📚 Simplifying Ratios | 化简比率
Ratios are used to compare quantities, and simplifying them makes relationships clearer and easier to work with. In this article, we will explore how to simplify ratios, including those with different units, fractions, decimals, and three or more terms. You will also learn how to express ratios in the form 1 : n, which is a common requirement in Cambridge Checkpoint examinations.
比率用于比较数量，化简比率能让关系更清晰、更易处理。本文将探讨如何化简比率，包括含不同单位、分数、小数以及三个或更多项的比率。你还将学习如何将比率表示为 1 : n 的形式，这是剑桥 Checkpoint 考试中的常见要求。
1. What is a Ratio? | 什么是比率？
A ratio shows the relative sizes of two or more values. It is written with a colon, for example 3 : 5. The order is important: 3 : 5 is not the same as 5 : 3. Ratios compare part to part, or part to whole, but they never have units written next to the numbers.
比率表示两个或多个值之间的大小关系，用冒号书写，例如 3 : 5。顺序很重要：3 : 5 与 5 : 3 不同。比率可以比较部分与部分，或部分与整体，但数字旁边从不写单位。
2. Writing Ratios in Their Simplest Form | 将比率写成最简形式
To simplify a ratio, divide each term by their highest common factor (HCF). For example, the ratio 8 : 12 can be simplified by dividing both numbers by 4, giving 2 : 3. Always check that the numbers left have no common factor other than 1.
要化简比率，将每一项除以它们的最大公因数 (HCF)。例如，比率 8 : 12 可以通过将两个数都除以 4 来化简，得到 2 : 3。务必检查剩下的数字除了 1 以外没有其他公因数。
3. Ratio with Different Units | 含不同单位的比率
When the quantities are given in different units, you must first convert them to the same unit. For instance, to simplify the ratio 50 cm to 2 m, change 2 m to 200 cm, then write 50 : 200. Simplify by dividing both by 50 to get 1 : 4. Do not leave mixed units in a ratio.
当给出的数量单位不同时，必须先将它们转换为相同的单位。例如，要化简比率 50 cm : 2 m，将 2 m 转换为 200 cm，然后写成 50 : 200。将两项都除以 50，得到 1 : 4。切勿在比率中保留混合单位。
4. Simplifying Ratios with Fractions | 化简含分数的比率
To simplify a ratio containing fractions, multiply every term by the lowest common denominator (LCD) of the fractions. For example, simplify ½ : ⅓. The LCD of 2 and 3 is 6. Multiply both terms by 6: ½ × 6 = 3, and ⅓ × 6 = 2, so the simplified ratio is 3 : 2.
若要化简含有分数的比率，将每一项都乘以分数的最小公分母 (LCD)。例如，化简 ½ : ⅓。2 和 3 的最小公分母是 6。将两项都乘以 6：½ × 6 = 3，⅓ × 6 = 2，因此化简后的比率为 3 : 2。
5. Simplifying Ratios with Decimals | 化简含小数的比率
Multiply all terms by a power of 10 that will make every term a whole number. For the ratio 0.4 : 1.2, multiply by 10 to get 4 : 12, then simplify to 1 : 3. For 0.25 : 0.5, multiply by 100 to get 25 : 50, which simplifies to 1 : 2.
将每一项都乘以 10 的幂，使每一项都变成整数。对于比率 0.4 : 1.2，乘以 10 得到 4 : 12，再化简为 1 : 3。对于 0.25 : 0.5，乘以 100 得到 25 : 50，化简为 1 : 2。
6. Expressing Ratios as 1 : n | 将比率表示为 1 : n
This form is often used to show a direct comparison. Given a ratio a : b, divide both sides by a to obtain 1 : (b ÷ a). For 4 : 10, divide by 4: the left side becomes 1, the right side becomes 10 ÷ 4 = 2.5, so the ratio is 1 : 2.5. You can also show this as a fraction 1 : ⁵⁄₂ if exact form is required.
这种形式常用来显示直接比较。给定比率 a : b，将两边同时除以 a，得到 1 : (b ÷ a)。对于 4 : 10，除以 4：左边变为 1，右边变为 10 ÷ 4 = 2.5，因此比率为 1 : 2.5。如果需要精确形式，你也可以将其表示为分数 1 : ⁵⁄₂。
7. Simplifying Three-Term Ratios | 化简三项比率
Ratios can have three or more terms, like 6 : 9 : 12. Find the HCF of all numbers—here it is 3—and divide each term by it to get 2 : 3 : 4. The same rules apply: convert to same units first if needed, and clear any fractions or decimals.
比率可以有三项或更多项，例如 6 : 9 : 12。找出所有数字的最大公因数（这里是 3），然后将每一项除以它，得到 2 : 3 : 4。同样的规则也适用：如果需要，先转换成相同单位，并清除所有分数或小数。
8. Using Ratios to Solve Problems | 用比率解决实际问题
Once a ratio is simplified, it can be used to find unknown quantities. If the ratio of boys to girls is 3 : 4 and there are 21 boys, set up a proportion: 3 parts = 21, so 1 part = 7. Then 4 parts = 28 girls. Always link the ratio to the total number of parts when finding the whole.
比率一旦化简，就可以用来求未知量。如果男生与女生的比率是 3 : 4，且男生有 21 人，则可列出比例：3 份对应 21，因此 1 份 = 7，那么 4 份 = 28 名女生。在求总数时，始终要将比率与总份数联系起来。
9. Common Mistakes to Avoid | 常见错误与防范
Many students forget to convert units, leading to wrong simplifications. Others try to subtract instead of divide, or cancel numbers incorrectly across a three-term ratio. Remember, ratios are multiplied or divided, never added or subtracted term-by-term. Always check your work by scaling back to the original numbers.
许多学生忘记转换单位，导致化简错误。也有人试图用减法而不是除法，或在三项比率中错误地约分。记住，比率是乘或除的，绝不能逐项加减。务必通过还原为原数字来检查你的答案。
10. Practice Questions and Checkpoint Tips | 练习题目与考试技巧
Try simplifying these: (a) 16 : 24, (b) 0.6 : 1.8, (c) ¼ : ¾, (d) 40 cm : 1 m, (e) 5 : 2 : 7 (already simplest). In the exam, express your final answer in the simplest integer form unless asked otherwise. For 1 : n questions, watch out for the order: 1 : n is not n : 1.
试着化简下列比率：(a) 16 : 24, (b) 0.6 : 1.8, (c) ¼ : ¾, (d) 40 cm : 1 m, (e) 5 : 2 : 7（已最简）。考试中，除非另有要求，否则应以最简整数形式给出最终答案。对于 1 : n 的问题，注意顺序：1 : n 与 n : 1 不同。
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