📚 Mastering Ratios and Proportions | 掌握比率与比例
Ratios and proportions form the backbone of many IGCSE Edexcel number and algebra problems. From dividing sweets among friends to working with map scales and mixing concrete, a solid grasp of ratios is essential. In this article, we break down every sub-topic you need for the exam, from simplifying ratios to solving direct and inverse proportion problems, alongside common pitfalls and tips.
比率与比例是许多IGCSE Edexcel 数学中数字与代数问题的基础。从给朋友分糖果到使用地图比例尺、混合混凝土，扎实掌握比率知识至关重要。本文将分解考试所需的每一个子主题，从化简比率到解决正比例与反比例问题，同时指出常见错误和实用技巧。
1. What Is a Ratio? | 什么是比率？
A ratio compares two or more quantities of the same kind. It tells us how much of one thing there is compared to another. The order is crucial: if recipe A uses 2 cups of flour to 1 cup of sugar, we write the ratio of flour to sugar as 2 : 1. The ratio has no units, because it expresses a relative size.
比率用于比较两个或多个同类量。它表示一种东西相对于另一种东西的数量。顺序非常关键：如果食谱 A 用了 2 杯面粉和 1 杯糖，那面粉与糖的比写作 2 : 1。比率没有单位，因为它表示的是相对大小。
an be written in three forms: using a colon (a : b), as a fraction (a/b), or with the word 'to' ee . In the Edexcel IGCSE exam, you will mainly see the colon notation. Remember that the ratio a : b is different from b : a unless a = b.
比率有三种书写形式：用冒号（a : b）、用分数（a/b）或用单词“to”。在 Edexcel IGCSE 考试中，最常见的是冒号表示法。请记住，比率 a : b 与 b : a 是不同的，除非 a = b。
- Example: In a class of 30 students, the ratio of boys to girls is 17 : 13. This means for every 17 boys there are 13 girls.
- 示例：在一个 30 人的班级里，男生与女生的比是 17 : 13。这意味着每 17 个男生对应 13 个女生。
2. Simplifying Ratios | 化简比率
To simplify a ratio, divide all parts by their greatest common divisor (GCD). The process is just like simplifying a fraction, but you apply it to each term in the ratio. Simplifying makes the ratio easier to work with and compare.
化简比率的方法是将所有部分除以它们的最大公约数（GCD）。这个过程和约分相似，但要对比率中的每一项分别进行。化简后的比率更容易使用和比较。
If the ratio contains decimals or fractions, first multiply each term by the same power of 10 (for decimals) or by the lowest common multiple of the denominators (for fractions) to obtain integers, then simplify.
如果比率中含有小数或分数，首先把所有项都乘以相同的 10 的幂（针对小数），或者乘以所有分母的最小公倍数（针对分数），将其化为整数比，然后再化简。
| Original Ratio | Step | Simplified Ratio |
|---|---|---|
| 36 : 24 | Divide by 12 | 3 : 2 |
| 1.5 : 3.5 | Multiply by 2 → 3 : 7 | 3 : 7 (already simplest) |
| ½ : ⅓ | Multiply by 6 → 3 : 2 | 3 : 2 |
In the exam, always reduce the ratio to its lowest terms unless stated otherwise. A common pitfall is leaving a common factor. Even if the numbers are large, check if they share any prime factor.
考试时，除非题目另有说明，否则始终要把比率化为最简形式。一个常见的陷阱是漏掉了公约数。即使数字很大，也要检查它们是否有共同的质因数。
3. Dividing a Quantity in a Given Ratio | 按给定比例分配数量
To divide a quantity in the ratio a : b, first find the total number of parts: a + b. Then each part is worth the total quantity divided by the total parts. The share for the first part is a times the value of one part, and the share for the second part is b times that value.
按 a : b 的比例分配一个数量，首先要计算总份数：a + b。然后，每份的价值 = 总数 ÷ 总份数。第一部分的份额就是 a × 每份价值，第二部分的份额是 b × 每份价值。
This method extends naturally to ratios with three or more terms, such as dividing £600 among A, B, and C in the ratio 2 : 3 : 5. The total parts = 2 + 3 + 5 = 10. One part = £360. Then A gets 2 × 360 = £720, B gets £1080, and C gets £1800.
这种方法可以自然地推广到三项或三项以上的比率，例如按 2 : 3 : 5 的比例把 3600 英镑分给 A、B 和 C。总份数 = 2 + 3 + 5 = 10。每份 = 360 英镑。那么 A 得到 2 × 360 = 720 英镑，B 得到 1080 英镑，C 得到 1800 英镑。
A typical Edexcel IGCSE question: 'Anna and Ben share £84 in the ratio 5 : 7. How much does Anna receive?' Total parts = 12, one part = £7, Anna gets 5 × 7 = £35. Always check the sum of the individual shares equals the given total.
