Ratio and Proportion | 比率与比例

📚 Ratio and Proportion | 比率与比例

Ratio and proportion are fundamental concepts in mathematics that describe relationships between quantities. A ratio compares two or more numbers, showing how much of one thing there is compared to another. Proportion, on the other hand, tells us that two ratios are equal. In KS3 Cambridge Mathematics, students learn to simplify ratios, divide quantities in a given ratio, and solve problems involving direct proportion, scale factors, and rates. Mastering these skills helps us make sense of everyday situations, from mixing ingredients in a recipe to working out speeds and currency exchanges. This article will guide you through the key ideas step by step, with clear examples and bilingual explanations.

比率与比例是数学中描述数量之间关系的基本概念。比率比较两个或多个数字，显示一种事物相对于另一种事物的多少。比例则告诉我们两个比率相等。在剑桥KS3数学中，学生将学习如何化简比、按给定比例分配数量，并解决涉及正比例、比例因子和速率的问题。掌握这些技能有助于我们理解日常生活中的各种情境，从调配食谱中的配料到计算速度和货币兑换。本文将逐步引导你掌握这些关键概念，配有清晰的例子和中英双语解释。

1. Understanding Ratio | 理解比率

A ratio is a way to compare two or more quantities. We write a ratio using a colon, for example, 3 : 2. This means for every 3 parts of the first quantity, there are 2 parts of the second. The order is very important: a ratio of 3 : 2 is not the same as 2 : 3. Ratios have no units because they compare quantities with the same unit. For instance, if a class has 12 boys and 8 girls, the ratio of boys to girls is 12 : 8, which simplifies to 3 : 2.

比率是比较两个或多个数量的一种方式。我们使用冒号来表示比，例如 3 : 2。这意味着第一个量每有 3 份，第二个量就有 2 份。顺序非常重要：3 : 2 与 2 : 3 不一样。比率没有单位，因为它们比较的是相同单位的量。例如，如果一个班级有 12 个男生和 8 个女生，男生与女生的比率是 12 : 8，化简后为 3 : 2。

We often use ratios to describe real-life scenarios. Imagine mixing orange squash with water. If the instructions say ‘mix 1 part squash to 4 parts water’, the ratio of squash to water is 1 : 4. This tells us that for every 1 ml of squash, we need 4 ml of water. The total amount of drink would then be made of 5 parts altogether.

我们经常用比率来描述现实生活中的场景。想象一下用浓缩橙汁兑水。如果说明书上写着“将1份浓缩液与4份水混合”，那么浓缩液与水的比率就是 1 : 4。这告诉我们每需要 1 毫升浓缩液，就需要 4 毫升水。最终饮料的总量总共由 5 份组成。

2. Simplifying Ratios | 化简比率

To simplify a ratio, we divide each part by the highest common factor (HCF) of all the numbers. For example, the ratio 15 : 10 : 5 can be simplified by dividing each term by 5, giving 3 : 2 : 1. Simplifying ratios makes them easier to understand, just like simplifying fractions. It is essential that all parts of the ratio are whole numbers in their simplest form.

要化简比率，我们需要将每一项除以所有数字的最大公因数（HCF）。例如，比率 15 : 10 : 5 可以通过每项除以 5 化简为 3 : 2 : 1。化简比率就像化简分数一样，能使其更易于理解。化简后比率的每一项都应为最简形式的整数，这一点很重要。

Sometimes a ratio may contain decimals or fractions. To simplify, we first multiply all terms by the same power of 10 (for decimals) or by the lowest common multiple of denominators (for fractions) to get whole numbers, then reduce. For instance, 0.8 : 1.2 becomes 8 : 12 after multiplying by 10, which simplifies to 2 : 3.

有时比率可能包含小数或分数。要化简，我们可以先将所有项乘以相同的 10 的幂（对于小数）或分母的最小公倍数（对于分数），得到整数后再进行约分。例如，0.8 : 1.2 乘以 10 变成 8 : 12，化简后为 2 : 3。

3. Equivalent Ratios | 等价比率

Equivalent ratios are ratios that represent the same relationship between quantities. We can find equivalent ratios by multiplying or dividing each term by the same non-zero number. Just like equivalent fractions, the value of the ratio does not change. For example, 1 : 2 is equivalent to 2 : 4, 3 : 6, and 10 : 20.

