📚 Understanding Ratios | 理解比率
A ratio is a way of comparing two or more quantities. In the IGCSE Mathematics course, ratios are used to solve sharing problems, work with recipes, interpret map scales and understand direct proportion. A ratio tells us how much of one quantity there is compared with another quantity.
比率是比较两个或多个数量的一种方式。在 IGCSE 数学课程中，比率用于解决分配问题、处理食谱、理解地图比例尺以及正比例关系。比率告诉我们一个数量相对于另一个数量有多少。
1. What Is a Ratio? | 什么是比率？
A ratio compares two or more quantities of the same kind. The notation a : b is read as "a to b". The order in which the quantities are written is very important because 3 : 4 is not the same as 4 : 3.
比率用于比较两个或多个同类量。记号 a : b 读作“a 比 b”。书写数量的顺序非常重要，因为 3 : 4 与 4 : 3 的含义不同。
a : b = a ÷ b = a/b
For example, if a class has 12 boys and 18 girls, the ratio of boys to girls is written as 12 : 18. This ratio can be simplified to 2 : 3 by dividing both sides by 6.
例如，如果一个班有 12 名男生和 18 名女生，男生与女生的比率写作 12 : 18。两边同时除以 6，可以将该比率化简为 2 : 3。
2. Writing Ratios from Contexts | 从情境中写出比率
When writing a ratio from a word problem, always match the order given in the question. If the question says "red to blue", the red quantity must come first. Also make sure both quantities are measured in the same units before writing the ratio.
从文字题中写出比率时，一定要按照题目给出的顺序来写。如果题目说“红色比蓝色”，那么红色数量必须写在前面。同时，在写出比率之前要确保两个数量使用相同的单位。
- Example 1: A fruit bowl has 8 apples and 12 oranges. The ratio of apples to oranges is 8 : 12, which simplifies to 2 : 3.
- Example 2: A jug contains 1 litre of juice and 250 ml of water. Write both in millilitres first: 1000 ml : 250 ml, which simplifies to 4 : 1.
例 1：一个水果碗里有 8 个苹果和 12 个橙子。苹果与橙子的比率为 8 : 12，化简为 2 : 3。
例 2：一个壶里有 1 升果汁和 250 毫升水。先把两者都写成毫升：1000 ml : 250 ml，化简为 4 : 1。
3. Simplifying Ratios | 化简比率
To simplify a ratio, divide both terms by their highest common factor. This is similar to simplifying a fraction. A ratio is fully simplified when the only common factor of the two terms is 1.
化简比率时，将两项同时除以它们的最大公因数。这与化简分数类似。当两项的唯一公因数为 1 时，该比率就是最简形式。
For example, 24 : 36 can be simplified by dividing both sides by 12, giving 2 : 3. The ratio 15 : 25 simplifies to 3 : 5 because both 15 and 25 are divisible by 5.
例如，24 : 36 可以两边同时除以 12，得到 2 : 3。比率 15 : 25 化简为 3 : 5，因为 15 和 25 都能被 5 整除。
4. Equivalent Ratios | 等价比率
Equivalent ratios are obtained by multiplying or dividing both terms of a ratio by the same non-zero number. Equivalent ratios represent the same comparison, just as equivalent fractions represent the same value.
等价比率是将比率的两个项同时乘以或除以同一个非零数而得到的。等价比率表示同一种比较关系，就像等价分数表示同一个值一样。
| Multiply by 2 | 2 : 3 | 4 : 6 |
| Multiply by 3 | 2 : 3 | 6 : 9 |
| Multiply by 5 | 2 : 3 | 10 : 15 |
The ratios 2 : 3, 4 : 6, 6 : 9 and 10 : 15 are all equivalent. They look different, but they show the same relationship.
比率 2 : 3、4 : 6、6 : 9 和 10 : 15 都是等价的。它们看起来不同，但表示相同的关系。
5. Ratios with Fractions and Decimals | 含分数和小数的比率
Sometimes a ratio contains fractions or decimals. To simplify such a ratio, first multiply every term by a common multiple to clear the fractions or decimals, then simplify as usual.
有时比率中包含分数或小数。要化简这样的比率，首先把每一项都乘以一个公倍数以消去分数或小数，然后按常规方法化简。
Example 1: Simplify 1/2 : 3/4. Multiply both terms by 4: 4 × 1/2 = 2 and 4 × 3/4 = 3. The simplified ratio is 2 : 3.
