📚 Understand Ratios | 理解比与比率
In IGCSE Mathematics, a ratio compares two or more quantities of the same kind, showing their relative sizes. This lesson introduces ratio notation, equivalent ratios, simplifying, sharing in a ratio, and real-life applications.
在 IGCSE 数学中，比用于比较两个或多个同类数量，表示它们之间的相对大小。本课介绍比的写法、等比、化简、按比例分配以及实际应用。
1. What Is a Ratio? | 什么是比？
A ratio is a comparison of two or more quantities. It tells us how many times one quantity contains another or how two quantities relate. The order of a ratio is very important because the ratio 'apples to oranges' is different from 'oranges to apples'.
比是两个或多个数量之间的比较。它表示一个数量包含另一个数量的多少倍，或两个数量之间的关系。比的顺序非常重要，因为“苹果与橙子的比”和“橙子与苹果的比”是不同的。
Example: A fruit bowl has 3 apples and 2 oranges. The ratio of apples to oranges is 3 : 2. The ratio of oranges to apples is 2 : 3.
例：一个果盘里有 3 个苹果和 2 个橙子。苹果与橙子的比是 3 : 2。橙子与苹果的比是 2 : 3。
The quantities in a ratio must be of the same kind. For example, we can compare lengths to lengths or masses to masses, but we cannot directly compare 3 metres to 2 kilograms.
比中的数量必须是同类数量。例如，我们可以比较长度与长度、质量与质量，但不能直接比较 3 米与 2 千克。
2. Three Ways to Write a Ratio | 比的三种写法
A ratio can be written in three equivalent forms: using a colon (a : b), using the word 'to' (a to b), or as a fraction (a/b). All three forms represent the same comparison.
比有三种等价的写法：用冒号（a : b）、用单词 “to”（a to b）或用分数（a/b）。这三种形式表示相同的比较。
Example: If a class has 5 girls and 3 boys, the ratio of girls to boys can be written as 5 : 3, 5 to 3, or 5/3.
例：如果一个班级有 5 个女生和 3 个男生，女生与男生的比可以写作 5 : 3、5 to 3 或 5/3。
a : b = a to b = a/b
The colon form is the most common in IGCSE exam answers. The fraction form is useful when we connect ratios to fractions and proportions.
在 IGCSE 考试答案中，冒号形式最为常见。分数形式在将比与分数、比例联系起来时很有用。
3. Part-to-Part and Part-to-Whole Ratios | 部分与部分、部分与整体
A part-to-part ratio compares two separate parts of a whole. A part-to-whole ratio compares one part to the total number of items or the whole group.
部分与部分的比是比较整体中的两个不同部分。部分与整体的比是将一个部分与物品总数或整体进行比较。
rls is 12 : 8. The part-to-whole ratio of boys to all students is 12 : (12 + 8) = 12 : 20.
例：一个班级有 12 个男生和 8 个女生。男生与女生的部分对部分比是 12 : 8。男生与全班学生的部分对整体比是 12 : (12 + 8) = 12 : 20。
Part-to-whole ratios are very useful when a question asks for a fraction or percentage of a total. The denominator of the fraction becomes the total number of parts.
当题目要求总数的分数或百分比时，部分与整体比非常有用。分数的分母就是总份数。
4. Equivalent Ratios | 等比
Equivalent ratios are ratios that express the same relationship. You can find an equivalent ratio by multiplying or dividing both terms by the same positive number.
等比是表示相同关系的比。你可以将两项同时乘以或除以同一个正数来得到等比。
Example: 2 : 3 is equivalent to 4 : 6 because we multiply both terms by 2. It is also equivalent to 6 : 9 because we multiply both terms by 3.
例：2 : 3 与 4 : 6 等价，因为我们将两项同时乘以 2。它也与 6 : 9 等价，因为我们将两项同时乘以 3。
| Operation | First term | Second term | Ratio |
| Original | 2 | 3 | 2 : 3 |
| Multiply by 2 | 4 | 6 | 4 : 6 |
| Multiply by 3 | 6 | 9 | 6 : 9 |
| Divide by 1 | 2 | 3 | 2 : 3 |
Equivalent ratios are used to solve many IGCSE problems, such as mixing ingredients, scaling up recipes, or comparing prices.
等比可用于解决许多 IGCSE 问题，例如混合配料、按比例放大配方或比较价格。
5. Simplifying Ratios | 化简比
To simplify a ratio, divide both terms by their greatest common divisor (GCD). This produces the simplest whole-number form of the ratio.