等价比率是指表示数量之间相同关系的比率。我们可以通过将每一项乘以或除以同一个非零数来找到等价比率。就像等价分数一样，比率的值不会改变。例如，1 : 2 等价于 2 : 4、3 : 6 和 10 : 20。

Finding equivalent ratios is useful when we need to scale a quantity up or down while keeping the same proportions. If a model car is built to a scale of 1 : 50, then the real car’s length of 450 cm would be represented by 9 cm on the model (because 450 ÷ 50 = 9). Here the ratio 9 : 450 simplifies to 1 : 50, confirming the equivalence.

当我们需要按相同比例放大或缩小某个量时，找到等价比率非常有用。如果一辆模型汽车的比例为 1 : 50，那么真车 450 厘米的长度在模型上就是 9 厘米（因为 450 ÷ 50 = 9）。这里 9 : 450 化简为 1 : 50，证明了等价关系。

4. Dividing a Quantity in a Given Ratio | 按给定比率分配数量

To share a quantity in a given ratio, we first find the total number of parts by adding all the terms of the ratio. Then we divide the total quantity by the total number of parts to find the value of one part. Finally, we multiply this value by each term of the ratio to find the individual shares.

要按给定比率分配一个数量，我们首先将比率的所有项相加，得出总份数。然后用总数量除以总份数，得出每一份的值。最后，将这个值分别乘以比率中的每一项，得出各自应得的数量。

For example, suppose Jack, Kim, and Li share £300 in the ratio 3 : 4 : 5. The total number of parts is 3 + 4 + 5 = 12. One part is worth £300 ÷ 12 = £25. Therefore Jack gets 3 × £25 = £75, Kim gets 4 × £25 = £100, and Li gets 5 × £25 = £125. Always check that the individual shares add up to the total original amount.

举个例子，假设 Jack、Kim 和 Li 按 3 : 4 : 5 的比例分享 300 英镑。总份数为 3 + 4 + 5 = 12。一份的价值是 300 ÷ 12 = 25 英镑。因此 Jack 得到 3 × 25 = 75 英镑，Kim 得到 4 × 25 = 100 英镑，Li 得到 5 × 25 = 125 英镑。务必检查每人得到的份额之和等于原始总金额。

5. Introduction to Proportion | 比例入门

Proportion tells us that two ratios are equal. If two quantities are in direct proportion, then as one quantity increases, the other increases at the same rate. We can write a proportion as an equation of two ratios, such as a : b = c : d, which is read as ‘a is to b as c is to d’. In this case, the cross-products are equal: a × d = b × c.

比例告诉我们两个比率相等。如果两个量成正比例，那么当一个量增加时，另一个量以相同的速率增加。我们可以将比例写为两个比率相等的等式，如 a : b = c : d，读作“a 比 b 等于 c 比 d”。在这种情况下，交叉乘积相等：a × d = b × c。

For instance, if 3 pens cost 45 pence, we can set up a proportion to find the cost of 8 pens. The ratio of pens to cost is 3 : 45 and 8 : x . Solving 3 × x = 8 × 45 gives x = 120 pence. This method is sometimes called the unitary method or cross‑multiplication.

例如，如果 3 支笔售价 45 便士，我们可以列出比例来求 8 支笔的价格。笔与价格的比例为 3 : 45 和 8 : x。解方程 3 × x = 8 × 45 得出 x = 120 便士。这种方法有时被称为归一法或交叉相乘法。

6. Direct Proportion and the Constant of Proportionality | 正比例与比例常数

When two quantities, y and x, are in direct proportion, we can express this relationship as y = k x, where k is the constant of proportionality. The graph of a direct proportion is a straight line passing through the origin. The value of k tells us how steep the line is; for example, if y = 3 x, then for every 1 unit increase in x, y increases by 3 units.

