📚 Ratio and Proportion: KS3 Cambridge Revision Guide | 比率与比例：KS3 剑桥复习指南
Ratio and proportion appear throughout the Cambridge KS3 Mathematics course and are used to compare quantities, share amounts fairly, and solve real-life problems involving scale, speed, and cost. This revision guide covers the key ideas you need to understand, including simplifying ratios, sharing in a given ratio, and using the unitary method for direct proportion.
比率和比例贯穿剑桥 KS3 数学课程，用于比较数量、公平分配金额，以及解决涉及比例尺、速度和成本的实际问题。本复习指南涵盖你需要掌握的关键概念，包括化简比率、按给定比率分配以及使用单位法解决正比例问题。
1. What is a ratio? | 什么是比率？
A ratio compares two or more quantities. It shows how many times one quantity contains another. For example, if a class has 8 boys and 12 girls, the ratio of boys to girls is 8 : 12. We read this as '8 to 12'.
比率用于比较两个或多个量。它表示一个量包含另一个量的多少倍。例如，如果班上有 8 名男生和 12 名女生，男生与女生的比率是 8 : 12。我们将其读作 '8 比 12'。
The order of a ratio is very important. The ratio of boys to girls is 8 : 12, but the ratio of girls to boys is 12 : 8. These two ratios are not the same because they compare the quantities in opposite directions.
比率的顺序非常重要。男生与女生的比率是 8 : 12，而女生与男生的比率是 12 : 8。这两个比率并不相同，因为它们以相反的方向比较数量。
Ratios can also compare more than two quantities. For example, if a recipe uses flour, butter, and sugar in the ratio 4 : 2 : 1, it means four parts flour, two parts butter, and one part sugar.
比率还可以比较两个以上的量。例如，如果一份食谱使用面粉、黄油和糖的比率为 4 : 2 : 1，这意味着四份面粉、两份黄油和一份糖。
2. Simplifying ratios | 化简比率
Ratios can be simplified in the same way as fractions. Divide both sides by the highest common factor (HCF). For example, to simplify 12 : 18, the HCF of 12 and 18 is 6, so divide both sides by 6.
比率可以像分数一样化简。将两边同时除以最大公因数 (HCF)。例如，要化简 12 : 18，12 和 18 的最大公因数是 6，因此两边同时除以 6。
12 : 18 → 2 : 3
Always check that the simplified ratio uses only whole numbers. A ratio is fully simplified when there is no common factor other than 1 between the parts.
始终检查化简后的比率是否只使用整数。当各部分之间除了 1 之外没有其他公因数时，比率即为最简形式。
If quantities have different units, convert them to the same unit before simplifying. For example, 30 cm : 1 m must be written as 30 cm : 100 cm first. This then simplifies to 3 : 10.
如果数量带有不同单位，请先将其转换为相同单位再化简。例如，30 cm : 1 m 必须先写成 30 cm : 100 cm，然后化简为 3 : 10。
Ratios with decimals or fractions can also be simplified by multiplying all parts by the same number to make them whole numbers. For example, 0.5 : 2 can be multiplied by 2 to get 1 : 4.
带有小数或分数的比率也可以通过将所有部分乘以同一个数化为整数后进行化简。例如，0.5 : 2 可以乘以 2 得到 1 : 4。
3. Equivalent ratios | 等效比率
Equivalent ratios are made by multiplying or dividing both sides of a ratio by the same non-zero number. They represent the same relationship between quantities.
等效比率通过将比率的各边同时乘以或除以同一个非零数得到。它们表示数量之间相同的关系。
For example, 1 : 2, 2 : 4, 3 : 6, and 10 : 20 are all equivalent because each one can be obtained from 1 : 2 by multiplying both sides by 2, 3, or 10.
例如，1 : 2、2 : 4、3 : 6 和 10 : 20 都是等效的，因为它们都可以通过将 1 : 2 的两边分别乘以 2、3 或 10 得到。
term. For example, if 3 : 5 = 9 : x, the first term is multiplied by 3, so x = 5 × 3 = 15.
要在等效比率中找到缺失项，先确定第一项乘以或除以了什么数，然后对第二项进行同样的操作。例如，如果 3 : 5 = 9 : x，第一项乘以了 3，因此 x = 5 × 3 = 15。
Understanding equivalent ratios is essential for solving problems where you need to scale quantities up or down while keeping the same proportion.
理解等效比率对于解决需要在保持相同比例的情况下放大或缩小数量的问题至关重要。
4. Ratio and fractions | 比率与分数
of parts is 3 + 5 = 8. Boys make up 3/8 of the group, and girls make up 5/8 of the group.
