📚 Understanding Ratios and Proportions | 理解比与比例
Ratios and proportions are fundamental tools in mathematics that help us compare quantities and solve real-world problems. They appear in almost every area of IGCSE Mathematics, from geometry and maps to finance and science.
比与比例是数学中帮助我们比较数量、解决实际问题的基础工具。它们几乎出现在 IGCSE 数学的每一个领域，从几何、地图到金融和科学。
1. What is a Ratio? | 什么是比
A ratio is a way of comparing two or more quantities of the same kind. It shows how many times one value contains another. For example, if a fruit basket contains 3 apples and 2 oranges, the ratio of apples to oranges is 3 : 2.
比是比较两个或多个同类型数量的一种方式。它表示一个数值包含另一个数值多少次。例如，如果一个水果篮里有 3 个苹果和 2 个橙子，那么苹果与橙子的比是 3 : 2。
The order of the numbers in a ratio is extremely important. "3 : 2" is not the same as "2 : 3". The ratio 3 : 2 means there are 3 parts of the first quantity for every 2 parts of the second quantity.
比中数字的顺序极其重要。"3 : 2" 与 "2 : 3" 不同。比 3 : 2 表示第一个数量每有 3 份，第二个数量就有 2 份。
- Ratios can be written using a colon: 3 : 2
- Ratios can be written as fractions: 3/2
- Ratios can be written in words: "3 to 2"
- 比可以用冒号表示：3 : 2
- 比可以用分数表示：3/2
- 比可以用文字表示："3 比 2"
In IGCSE, you should always give ratios in their simplest form, with whole numbers, unless the question specifically asks for a different form.
在 IGCSE 中，除非题目特别要求其他形式，你应当始终给出最简整数比。
2. Simplifying Ratios | 化简比
To simplify a ratio, divide every part of the ratio by the same number. Continue dividing until no common factor remains between the parts.
要化简一个比，将比中的每一项都除以同一个数。继续除以直到各项之间没有公因数为止。
he ratio 3 : 4 is in its simplest form because 3 and 4 have no common factor other than 1.
例如，化简比 12 : 16。两个数都能被 4 整除，所以 12 : 16 = 3 : 4。比 3 : 4 是最简形式，因为 3 和 4 除 1 外没有公因数。
| Original ratio | Divide by | Simplified ratio |
| 24 : 36 | 12 | 2 : 3 |
| 5 : 15 | 5 | 1 : 3 |
| 100 : 25 | 25 | 4 : 1 |
When the ratio contains fractions or decimals, multiply every term by a suitable power of 10 (or by the denominator) first, then simplify.
当比中含有分数或小数时，先将每一项乘以合适的 10 的幂（或乘以分母），然后再化简。
Example: 0.4 : 1.2 = 4 : 12 = 1 : 3
instance, to compare 40 cm and 2 m, convert 2 m to 200 cm first, giving 40 : 200 = 1 : 5.
写比之前务必检查单位是否相同。例如，比较 40 cm 和 2 m 时，先把 2 m 换算成 200 cm，得到 40 : 200 = 1 : 5。
3. Dividing Quantities in a Given Ratio | 按给定比分配数量
Dividing a quantity in a given ratio means splitting it into parts according to the ratio. The key steps are: find the total number of parts, work out the value of one part, then multiply by each part of the ratio.
按给定比分配数量是指按照比将总量分成若干部分。关键步骤是：求出总份数，算出每份的值，再分别乘以比中的每一项。
Share £60 in the ratio 2 : 3
Total parts = 2 + 3 = 5. One part = £60 ÷ 5 = £12. The first share = 2 × £12 = £24. The second share = 3 × £12 = £36.
总份数 = 2 + 3 = 5。每份 = £60 ÷ 5 = £12。第一份 = 2 × £12 = £24。第二份 = 3 × £12 = £36。
Check your answer: £24 + £36 = £60, and the ratio 24 : 36 simplifies to 2 : 3, so the answer is correct.
检查答案：£24 + £36 = £60，且比 24 : 36 化简为 2 : 3，所以答案正确。
total parts = 1 + 2 + 3 = 6, one part = 120 ÷ 6 = 20, so the portions are 20, 40, and 60.
这种方法同样适用于三项比。例如，将 120 颗糖果按 1 : 2 : 3 分给三个孩子，总份数 = 1 + 2 + 3 = 6，每份 = 120 ÷ 6 = 20，所以三份分别是 20、40 和 60。
4. Direct Proportion | 正比例
ne quantity doubles, the other doubles too. If y is directly proportional to x, we write y ∝ x, or y = kx, where k is a constant called the constant of proportionality.
如果两个量以相同速率增加或减少，那么它们成正比例。换句话说，当一个量翻倍时，另一个量也翻倍。如果 y 与 x 成正比例，我们写作 y ∝ x，或 y = kx，其中 k 是常数，称为比例常数。
y = kx
To solve a direct proportion problem, first find k using the known pair of values, then use k to find the unknown value.