典型的 Edexcel IGCSE 考题：“Anna 和 Ben 按 5 : 7 的比例分配 84 英镑。Anna 得到多少钱？”总份数 = 12，每份 = 7 英镑，Anna 得到 5 × 7 = 35 英镑。务必验算各部分之和等于给定的总数。
4. Direct Proportion | 正比例
nstant of proportionality. The graph of y against x is a straight line through the origin.
当两个量以相同的速率同时增加或减少时，它们就成正比例关系。数学上，如果 y = kx（其中 k 是比例常数），那么 y 与 x 成正比例。y 对 x 的图像是一条通过原点的直线。
alues to find k, then use the formula to find unknown values. For instance, if 5 pens cost £ the cost (C) is directly proportional to the number of pens (n): C = 0.5n. So 12 pens cost £6.00.
解题时，先利用已知的一对数值求出 k，然后再用公式求未知量。例如，5 支钢笔 2.50 英镑，则价格（C）与钢笔数量（n）成正比例：C = 0.5n。因此 12 支钢笔的价格是 6.00 英镑。
In IGCSE, direct proportion often appears in currency conversion, recipe scaling, and speed–distance problems (with constant speed). Always look for the phrase 'is proportional to' or 'varies directly as'.
在 IGCSE 中，正比例常出现在货币兑换、食谱缩放以及速度–距离问题中（当速度恒定时）。要留意“与…成比例”或“随…直接变化”这些表述。
5. Inverse Proportion | 反比例
such a way that their product remains constant. The general formula is y = k/x, or xy = k.
反比例意味着一个量增加时，另一个量会减少，且两者的乘积保持不变。一般公式为 y = k/x，即 xy = k。
kers (w). Find k using 3 × 12 = 36. So t = 36/w. Then 4 workers would take 36/4 = 9 hours.
例如，3 个工人筑一堵墙需要 12 小时，那么所需时间（t）与工人数量（w）成反比例。利用 3 × 12 = 36 求出 k。所以 t = 36/w。那么 4 个工人需要 36/4 = 9 小时。
The graph of inverse proportion is a hyperbola. Candidates often confuse direct and inverse proportion; remember that in inverse proportion, doubling one variable halves the other, whereas in direct proportion doubling one doubles the other.
反比例的图像是双曲线。考生经常混淆正比例和反比例；记住，在反比例中，一个变量翻倍会使另一个减半，而在正比例中，一个变量翻倍另一个也翻倍。
6. Map Scales and Ratios | 地图比例尺与比率
A map scale is a ratio that compares a distance on the map to the actual distance on the ground. If a scale is 1 : 50 000, then 1 cm on the map represents 50 000 cm (or 0.5 km) in reality. In IGCSE you will often convert between map and real distances using this ratio.
地图比例尺是将图上的距离与实际地面距离进行比较的比率。如果比例尺为 1 : 50 000，图上 1 cm 代表实际 50 000 cm（即 0.5 km）。在 IGCSE 中，你常常需要用这个比率在图上距离和实际距离之间进行换算。
To find the actual distance, multiply the map distance by the scale factor. To find the map distance, divide the real distance by the scale factor. Always ensure units are consistent; convert to the same unit before solving.
求实际距离时，将图上距离乘以比例尺因子。求图上距离时，将实际距离除以比例尺因子。务必保证单位一致，解题前先转化为相同的单位。
Map scale problems often lead to area scale questions: if the linear scale is 1 : n, the area scale is 1 : n². This trips up many students, so be careful.
比例尺问题常会引出面积比例尺的题目：如果线段比例尺是 1 : n，那么面积比例尺就是 1 : n²。这一点常让学生出错，一定要小心。
7. The Unitary Method | 归一法
The unitary method involves finding the value of one unit first, and then multiplying up to find the required value. It is extremely useful for both direct and inverse proportion and for ratio word problems.
归一法就是先求出一个单位对应的数值，然后再乘以相应倍数求出所需值。这种方法对于正比例和反比例问题以及比率的文字题都极其有用。
For direct proportion example: 8 notebooks cost £6.40. Find the cost of 20 notebooks. First, find cost of 1 notebook: £6.40 ÷ 8 = £0.80. Then cost of 20 = 20 × 0.80 = £16.00.
正比例的例子：8 个笔记本售价 6.40 英镑，求 20 个笔记本的价格。先算 1 个笔记本的价格：6.40 ÷ 8 = 0.80 英镑，然后 20 个的价格 = 20 × 0.80 = 16.00 英镑。
es = 40 ÷ 8 = 5 hours. In essence, after finding the total work, divide by the new number.
