Ratio and Proportion | 比与比例

📚 Ratio and Proportion | 比与比例

A ratio compares two or more quantities in the same units. It shows how much of one quantity there is compared with another quantity. For example, if a fruit drink mixes 2 parts orange juice with 3 parts water, the ratio is written as 2:3. This means that for every 2 units of orange juice, there are 3 units of water.

比用于比较两个或多个同单位的数量。它表示一个数量相对于另一个数量有多少。例如，如果一种果汁饮料按 2 份橙汁和 3 份水混合，这个比就写作 2:3。这意味着每对应 2 份橙汁，就有 3 份水。

1. What is a Ratio? | 什么是比？

A ratio has no units and is written with a colon between the numbers. The order is very important: a ratio of 2:3 is not the same as 3:2. In the example above, 2:3 tells us that the drink has 2 parts juice and 3 parts water, while 3:2 would mean 3 parts juice and 2 parts water.

比没有单位，并用冒号分隔数字。顺序非常重要：2:3 与 3:2 是不同的。在上面的例子中，2:3 表示饮料中有 2 份果汁和 3 份水，而 3:2 则表示有 3 份果汁和 2 份水。

Ratios can compare more than two quantities. For example, a ratio of 1:2:4 could compare cement, sand and gravel in a concrete mix. The ratio tells us the relative size of each part, but it does not tell us the actual amount unless we know one of the real quantities.

比可以比较两个以上的数量。例如，1:2:4 可以表示混凝土中的水泥、沙子和石子的比例。这个比告诉我们每一部分的相对大小，但除非知道其中一个实际数量，否则它不能告诉我们实际的总量。

2. Simplifying Ratios | 化简比

To simplify a ratio, divide all parts by their highest common factor. For example, the ratio 8:12 can be simplified by dividing both numbers by 4, giving 2:3. A simplified ratio is usually easier to read and compare with other ratios.

化简比时，要用所有部分的最大公因数去除。例如，8:12 可以通过把两个数都除以 4 来化简，得到 2:3。化简后的比通常更容易阅读，也更容易与其他比进行比较。

If the ratio contains decimals or fractions, multiply all parts by the same number to make them whole numbers first. For example, 0.5:1.5 can be multiplied by 10 to give 5:15, which then simplifies to 1:3.

如果比中含有小数或分数，先把所有部分乘以同一个数使它们变成整数。例如，0.5:1.5 可以先乘以 10 得到 5:15，再化简为 1:3。

Original ratio	Simplified ratio
8:12	2:3
15:25	3:5
0.5:2	1:4

3. Equivalent Ratios | 等价比

Equivalent ratios are ratios that show the same relationship. You can find an equivalent ratio by multiplying or dividing both parts by the same number. For instance, 1:2 is equivalent to 2:4, 3:6, and 10:20.

等价比是表示相同关系的比。你可以把比的两个部分同时乘或除以同一个数来得到等价比。例如，1:2 与 2:4、3:6 和 10:20 都是等价比。

This idea is useful when comparing mixtures or working with recipes. If a recipe uses flour and sugar in the ratio 3:1, then using 6 cups of flour and 2 cups of sugar keeps the same ratio. The taste and texture of the mixture will be the same if all ingredients are scaled equally.

这个概念在比较混合物或处理配方时很有用。如果一个配方中面粉和糖的比例是 3:1，那么使用 6 杯面粉和 2 杯糖就保持了相同的比。如果所有原料都按相同比例缩放，混合物的味道和口感不会改变。

4. Dividing a Quantity in a Given Ratio | 按给定比例分配数量

To divide a quantity in a given ratio, first add the parts of the ratio to find the total number of parts. Then divide the total quantity by this number to find the value of one part. Finally, multiply the value of one part by each ratio number.

要按给定比例分配一个数量，首先把比中的各部分相加，求出总份数。然后用总数量除以这个总份数，求出一份的值。最后用一份的值分别乘以比中的每个数。

For example, divide £80 in the ratio 3:5. The total number of parts is 3 + 5 = 8. One part is £80 ÷ 8 = £10. Therefore the two amounts are 3 × £10 = £30 and 5 × £10 = £50.

例如，按 3:5 的比例分配 80 英镑。总份数是 3 + 5 = 8。一份是 80 ÷ 8 = 10 英镑。因此两个金额分别是 3 × 10 = 30 英镑和 5 × 10 = 50 英镑。

Total parts = 3 + 5 = 8 One part = 80 ÷ 8 = 10 Amounts = 3×10 and 5×10 = £30 and £50

5. Ratio and Fractions | 比与分数

A ratio can be written as a fraction to show what proportion of the total belongs to each part. In the ratio 3:5, there are 8 parts in total. The first quantity is 3/8 of the total, and the second quantity is 5/8 of the total.

比可以写成分数，用来表示每一部分占总量的比例。在 3:5 中，总共有 8 份。第一个数量占总量的 3/8，第二个数量占总量的 5/8。

This is useful when answering questions such as ‘What fraction of the class are boys?’ If the ratio of boys to girls is 4:5, the total number of parts is 9, so boys make up 4/9 of the class and girls make up 5/9.