当两个量 y 和 x 成正比例时，我们可以用 y = k x 来表示这种关系，其中 k 为比例常数。正比例关系的图像是一条过原点的直线。k 值告诉我们直线的陡峭程度；例如，如果 y = 3 x，那么 x 每增加 1 个单位，y 就增加 3 个单位。

To find k, we can use any pair of corresponding values (x, y) and solve k = y ÷ x. Suppose a car uses fuel at a steady rate: it travels 180 km using 15 litres. The constant of proportionality (km per litre) is k = 180 ÷ 15 = 12. So the formula is distance = 12 × litres. This constant is often called a rate.

要找到 k，我们可以使用任意一组对应的值 (x, y) 并解出 k = y ÷ x。假设一辆汽车以稳定的速率消耗燃油：行驶 180 公里用掉 15 升汽油。比例常数（每升公里数）为 k = 180 ÷ 15 = 12。因此公式为距离 = 12 × 升数。这个常数常被称为速率。

7. Using the Unitary Method | 使用归一法

The unitary method is a powerful technique for solving proportion problems. We first find the value of one unit (the ‘per one’ quantity) and then multiply it by the desired number of units. This approach works for direct proportion, rates, and best‑buy comparisons. It is especially helpful when the numbers do not easily form a simple proportion equation.

归一法是解决比例问题的一种很有效的方法。我们先求出一个单位的值（“每个”的数量），然后再乘以我们想要的单位数。这种方法适用于正比例、速率和最佳购买比较。当数字不易构成简单的比例等式时，这种方法尤其有用。

Example: A machine prints 1200 pages in 5 minutes. How many pages can it print in 8 minutes? First find the rate per minute: 1200 ÷ 5 = 240 pages per minute. Then multiply by 8: 240 × 8 = 1920 pages. Notice that we used the constant of proportionality without explicitly writing an equation.

例子：一台机器在 5 分钟内打印 1200 页。它在 8 分钟内能打印多少页？先求每分钟的速率：1200 ÷ 5 = 240 页每分钟。然后乘以 8：240 × 8 = 1920 页。请注意，我们使用了比例常数，而无需明确写出方程。

8. Scale Drawings and Maps | 比例图与地图

A scale drawing or map uses a ratio to represent real‑life distances in a smaller size. The scale is usually given as a ratio, such as 1 : 25000, meaning that 1 cm on the map represents 25000 cm (or 250 m) in reality. We can use direct proportion to convert between map distances and real distances.

比例图或地图使用比率将真实距离缩小表示。比例通常以比率的形式给出，例如 1 : 25000，意味着地图上的 1 厘米代表实际距离 25000 厘米（即 250 米）。我们可以使用正比例在地图距离和实际距离之间进行转换。

To find the real length, multiply the map length by the scale factor. If a model train is built to scale 1 : 87 and the model is 15 cm long, the real train is 15 × 87 = 1305 cm, or 13.05 m long. Conversely, to find a model length, divide the real length by the scale factor.

要找到实际长度，需将地图长度乘以比例因子。如果一辆模型火车的制作比例为 1 : 87，模型长 15 厘米，那么真车长度为 15 × 87 = 1305 厘米，即 13.05 米。反之，要求模型长度，则用实际长度除以比例因子。

9. Ratio and Proportion in Recipes | 食谱中的比率与比例

Recipes are a perfect example of using ratios and proportion. The ingredients are listed in fixed ratios. If we want to make more or fewer servings, we scale the amounts up or down proportionally. This involves multiplying each ingredient by a scale factor (the ratio of the new number of servings to the original number of servings).

食谱是运用比率和比例的一个绝佳例子。配料以固定的比率列出。如果我们想制作更多或更少的份数，就需要按比例放大或缩小每种配料的分量。这需要将每种配料的量乘以一个比例因子（即新份数与原始份数的比值）。

For example, a recipe for 4 people requires 250 g of flour and 100 g of sugar. For 6 people, the scale factor is 6 ÷ 4 = 1.5. So flour needed = 250 × 1.5 = 375 g and sugar = 100 × 1.5 = 150 g. The ratio of flour to sugar remains 250 : 100 = 5 : 2, whether for 4 or 6 servings.