对于反比例，只要保持乘积不变，归一法同样适用。5 台机器生产一批产品需要 8 小时，求 8 台机器需要多长时间。乘积恒定 = 5 × 8 = 40 机器小时。8 台机器所需时间 = 40 ÷ 8 = 5 小时。本质上，求出总工作量后再除以新数量即可。
8. Word Problems with Ratios | 比率文字题
Ratio word problems often describe real-life scenarios like mixing juice, sharing profits, or finding missing values in a given ratio. The key is to identify the parts and the total, and then set up an equation.
比率文字题通常描述现实场景，如混合果汁、分配利润或在给定的比率中寻找缺失值。关键是要找出份数和总数，然后列出方程。
For example: ' bag is 3 : 5. There are 15 more blue marbles than red. How many marbles are there in total?' re, resulting in x = 5 and total = 8x = 40. Always ensure solutions make sense in context.
例如：“一个袋子里红色弹珠与蓝色弹珠的比是 3 : 5，蓝色弹珠比红色多 15 个。袋子里总共有多少个弹珠？”设红色弹珠 3x，蓝色弹珠 5x。那么 5x − 3x = 15 → 2x = 15 → x = 7.5。但弹珠数必须是整数，因此要重新审题。或许是多 10 个，这样就能得到 x = 5，总数 = 8x = 40。始终要确保答案在给定的语境下合理。
Using bar models or algebra can make these problems much clearer. In the Edexcel paper, you must show your working clearly, especially when there are multiple steps.
使用条形模型或代数方法可以使这类问题更清晰。在 Edexcel 试卷中，你必须清楚地展示解题步骤，特别是当问题涉及多个步骤时。
9. Common Mistakes to Avoid | 需要避免的常见错误
Even strong candidates can lose marks on ratio questions by making avoidable slips. Below are the most frequent ones:
- Forgetting to simplify the ratio completely, e.g. leaving 6 : 8 instead of 3 : 4.
- Mixing up the order of terms when the question says 'ratio of A to B' but you write B : A.
- Using the total parts incorrectly when dividing a quantity: adding only two parts when there are three.
- Confusing direct and inverse proportion, especially under time pressure.
- Not converting units before using a map scale, e.g. using km directly with cm.
- Rounding off intermediate values too early, leading to an inaccurate final answer.
即使能力较强的考生也会在比率问题中因为可避免的失误而丢分。以下是最常见的错误：
- 忘记将比率彻底化简，例如将 6 : 8 留在那里而不是化成 3 : 4。
- 当题目说“A 与 B 的比率”时，却把各项顺序搞反，写成了 B : A。
- 在分配数量时错误使用总份数，比如有三项却只加了两项。
- 尤其在时间紧张的时候，混淆正比例和反比例。
- 使用地图比例尺之前没有统一单位，例如直接把千米和厘米混用。
- 过早对中间值进行四舍五入，导致最终答案不准确。
10. Exam Tips and Tricks | 考试技巧与窍门
In the Edexcel IGCSE Mathematics exam, ratio and proportion questions often carry a lot of marks and appear both on the non-calculator and calculator papers. Use these strategies to maximize your score:
在 Edexcel IGCSE 数学考试中，比率与比例题往往占分较多，且同时出现在非计算器和计算器卷中。使用以下策略来最大化你的分数：
- Read carefully: Underline the order of the ratio and the total quantity, if given.
- Use the check method: After dividing, add up all shares to confirm they equal the original total.
- Write k for proportion: Immediately write y = kx or y = k/x, find k, then use it.
- Draw a bar model: For complex sharing problems, a quick sketch helps visualise the parts.
- Look out for 'in the same ratio': This means apply the same multiplier to each term.
- Convert units first: If the scale is 1 : 250 000, convert real distance to cm before doing division.
- Double-check simplifications: Ensure you have divided by the GCD and not just a common factor.
- 仔细读题： 划出比率的顺序和总数（如果已给出）。
- 使用验算方法： 分配完成后，把各份加起来，确认它们等于原始总数。
- 写出比例常数 k： 立即写出 y = kx 或 y = k/x，求出 k，然后再使用它。
- 画条形模型： 对于复杂的分配问题，快速画个草图有助于把份数形象化。
- 注意“以相同的比率”： 这意味着将每一项都乘以相同的倍数。
- 先换算单位： 如果比例尺是 1 : 250 000，在做除法之前先把实际距离换算成厘米。
- 再检查一遍化简： 确保你已经除以了最大公约数，而不只是某个公约数。
Above all, practice past paper questions regularly. Ratio problems can become very predictable once you have seen a few variations.
最重要的是，要定期练习往年真题。一旦你见过了几种不同的题型，比率问题就会变得非常容易预测。
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