📚 Cambridge KS3 Maths p253 Practice: Ratio, Proportion and Unitary Method | 剑桥 KS3 数学 p253 练习：比例、比例关系与单位法
In this Cambridge KS3 Mathematics revision article, we focus on the key ideas found on page 253 of the practice booklet: ratio, proportion and the unitary method. These topics appear regularly in Checkpoint tests and provide a foundation for later work on rates, scales and algebraic proportion.
在这篇剑桥 KS3 数学复习文章中，我们聚焦练习册第 253 页的核心内容：比例、比例关系与单位法。这些主题在 Checkpoint 考试中经常出现，并为后续的速率、比例尺和代数比例学习打下基础。
1. Understanding Ratio | 理解比
A ratio compares two or more quantities in the same unit. The order of a ratio is important, because the ratio 2 : 3 is not the same as 3 : 2. We read 2 : 3 as 'two to three' and it means that for every 2 parts of the first quantity there are 3 parts of the second quantity.
比用来比较两个或多个相同单位的量。比的顺序很重要，因为 2 : 3 和 3 : 2 不相同。我们将 2 : 3 读作“二比三”，它表示第一个量每有 2 份，第二个量就有 3 份。
2 : 3 means 2 parts to 3 parts
In a question, the quantities must be measured in the same unit before a ratio is written. For example, 50 cm to 1 m must first be written as 50 cm to 100 cm, giving the ratio 50 : 100.
在题目中，写出比之前必须将量使用相同的单位。例如 50 厘米比 1 米必须先写成 50 厘米比 100 厘米，从而得到比 50 : 100。
2. Simplifying Ratios | 化简比
To simplify a ratio, divide every part by the highest common factor. For example, 8 : 12 simplifies to 2 : 3 because both 8 and 12 are divisible by 4. A simplified ratio has no common factor except 1 and is usually easier to interpret.
化简比时，用最大公因数去除比的每一部分。例如 8 : 12 化简为 2 : 3，因为 8 和 12 都能被 4 整除。化简后的比除 1 外没有其他公因数，通常更容易理解。
8 : 12 = (8 ÷ 4) : (12 ÷ 4) = 2 : 3
When simplifying, divide all parts by the same number. The ratio 15 : 25 : 40 simplifies by dividing each part by 5 to give 3 : 5 : 8.
化简时，每一部分都除以同一个数。比 15 : 25 : 40 将每一部分除以 5，化简为 3 : 5 : 8。
3. Writing Ratios in the Form 1 : n | 将比写成 1 : n 形式
Sometimes a ratio is written in the unit form 1 : n. To do this, divide both sides by the first number. For instance, 3 : 5 becomes 1 : 5/3, which is 1 : 1.67 to two decimal places. This form is useful when comparing several ratios.
有时需要把比写成单位形式 1 : n。做法是两边都除以第一个数。例如 3 : 5 变成 1 : 5/3，即 1 : 1.67（保留两位小数）。这种形式在比较多个比例时很有用。
3 : 5 = (3 ÷ 3) : (5 ÷ 3) = 1 : 1.67
If the first number is not the smallest, the same method still works. For example, 7 : 2 becomes 1 : 2/7, which is approximately 1 : 0.29.
如果第一个数不是最小的，方法仍然相同。例如 7 : 2 变成 1 : 2/7，约等于 1 : 0.29。
4. Sharing in a Given Ratio | 按给定比例分配
ind the total number of parts, then work out the value of one part. For example, to divide £120 in the ratio 3 : 2, the total parts are 3 + 2 = 5, so one part is £120 ÷ 5 = £24. The shares are 3 × £24 = £72 and 2 × £24 = £48.
按比例分配数量时，先求出总份数，再计算一份的值。例如将 120 英镑按 3 : 2 分配，总份数是 3 + 2 = 5，因此一份是 120 ÷ 5 = 24 英镑。两份分别是 3 × 24 = 72 英镑和 2 × 24 = 48 英镑。
Total parts = 3 + 2 = 5; one part = £120 ÷ 5 = £24
Always check that the individual shares add back to the original amount. Here £72 + £48 = £120, so the division is correct.
一定要检查各份额加回去是否等于原来的总量。这里 72 英镑 + 48 英镑 = 120 英镑，因此分配正确。
5. Using the Unitary Method | 使用单位法
s the value of one unit first, then multiplies to find the required amount. If 6 pens cost £4.50, one pen costs £4.50 ÷ 6 = £0.75, so 10 pens cost 10 × £0.75 = £7.50. This method is central to ratio and proportion problems.
单位法先求一个单位的值，再乘以所需数量。如果 6 支笔花费 4.50 英镑，一支笔花费 4.50 ÷ 6 = 0.75 英镑，那么 10 支笔花费 10 × 0.75 = 7.50 英镑。此方法是比例和比例关系问题的核心。
1 pen = £4.50 ÷ 6 = £0.75; 10 pens = 10 × £0.75 = £7.50
The unitary method works in two stages: divide to find one unit, then multiply to find any number of units. It is especially useful when the question does not give a simple whole-number multiplier.