解正比例问题时，先用已知的一对数值求出 k，再用 k 求未知值。
Example: If y = 8 when x = 2, and y is directly proportional to x, find y when x = 5.
例：如果 y = 8 时 x = 2，且 y 与 x 成正比例，求 x = 5 时 y 的值。
k = 8 ÷ 2 = 4, so y = 4 × 5 = 20.
k = 8 ÷ 2 = 4，所以 y = 4 × 5 = 20。
In IGCSE, direct proportion also appears as "unitary method" problems: find the cost of one item first, then multiply. If 3 pens cost $1.50, then 1 pen costs $0.50, and 10 pens cost $5.00.
在 IGCSE 中，正比例也以"归一法"问题的形式出现：先求一件物品的价格，再相乘。如果 3 支笔卖 $1.50，那么 1 支笔卖 $0.50，10 支笔卖 $5.00。
5. Inverse Proportion | 反比例
Two quantities are in inverse proportion when one quantity increases while the other decreases at the same rate. For example, the time taken to complete a job is inversely proportional to the number of workers: more workers mean less time.
当一个量增大而另一个量以相同速率减小时，这两个量成反比例。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，用时越少。
y = k / x
x, then y = k ÷ x, or equivalently xy = k. The product of the two quantities is constant.
如果 y 与 x 成反比例，则 y = k ÷ x，等价于 xy = k。两个量的乘积是常数。
Example: 4 workers can build a wall in 6 days. How long would 8 workers take?
例：4 名工人 6 天能砌好一堵墙。8 名工人需要多少天？
Total work = 4 × 6 = 24 worker-days. With 8 workers, time = 24 ÷ 8 = 3 days.
总工作量 = 4 × 6 = 24 个工日。8 名工人时，时间 = 24 ÷ 8 = 3 天。
f workers halves the time. Check that 4 × 6 = 8 × 3 = 24, confirming the constant product.
注意，工人数翻倍使时间减半。验证 4 × 6 = 8 × 3 = 24，确认乘积恒定。
6. Proportional Division in Word Problems | 比例应用题
Many IGCSE word problems combine ratios with percentages, fractions, or money. The key is to translate the English into a mathematical ratio and then apply the standard method.
许多 IGCSE 应用题将比与百分比、分数或金钱相结合。关键是将文字翻译成数学比，然后应用标准方法。
Example: The angles of a triangle are in the ratio 2 : 3 : 4. Find each angle.
例：一个三角形的三个内角之比为 2 : 3 : 4。求各个角的度数。
4 = 9. Sum of angles = 180°. One part = 180° ÷ 9 = 20°. The angles are 40°, 60°, and 80°.
总份数 = 2 + 3 + 4 = 9。内角和 = 180°。每份 = 180° ÷ 9 = 20°。三个角分别为 40°、60° 和 80°。
Another common type involves "difference" he difference of 2 parts = 14, so one part = 7. The numbers are 5 × 7 = 35 and 3 × 7 = 21.
另一种常见题型涉及"差值"。如果两个数之比为 5 : 3，且它们的差为 14，那么 2 份 = 14，所以每份 = 7。两个数分别为 5 × 7 = 35 和 3 × 7 = 21。
When a ratio includes a total, such as "a sum of money is divided between A and B in the ratio 4 : 7 and B receives $28 more than A", find the value of one part first using the difference in parts.
当比中包含总量时，例如"一笔钱按 4 : 7 分给 A 和 B，且 B 比 A 多 $28"，先利用份数差求出每份的值。
7. Scaling and Map Scales | 缩放与地图比例尺
Ratios are used to represent scale drawings and maps. A scale of 1 : 100 means that 1 cm on the drawing represents 100 cm (1 m) in real life.
比用于表示比例尺和缩放图。比例尺 1 : 100 表示图上的 1 cm 代表实际生活中的 100 cm（即 1 m）。
map distance : real distance = scale ratio
Example: A map has a scale of 1 : 50,000. Two towns are 4 cm apart on the map. What is the actual distance in kilometres?
例：一张地图的比例尺为 1 : 50,000。两个城镇在地图上相距 4 cm。实际距离是多少千米？
00,000 cm. Convert to metres: 200,000 cm = 2,000 m. Convert to kilometres: 2,000 m = 2 km.
实际距离 = 4 × 50,000 = 200,000 cm。换算成米：200,000 cm = 2,000 m。换算成千米：2,000 m = 2 km。
For area scaling, remember that if lengths are in the ratio 1 : n, then areas are in the ratio 1 : n², and volumes are in the ratio 1 : n³.
对于面积缩放，记住如果长度之比为 1 : n，那么面积之比为 1 : n²，体积之比为 1 : n³。
| Linear scale | Area scale | Volume scale |
| 1 : 2 | 1 : 4 | 1 : 8 |
| 1 : 10 | 1 : 100 | 1 : 1000 |
If a small model car has a scale of 1 : 20 and the model's length is 12 cm, the real car is 12 × 20 = 240 cm = 2.4 m long.