例 1：化简 1/2 : 3/4。将两项都乘以 4：4 × 1/2 = 2，4 × 3/4 = 3。化简后的比率为 2 : 3。
Example 2: Simplify 1.5 : 2.5. Multiply both terms by 10 to get 15 : 25, then divide both by 5 to get 3 : 5.
例 2：化简 1.5 : 2.5。将两项都乘以 10，得到 15 : 25，然后两边都除以 5，得到 3 : 5。
6. Three-Part Ratios | 三部分比率
A ratio can compare three or more quantities. A three-part ratio is written as a : b : c. To simplify it, divide all three terms by a common factor. To find the total number of parts, add the three terms.
比率可以比较三个或更多数量。三部分比率写作 a : b : c。化简时将所有三项都除以一个公因数。要求总份数，将三项相加即可。
For example, the ratio 8 : 12 : 20 simplifies to 2 : 3 : 5 because each term can be divided by 4. The total number of parts is 2 + 3 + 5 = 10.
例如，比率 8 : 12 : 20 化简为 2 : 3 : 5，因为每一项都能被 4 整除。总份数为 2 + 3 + 5 = 10。
7. Dividing a Quantity in a Given Ratio | 按给定比率分配数量
To share a quantity in a given ratio, first work out the total number of parts by adding the ratio terms. Then divide the total quantity by the total number of parts to find the value of one part. Finally, multiply the value of one part by each ratio term.
按给定比率分配数量时，首先将比率的各项相加，求出总份数。然后用总数量除以总份数，求出一份的值。最后用一份的值分别乘以比率的每一项。
Example: Divide £60 in the ratio 2 : 3.
例：按 2 : 3 的比率分配 £60。
Total parts = 2 + 3 = 5
One part = £60 ÷ 5 = £12
The two shares are 2 × £12 = £24 and 3 × £12 = £36. The amounts add up to £60, which confirms the answer is correct.
两份分别为 2 × £12 = £24 和 3 × £12 = £36。两个数相加等于 £60，这验证了答案是正确的。
8. Scale Drawings and Maps | 比例图和地图
A map scale is a ratio that compares a length on a map to the actual length on the ground. For example, a scale of 1 : 50 000 on the map represents 50 000 cm in real life. Since 100 cm = 1 m, 50 000 cm equals 500 m.
地图比例尺是一种比率，用于比较地图上的长度与实际地面上的长度。例如，比例尺 1 : 50 000 表示地图上的 1 厘米代表实际中的 50 000 厘米。因为 100 厘米 = 1 米，所以 50 000 厘米等于 500 米。
Example: On a map with scale 1 : 250 000, two towns are 8 cm apart. The real distance is 8 × 250 000 = 2 000 000 cm = 20 km.
例：在比例尺为 1 : 250 000 的地图上，两个城镇相距 8 厘米。实际距离为 8 × 250 000 = 2 000 000 厘米 = 20 公里。
9. Ratios and Direct Proportion | 比率与正比例
Ratios are closely linked to direct proportion. If two quantities are in direct proportion, their ratio stays constant when both quantities are scaled up or down. Recipes and ingredient mixtures are common examples.
比率与正比例密切相关。如果两个量成正比例，那么当两个量同时扩大或缩小时，它们的比率保持不变。食谱和配料混合物是常见的例子。
For example, a pancake mixture uses flour and milk in the ratio 3 : 2. If you double the recipe, you need 6 cups of flour and 4 cups of milk. The ratio 6 : 4 is equivalent to 3 : 2.
例如，煎饼混合物中面粉和牛奶的比率为 3 : 2。如果将食谱翻倍，就需要 6 杯面粉和 4 杯牛奶。比率 6 : 4 与 3 : 2 等价。
10. Common Errors and Exam Tips | 常见错误与考试提示
Do not confuse a ratio with a fraction of the whole. If the ratio of boys to girls is 2 : 3, then boys make up 2/5 of the total and girls make up 3/5 of the total, not 2/3 and 3/2.
不要将比率与整体中的分数混淆。如果男生与女生的比率为 2 : 3，那么男生占总数的 2/5，女生占总数的 3/5，而不是 2/3 和 3/2。
- Always write both quantities in the same units before forming a ratio.
- Always simplify a ratio unless the question asks for a different form.
- Check that the parts of a shared quantity add up to the original total.
- For scale and recipe problems, use equivalent ratios to find missing values.
在形成比率之前，始终将两个数量写成相同的单位。
除非题目要求其他形式，否则始终化简比率。
检查分配数量的各部分相加是否等于原来的总数。
对于比例尺和食谱问题，使用等价比率来求缺失值。
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