单位法分两步：先除以数量求一个单位的值，再乘以任意数量求总价值。当题目给出的倍数不是简单整数时，这个方法特别有用。
6. Direct Proportion | 正比例
direct proportion when they increase or decrease at the same rate. If 1 kg of apples costs £2.40, then 3 kg costs £2.40 × 3 = £7.20. The cost and mass keep the same ratio, so doubling one doubles the other.
当两个量以相同速率增加或减少时，它们成正比例。如果 1 千克苹果花费 2.40 英镑，那么 3 千克花费 2.40 × 3 = 7.20 英镑。花费和质量保持相同的比，因此一个量翻倍，另一个量也翻倍。
Cost = constant × mass; 3 × £2.40 = £7.20
Direct proportion can be recognised from a table of values: as one quantity is multiplied by a number, the other quantity is multiplied by the same number. If 2 kg costs £4.80 and 4 kg costs £9.60, the cost per kilogram remains £2.40.
正比例可以从数值表中识别：当一个量乘以某个数时，另一个量也乘以相同的数。如果 2 千克花费 4.80 英镑，4 千克花费 9.60 英镑，那么每千克的价格仍然是 2.40 英镑。
7. Map Scales and Ratio | 地图比例尺与比
A map scale is a ratio that connects a distance on a map to the real distance on the ground. A scale of 1 : 50 000 means 1 cm on the map represents 50 000 cm in real life, which is 500 m. To find a real distance, multiply the map length by the scale factor.
地图比例尺是将地图上的距离与实际地面距离联系起来的一种比。比例尺 1 : 50 000 表示地图上 1 厘米代表实际 50 000 厘米，即 500 米。求实际距离时，将地图长度乘以比例系数。
1 : 50 000 means 1 cm = 50 000 cm = 500 m
s 1 : 50 000, the real distance is 4.5 × 50 000 = 225 000 cm, which is 2 250 m or 2.25 km.
如果地图上的长度是 4.5 厘米，比例尺为 1 : 50 000，则实际距离为 4.5 × 50 000 = 225 000 厘米，即 2 250 米或 2.25 千米。
8. Recipe Problems | 配方问题
the flour needed is 250 g × 1.5 = 375 g. Each ingredient is multiplied by the same factor.
食谱常常需要按比例增加或减少。如果一份供 4 人食用的食谱使用 250 克面粉，那么供 6 人食用时倍数是 6 ÷ 4 = 1.5，因此所需面粉为 250 × 1.5 = 375 克。每种配料都乘以相同的倍数。
Multiplier = 6 ÷ 4 = 1.5; flour = 250 g × 1.5 = 375 g
e for 2 people, the multiplier is 2 ÷ 4 = 0.5, so the flour needed is 250 g × 0.5 = 125 g.
减少食谱时，倍数小于 1。如果同一食谱供 2 人食用，倍数是 2 ÷ 4 = 0.5，因此所需面粉为 250 × 0.5 = 125 克。
9. Ratio and Fractions | 比与分数
A ratio can be written as fractions of the total. In the ratio 3 : 2, the total parts are 5, so the first quantity represents 3/5 of the whole and the second represents 2/5. This helps when a question asks for one part as a fraction of the total.
比可以写成分数形式，表示占总量的多少。在比 3 : 2 中，总份数是 5，因此第一个量占整体的 3/5，第二个量占 2/5。当题目要求将一部分表示成整体的几分之几时，这一方法很有帮助。
3 : 2 means 3/5 and 2/5 of the total
For example, if a class contains boys and girls in the ratio 3 : 2, then 3/5 of the class are boys and 2/5 are girls. If there are 30 students, there are 18 boys and 12 girls.
例如，如果一个班级中男生和女生的比是 3 : 2，那么班级的 3/5 是男生，2/5 是女生。如果共有 30 名学生，则有 18 名男生和 12 名女生。
10. Common Mistakes and Exam Tips | 常见错误与应试技巧
A common mistake is changing the order of the ratio or forgetting to add the parts before sharing. Always simplify first, write the ratio in the correct order and check that the shares add back to the original total. In Checkpoint questions, show every step clearly because method marks are awarded.
一个常见错误是改变比的顺序，或者在分配前忘记先求总份数。解题时一定要先化简，按正确顺序写出比，并检查各份额加回去是否等于原来的总量。在 Checkpoint 考试中，要把每一步写清楚，因为过程分也会计入。
- English: Always write the ratio in the order given in the question.
- 中文：始终按照题目给出的顺序写出比。
- English: Add parts to find the total before sharing a quantity.
- 中文：分配数量前先求出总份数。
- English: Check your answer by adding the shares together.
- 中文：把各份额加起来检查答案。
- English: Use the same units for both quantities before simplifying a ratio.
- 中文：化简比之前先统一两个量的单位。
Finally, do not confuse ratio with fraction of a part. A ratio 3 : 2 compares two separate quantities, whereas the fraction 3/2 compares the first quantity directly to the second as one value divided by another.
最后，不要将比与部分分数混淆。比 3 : 2 比较两个独立的量，而分数 3/2 是将第一个量直接除以第二个量，作为一个值来比较。
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