如果一辆模型车的比例尺为 1 : 20，模型长度为 12 cm，那么真车长度为 12 × 20 = 240 cm = 2.4 m。
8. Ratios and Fractions | 比与分数
total (because the total is 2 + 3 = 5 parts), and the second quantity is 3/5 of the total.
比可以用分数表示。例如，比 2 : 3 表示第一个数量占总量的 2/5（因为总量是 2 + 3 = 5 份），第二个数量占总量的 3/5。
To convert a ratio to a fraction, add the parts to get the denominator, and use each part as the numerator. To convert a fraction to a ratio, subtract the numerator from the denominator to get the second part.
将比转换为分数时，将各项相加得到分母，每一项作为分子。将分数转换为比时，用分母减去分子得到第二项。
glasses, then the ratio of glasses-wearers to non-glasses-wearers is 3 : (8 − 3) = 3 : 5.
例：如果一个班级有 3/8 的学生戴眼镜，那么戴眼镜与不戴眼镜的人数之比为 3 : (8 − 3) = 3 : 5。
Be careful: a ratio of 1 : 4 is not the same as a fraction 1/4 of the total; the total has 5 parts, so 1 : 4 means 1/5 of the total is the first quantity.
要小心：比 1 : 4 并不等于总量中的 1/4；总量有 5 份，所以 1 : 4 表示第一个数量占总量的 1/5。
9. Common Mistakes and Tips | 常见错误与技巧
Here are the most frequent errors students make in ratio and proportion questions, and how to avoid them.
以下是学生在比与比例题中最常见的错误，以及如何避免它们。
- Mixing the order: Always read the question carefully to see which quantity comes first. "The ratio of boys to girls" is different from "girls to boys".
- 顺序颠倒：仔细读题，确定哪个数量在前。"男生与女生之比"和"女生与男生之比"不同。
- Forgetting to simplify: Give the final ratio in simplest form unless asked otherwise.
- 忘记化简：除非另有要求，给出最简形式的比。
- Ignoring units: Convert all quantities to the same unit before forming a ratio.
- 忽略单位：在写比之前将所有量换算成相同单位。
- Using cross-multiplication incorrectly: In proportions like a/b = c/d, cross-multiply correctly: a × d = b × c.
- 错误使用交叉相乘：在比例式 a/b = c/d 中，正确交叉相乘：a × d = b × c。
- Not checking the answer: Always verify that the parts add up to the total and simplify back to the given ratio.
- 不检查答案：务必验证各部分相加等于总量，且能化简回给定比。
Tip: Write down the method step by step. In IGCSE exams, method marks are awarded even for incorrect numerical answers.
技巧：逐步写出方法。在 IGCSE 考试中，即使数值答案错误，也能获得方法分。
10. Practice Questions | 练习题
Try these questions yourself before looking at the answers.
先自己尝试以下题目，再查看答案。
Question 1: Simplify the ratio 18 : 24 : 30.
题目 1：化简比 18 : 24 : 30。
Question 2: Divide £84 in the ratio 3 : 4.
题目 2：将 £84 按 3 : 4 分配。
Question 3: A recipe uses flour and sugar in the ratio 5 : 2. If 200 g of sugar is used, how much flour is needed?
题目 3：一个食谱中面粉与糖之比为 5 : 2。如果用了 200 g 糖，需要多少面粉？
Question 4: y is inversely proportional to x. If y = 10 when x = 4, find y when x = 8.
题目 4：y 与 x 成反比例。如果 y = 10 时 x = 4，求 x = 8 时 y 的值。
Answers: 1) 3 : 4 : 5. 2) £36 and £48. 3) Flour = 200 × 5/2 = 500 g. 4) y = 40 ÷ 8 = 5.
答案：1) 3 : 4 : 5。2) £36 和 £48。3) 面粉 = 200 × 5/2 = 500 g。4) y = 40 ÷ 8 = 5。
Review any question you got wrong and try similar problems in your textbook. Practice is essential for mastering ratios and proportions.
仔细复习做错的题目，并在课本中寻找类似题目练习。反复练习是掌握比与比例的关键。
11. Conclusion | 总结
Ratios and proportions are more than just exam topics; they are life skills. From cooking and map reading to currency conversion and building design, the logic of proportional reasoning appears everywhere.
比与比例不仅仅是考试内容，更是生活技能。从烹饪、读地图到货币兑换和建筑设计，比例推理的逻辑无处不在。
Master the core steps: simplify ratios, divide quantities in a given ratio, and distinguish between direct and inverse proportion. With consistent practice, these questions will become fast and reliable marks in your IGCSE exam.
掌握核心步骤：化简比、按给定比分配数量、区分正比例和反比例。通过持续练习，这些问题将在 IGCSE 考试中成为快速且稳定的得分点。
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