📚 Ratio, Proportion and Scale Drawings: Cambridge KS3 Page 278 Practice | 比、比例与比例图：剑桥KS3第278页练习
This article explains the core ratio and proportion skills that frequently appear in Cambridge KS3 mathematics, including the type of question found on page 278. You will learn how to interpret ratio notation, simplify ratios, share quantities, solve direct and inverse proportion problems, and apply scale factors to maps and enlargements.
本文讲解剑桥 KS3 数学中经常出现的比与比例的核心技能，包括第278页常见的题型。你将学习如何解读比的表示法、化简比、按比分配、解决正比例和反比例问题，以及将比例因子应用到地图和图形放大中。
1. Understanding Ratio Notation | 理解比的表示法
A ratio compares two or more quantities of the same kind. It is written using a colon, for example 2:3. The order of the numbers is very important: 2:3 is not the same as 3:2. Ratios do not have units because they compare relative sizes.
比用于比较两个或两个以上同类量。它用冒号书写，例如 2:3。数字的顺序非常重要：2:3 与 3:2 不同。比没有单位，因为它比较的是相对大小。
If a class has 12 boys and 8 girls, the ratio of boys to girls is written as 12:8. This ratio can be simplified later, but the order must always be boys first and girls second.
如果一个班级有12名男生和8名女生，男生与女生的比写作 12:8。这个比之后可以化简，但顺序必须始终是男生在前、女生在后。
Ratios can also compare more than two quantities. For example, a concrete mixture may use cement, sand and gravel in the ratio 1:2:3. This means for every 1 part of cement there are 2 parts of sand and 3 parts of gravel.
比也可以比较两个以上的量。例如，混凝土混合物中水泥、沙子和石子的比可能是 1:2:3。这意味着每1份水泥对应2份沙子和3份石子。
2. Simplifying Ratios | 化简比
To simplify a ratio, divide every part by the highest common factor (HCF). For example, the ratio 24:36 can be simplified because both numbers are divisible by 12. Dividing both parts by 12 gives 2:3.
化简比时，用最大公因数去除以每一个部分。例如，比 24:36 可以化简，因为两个数都能被12整除。两部分都除以12得到 2:3。
If a ratio contains decimals or fractions, multiply all parts by a common number to produce whole numbers first. For 0.5:1.5, multiply both sides by 2 to obtain 1:3. For 1/2:3/4, multiply both sides by 4 to obtain 2:3.
如果比包含小数或分数，先将所有部分乘以同一个数化成整数。对于 0.5:1.5，两边同时乘以2得到 1:3。对于 1/2:3/4，两边同时乘以4得到 2:3。
Always check that the simplified ratio has no common factor other than 1. A ratio like 6:9 is not fully simplified because both numbers share a factor of 3. The simplified form is 2:3.
始终检查化简后的比除1以外没有其他公因数。像 6:9 这样的比没有完全化简，因为两个数都有公因数3。化简后的形式是 2:3。
- Do not confuse the order of the numbers.
- Do not leave decimals or fractions in a simplified ratio.
- Do not forget to divide every part by the same number.
- 不要混淆数字的顺序。
- 化简后的比中不要留下小数或分数。
- 不要忘记每一部分都要除以同一个数。
3. Sharing in a Given Ratio | 按给定比分配
To share a quantity in a given ratio, first find the total number of parts. For a ratio a:b, the total number of parts is a + b. 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 number in the ratio.
按给定比分配数量时，先求出总份数。对于比 a:b，总份数是 a + b。然后用总量除以总份数，得到一份的值。最后用一份的值乘以比中的每一个数。
Example: Share £60 in the ratio 2:3. The total number of parts is 2 + 3 = 5. One part is £60 ÷ 5 = £12. The two shares are 2 × £12 = £24 and 3 × £12 = £36.