这在回答 ‘班上有多少比例是男生？’ 这类问题时很有用。如果男生和女生的比是 4:5，总份数是 9，所以男生占全班的 4/9，女生占 5/9。

6. Direct Proportion | 正比例

Two quantities are in direct proportion when they increase or decrease at the same rate. If one quantity doubles, the other doubles. If one quantity is halved, the other is also halved. This can be written as y = kx, where k is the constant of proportionality.

当两个数量以相同的速率增加或减少时，它们成正比例。如果一个数量翻倍，另一个也翻倍；如果一个数量减半，另一个也减半。这可以写作 y = kx，其中 k 是比例常数。

For example, if 5 pens cost £2, then 10 pens cost £4 and 20 pens cost £8. The cost is directly proportional to the number of pens, and the constant of proportionality is 2 ÷ 5 = 0.4, so cost = 0.4 × number of pens.

例如，如果 5 支笔花费 2 英镑，那么 10 支笔花费 4 英镑，20 支笔花费 8 英镑。费用与笔的数量成正比例，比例常数是 2 ÷ 5 = 0.4，所以费用 = 0.4 × 笔的数量。

y = kx k = y ÷ x When x = 5, y = 2, so k = 2 ÷ 5 = 0.4

7. Best Buys and Unit Rates | 最佳购买与单位比率

Ratios and unit rates help us compare prices. To find the best buy, calculate the price per unit of measurement, such as price per gram, price per litre or price per item. The option with the lower unit price is usually better value.

比和单位比率可以帮助我们比较价格。要找到最划算的商品，需要计算每单位数量的价格，例如每克、每升或每件商品的价格。单位价格较低的选项通常更划算。

3 kg of potatoes costs £2.40, so the unit rate is £2.40 ÷ 3 = £0.80 per kg.

3 公斤土豆花费 2.40 英镑，所以单位价格是 2.40 ÷ 3 = 每公斤 0.80 英镑。
5 kg of potatoes costs £3.50, so the unit rate is £3.50 ÷ 5 = £0.70 per kg. The 5 kg bag is the better buy.

5 公斤土豆花费 3.50 英镑，所以单位价格是 3.50 ÷ 5 = 每公斤 0.70 英镑。5 公斤装更划算。

8. Maps and Scale Drawings | 地图与比例图

A scale ratio shows how a drawing or map compares with the real object. A scale of 1:50000 means that 1 cm on the map represents 50000 cm in real life. You can convert the real measurement into metres or kilometres to make it easier to understand.

比例尺表示图纸或地图与实际物体之间的比较关系。比例尺 1:50000 表示地图上的 1 厘米代表实际中的 50000 厘米。你可以把实际长度转换成米或千米，这样更容易理解。

For example, if a map has a scale of 1:50000, then 1 cm represents 50000 cm = 500 m = 0.5 km. A road measured as 3 cm on the map represents 3 × 0.5 = 1.5 km in real life.

例如，如果地图的比例尺是 1:50000，那么 1 厘米代表 50000 厘米 = 500 米 = 0.5 千米。地图上一条 3 厘米长的道路代表实际中的 3 × 0.5 = 1.5 千米。

9. Common Mistakes | 常见误区

One common mistake is reversing the order of a ratio. If the ratio of apples to bananas is 2:3, it is wrong to write 3:2 unless the question asks for bananas to apples. Always check which quantity is mentioned first.

一个常见错误是颠倒比的顺序。如果苹果与香蕉的比是 2:3，就不能写成 3:2，除非题目要求的是香蕉与苹果的比。一定要检查题目中先提到的是哪个数量。

Another mistake is forgetting to add all parts when dividing a quantity. In the ratio 2:3:5, the total number of parts is 10, not 8 or 5. Adding incorrectly will give the wrong value for one part and therefore the wrong final amounts.

另一个错误是在分配数量时忘记把所有部分相加。在 2:3:5 中，总份数是 10，而不是 8 或 5。加错总份数会导致每份的价值计算错误，从而得出错误的最终分配结果。

10. Exam-style Practice | 考试题型练习

Try these examples. First, divide £72 in the ratio 4:5. The total parts are 9, one part is £72 ÷ 9 = £8, so the amounts are £32 and £40.

试做以下例题。首先，按 4:5 的比例分配 72 英镑。总份数是 9，一份是 72 ÷ 9 = 8 英镑，所以两份金额分别是 32 英镑和 40 英镑。

Second, a drink is made with cordial and water in the ratio 1:6. How much cordial is needed for 2.1 litres of drink? The total parts are 7, so one part is 2.1 ÷ 7 = 0.3 litres. The cordial is 0.3 litres, and the water is 1.8 litres.

第二题，一种饮料由浓缩果汁和水按 1:6 的比例制成。制作 2.1 升饮料需要多少浓缩果汁？总份数是 7，所以一份是 2.1 ÷ 7 = 0.3 升。浓缩果汁为 0.3 升，水为 1.8 升。

11. Summary and Key Formulae | 总结与关键公式

In summary, a ratio compares quantities without units. Simplify ratios by dividing by the highest common factor. To divide a quantity, add the ratio parts, find one part, then multiply. Use direct proportion when two quantities change at the same rate.

总之，比是无单位地比较数量。化简比时要除以最大公因数。分配数量时，先把比的部分相加，求出一份的值，再分别相乘。当两个数量以相同速率变化时，使用正比例。

Total parts = sum of ratio numbers One part = total quantity ÷ total parts y = kx

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Ratio and Proportion | 比与比例