比率可以表示成整体的分数。如果男生与女生的比率是 3 : 5，总份数是 3 + 5 = 8。男生占整体的 3/8，女生占整体的 5/8。
This idea is useful when you know the total and need to find the actual number in each category. For example, if there are 40 students in total, the number of boys is:
当你知道总数并需要找出每个类别的实际数量时，这个想法非常有用。例如，如果总共有 40 名学生，男生人数为：
3/8 × 40 = 15
, the number of girls is 5/8 × 40 = 25. Notice that 15 + 25 = 40, which matches the total.
同样，女生人数为 5/8 × 40 = 25。注意 15 + 25 = 40，与总数相符。
Converting between ratios and fractions also helps in percentage problems. A ratio of 2 : 3 means the first part is 2/5 of the whole, which is 40%.
比率与分数之间的转换也有助于解决百分比问题。比率为 2 : 3 表示第一部分占整体的 2/5，即 40%。
5. Sharing in a given ratio | 按给定比率分配
To share an amount in a given ratio, first add the parts to find the total number of parts. Then divide the amount by this total to find the value of one part. Finally, multiply this value by each part of the ratio.
要按给定比率分配一个数量，首先将各份数相加得到总份数。然后用总数量除以总份数，得到一份的值。最后用一份的值乘以比率中的每一份。
For example, share £60 in the ratio 2 : 3. The total number of parts is 2 + 3 = 5. One part is £60 ÷ 5 = £12. The first share is 2 × £12 = £24, and the second share is 3 × £12 = £36.
例如，将 £60 按 2 : 3 的比率分配。总份数是 2 + 3 = 5。一份是 £60 ÷ 5 = £12。第一份是 2 × £12 = £24，第二份是 3 × £12 = £36。
Always check your answer by adding the shares together. Here £24 + £36 = £60, which matches the original amount, so the answer is correct.
始终通过将各份相加来检查答案。这里 £24 + £36 = £60，与原始金额一致，因此答案正确。
When the ratio has more than two parts, the method is the same. For example, share £90 in the ratio 1 : 2 : 3. Total parts = 1 + 2 + 3 = 6, so one part is £90 ÷ 6 = £15. The shares are £15, £30, and £45.
当比率包含三个或更多份时，方法相同。例如，将 £90 按 1 : 2 : 3 的比率分配。总份数 = 1 + 2 + 3 = 6，因此一份是 £90 ÷ 6 = £15。各份为 £15、£30 和 £45。
6. Proportion and the unitary method | 比例与单位法
Proportion compares a part to the whole, often written as a fraction, decimal, or percentage. It can also describe how one quantity changes with another. The unitary method is a powerful tool for proportion problems because it finds the value of one unit first.
比例将一个部分与整体进行比较，通常写成分数、小数或百分比。它还可以描述一个量如何随另一个量变化。单位法是解决比例问题的有力工具，因为它先求出一个单位的值。
For example, if 5 pens cost £3.50, find the cost of 8 pens. First find the cost of 1 pen: £3.50 ÷ 5 = £0.70. Then multiply by 8: £0.70 × 8 = £5.60.
例如，如果 5 支笔花费 £3.50，求 8 支笔的价格。先求出 1 支笔的价格：£3.50 ÷ 5 = £0.70。然后乘以 8：£0.70 × 8 = £5.60。
The unitary method works because the relationship is proportional. If the number of pens doubles, the cost also doubles, so finding one unit is enough to find any other number.
单位法之所以有效，是因为这种关系是成比例的。如果笔的数量翻倍，价格也翻倍，因此求出一支笔的价格就足以求出任何其他数量的价格。
00 g, one book weighs 900 ÷ 6 = 150 g. Ten books would weigh 150 × 10 = 1500 g, or 1.5 kg.
你还可以将这种方法用于减少的数量。例如，如果 6 本相同的书重 900 g，那么一本书重 900 ÷ 6 = 150 g。十本书重 150 × 10 = 1500 g，即 1.5 kg。
7. Direct proportion | 正比例
Two quantities are in direct proportion if they increase or decrease at the same rate. When one quantity doubles, the other also doubles. When one quantity is halved, the other is also halved. The ratio between corresponding values remains constant.
如果两个量以相同的速率增加或减少，则它们成正比例。当一个量翻倍时，另一个量也翻倍。当一个量减半时，另一个量也减半。对应值之间的比率保持不变。
For example, if a car travels at a constant speed, the distance travelled and the time taken are in direct proportion. If the speed is 60 km/h, then in 1 hour it travels 60 km, in 2 hours it travels 120 km, and in 3 hours it travels 180 km.
例如，如果汽车以恒定速度行驶，行驶的路程和所用的时间成正
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