化简比时，将两项同时除以它们的最大公因数（GCD）。这样可以得到比的最简整数形式。
is 4, so divide both terms by 4: 8 ÷ 4 = 2 and 12 ÷ 4 = 3. The simplified ratio is 2 : 3.
例：化简 8 : 12。8 和 12 的最大公因数是 4，所以将两项同时除以 4：8 ÷ 4 = 2，12 ÷ 4 = 3。化简后的比是 2 : 3。
If a ratio contains fractions or decimals, first multiply both terms to obtain whole numbers, then simplify as usual.
如果比含有分数或小数，先将两项乘以适当倍数得到整数，再按常规方法化简。
Example: Simplify 0.5 : 2. Multiply both terms by 2 to get 1 : 4. Example: Simplify 1/2 : 3/4. Multiply both terms by 4 to get 2 : 3.
例：化简 0.5 : 2。将两项同时乘以 2，得到 1 : 4。例：化简 1/2 : 3/4。将两项同时乘以 4，得到 2 : 3。
Always check that a simplified ratio uses the smallest possible positive whole numbers unless the question says otherwise.
除非题目另有说明，化简后的比应使用最小的正整数。
6. Ratios with Different Units | 不同单位的比
Ratios compare quantities of the same kind. If the quantities have different units, convert one or both quantities so that the units match before writing the ratio.
比用于比较同类数量。如果数量的单位不同，需要先将一个或两个数量换算成相同单位，再写出比。
Example: Write the ratio 2 m to 50 cm in its simplest form. Change 2 m to 200 cm, so the ratio becomes 200 : 50. Divide both terms by 50 to get 4 : 1.
例：写出 2 米与 50 厘米的最简比。将 2 米换算成 200 厘米，比变成 200 : 50。将两项同时除以 50，得到 4 : 1。
Example: Write the ratio 1.5 hours to 30 minutes in its simplest form. Change 1.5 hours to 90 minutes, so the ratio is 90 : 30. Divide both terms by 30 to get 3 : 1.
例：写出 1.5 小时与 30 分钟的最简比。将 1.5 小时换算成 90 分钟，比是 90 : 30。将两项同时除以 30，得到 3 : 1。
nclude 1 m = 100 cm, 1 km = 1000 m, 1 hour = 60 minutes, 1 kg = 1000 g, and 1 L = 1000 mL.
IGCSE 常见单位换算包括：1 米 = 100 厘米，1 千米 = 1000 米，1 小时 = 60 分钟，1 千克 = 1000 克，1 升 = 1000 毫升。
7. Using Ratio Tables | 比率表
A ratio table helps us generate equivalent ratios and solve problems. Each row is produced by multiplying or dividing both sides of the ratio by the same number.
比率表有助于找出等比并解决问题。表中的每一行都是将比的两项同时乘以或除以同一个数得到的。
Example: A recipe uses 3 cups of flour for every 1 cup of sugar. Build a ratio table for flour to sugar.
例：一个配方每使用 3 杯面粉就需要 1 杯糖。建立面粉与糖的比率表。
| Flour in cups | Sugar in cups | Ratio |
| 3 | 1 | 3 : 1 |
| 6 | 2 | 3 : 1 |
| 9 | 3 | 3 : 1 |
| 12 | 4 | 3 : 1 |
Ratio tables are especially useful when we need to scale a recipe, find missing values, or solve proportion problems.
当我们需要按比例调整配方、求缺失值或解决比例问题时，比率表尤其有用。
8. Sharing in a Given Ratio | 按比例分配
To share a quantity in a given ratio, first find the total number of parts. 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 term of the ratio.
按给定比例分配时，先求总份数。然后用总数除以总份数得到一份的值。最后用一份的值乘以比中的每一项。
Example: Share $120 between Ali and Ben in the ratio 3 : 2. Total parts = 3 + 2 = 5. One part = $120 ÷ 5 = $24. Ali receives 3 × $24 = $72, and Ben receives 2 × $24 = $48.
例：将 120 美元按 3 : 2 分给 Ali 和 Ben。总份数 = 3 + 2 = 5。一份 = 120 ÷ 5 = 24 美元。Ali 得到 3 × 24 = 72 美元，Ben 得到 2 × 24 = 48 美元。
Check: $72 + $48 = $120, so all money is shared. The answers must always add back to the original total.
检验：72 + 48 = 120 美元，所有钱都分配完毕。答案相加后必须等于原始总数。
s and the ratio of boys to girls is 5 : 7. Total parts = 5 + 7 = 12. One part = 360 ÷ 12 =
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