例如：按 2:3 分配 60 英镑。总份数是 2 + 3 = 5。一份是 60 ÷ 5 = 12 英镑。两份分别是 2 × 12 = 24 英镑和 3 × 12 = 36 英镑。
total parts are 8. One part is 120 ÷ 8 = 15 kg, so the shares are 15 kg, 45 kg and 60 kg.
对于像 1:3:4 这样的三部分比，将三个数字相加得到总份数。如果将120千克按 1:3:4 分配，总份数是8。一份是 120 ÷ 8 = 15 千克，因此三份分别是15千克、45千克和60千克。
4. Ratio and Fractions | 比与分数
A ratio can be converted into fractions. If a whole is split in the ratio 3:5, the total number of parts is 8. The first part represents 3/8 of the total, and the second part represents 5/8 of the total.
比可以转化为分数。如果总量按 3:5 分配，总份数是8。第一部分占总量的 3/8，第二部分占总量的 5/8。
| Ratio | Total parts | First fraction | Second fraction |
| 3:5 | 8 | 3/8 | 5/8 |
| 5:7 | 12 | 5/12 | 7/12 |
the ratio is 5:7, the first part is 5/12 × 72 = 30, and the second part is 7/12 × 72 = 42.
你可以用这些分数来求实际数量。如果总量是72，比是 5:7，则第一部分是 5/12 × 72 = 30，第二部分是 7/12 × 72 = 42。
5. Direct Proportion | 正比例
Two quantities are in direct proportion if their ratio remains constant. As one quantity increases, the other increases at the same rate. For example, if 3 apples cost £1.20, then 6 apples cost £2.40 because the number of apples has doubled and the cost has doubled.
如果两个量的比保持恒定，它们成正比例。一个量增加，另一个量以相同的速率增加。例如，如果3个苹果花费1.20英镑，那么6个苹果花费2.40英镑，因为苹果数量翻倍，价格也翻倍。
ortion problems using the unitary method. First find the value of 1 unit. If 3 apples cost £1.20, one apple costs £0.40. Then 10 apples cost 10 × £0.40 = £4.00.
你可以用单位法来解决正比例问题。先求1个单位的值。如果3个苹果花费1.20英镑，一个苹果花费0.40英镑。那么10个苹果花费 10 × 0.40 = 4.00 英镑。
in. The relationship can be written as y = kx, where k is the constant of proportionality.
在正比例中，y 对 x 的图像是一条经过原点的直线。这种关系可以写成 y = kx，其中 k 是比例常数。
6. Inverse Proportion | 反比例
Two quantities are inversely proportional if their product remains constant. When one quantity doubles, the other halves. For example, if 4 workers take 6 days to build a wall, then 8 workers take 3 days because the number of workers has doubled and the time has halved.
如果两个量的乘积保持恒定，它们成反比例。一个量加倍，另一个量减半。例如，如果4个工人建墙需要6天，那么8个工人需要3天，因为工人数量翻倍，时间减半。
6 days, the product is 4 × 6 = 24 worker-days. For 8 workers, the time is 24 ÷ 8 = 3 days.
解决反比例问题时，先将第一对数值相乘得到常数乘积。然后用这个常数除以另一个量的已知值。对于4个工人和6天，乘积是 4 × 6 = 24 个工日。对于8个工人，时间是 24 ÷ 8 = 3 天。
7. Scale Drawings and Maps | 比例图与地图
A scale drawing shows a real object with all lengths reduced or enlarged by the same ratio. A map scale such as 1:50000 means that 1 cm on the map represents 50000 cm in real life.
比例图按相同比例缩小或放大实物的长度。地图比例尺如 1:50000 表示地图上的1厘米代表实际中的50000厘米。
distance is 4 × 50000 = 200000 cm. Convert this to kilometres: 200000 cm = 2000 m = 2 km.
求实际距离时，将图上距离乘以比例因子。对于 1:50000 的比例尺和4厘米的图上距离，实际距离是 4 × 50000 = 200000 厘米。将其转换为千米：200000 厘米 = 2000 米 = 2 千米。
Always convert units carefully. Common
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