ratio questions in the IGCSE exam typically range from 2 to 5 marks, and they reward clear, logical working. Master the fundamentals above, and you will find that even the most intimidating proportion problems reduce to simple, manageable steps.
坚持用历年真题并按考试时间进行限时训练。IGCSE 考试中比率题的分值通常在 2 到 5 分之间，清晰的逻辑过程能帮助你拿分。掌握上述基础知识后，即使是最复杂的比例问题也能化简为简单、可控的步骤。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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