例如，一份供 4 人食用的食谱需要 250 克面粉和 100 克糖。如果要供 6 人食用，比例因子为 6 ÷ 4 = 1.5。因此所需面粉为 250 × 1.5 = 375 克，糖为 100 × 1.5 = 150 克。不论供 4 人还是 6 人，面粉与糖的比率依然保持 250 : 100 = 5 : 2。

10. Rates and Units of Measure | 速率与计量单位

Rates are a special kind of ratio that compare two quantities with different units, such as speed (km/h), density (kg/m³), or price per litre (¢/L). When dealing with rates, we often use the formula speed = distance ÷ time, or more generally, rate = amount ÷ time. Understanding rates helps us solve problems involving travel, flow, and consumption.

速率是一种特殊的比率，用于比较两个单位不同的量，例如速度（公里/小时）、密度（千克/立方米）或每升价格（分/升）。处理速率时，我们常用公式速度 = 距离 ÷ 时间，或更一般地，速率 = 数量 ÷ 时间。理解速率有助于我们解决涉及旅行、流量和消耗的问题。

To compare different rates or find the best value, we reduce each rate to a ‘per unit’ basis. For example, if a 2 kg bag of rice costs £3.20 and a 5 kg bag costs £7.50, we find the price per kilogram: £3.20 ÷ 2 = £1.60/kg and £7.50 ÷ 5 = £1.50/kg. The larger bag offers better value per kilogram.

要比较不同的速率或寻找最优价值，我们可以将每种速率换算为“每单位”的基础。例如，一袋 2 公斤的米售价 3.20 英镑，一袋 5 公斤的米售价 7.50 英镑，我们可以求出每公斤的价格：3.20 ÷ 2 = 1.60 英镑/公斤，7.50 ÷ 5 = 1.50 英镑/公斤。大袋米在每公斤价格上更划算。

11. Solving Proportion Problems with Tables | 使用表格解决比例问题

Organising information in a ratio table helps visualise the relationship between quantities. A ratio table lists equivalent ratios in columns or rows. By adding or multiplying columns, we can find missing values. This method is particularly useful when solving more complex proportion puzzles without a calculator.

用比率表格整理信息有助于直观地理解数量之间的关系。比率表格按列或行列出等价比率。通过将列相加或相乘，我们可以找出缺失的数值。这种方法在没有计算器的情况下解决较复杂的比例谜题时尤其有用。

Consider making a table for the relationship between oranges and juice: 3 oranges give 240 ml of juice. We want to know how much juice 7 oranges yield.

Oranges	3	1	7
Juice (ml)	240	80	560

First find the unit rate by dividing 240 by 3 to get 80 ml per orange, then multiply by 7. The table clearly shows the step from 3 oranges to 1 orange (÷3), then to 7 oranges (×7). This builds confidence in proportional reasoning.

先用 240 除以 3 得到每只橙子 80 毫升的单位速率，然后乘以 7。表格清晰地显示了从 3 只橙子到 1 只橙子（除以 3），再到 7 只橙子（乘以 7）的步骤。这可以增强比例推理的信心。

12. Common Misconceptions and Tips | 常见误区与提示

One common mistake is to confuse the order of terms in a ratio. Remember that a ratio of boys to girls is different from girls to boys. Another pitfall is to add the same amount to both sides of a ratio, which changes the proportion. Only multiplication or division preserves the ratio, not addition or subtraction.

一个常见的错误是混淆比率中各项的顺序。请记住，男生与女生的比率和女生与男生的比率是不同的。另一个陷阱是给比率的两边加上相同的数量，这会改变比例。只有乘法和除法能保持比率不变，加法和减法不行。

When simplifying, always check that all terms are whole numbers and have no common factor other than 1. Also, be careful with mixed units: always convert to the same unit before writing a ratio. For instance, when comparing 1.5 m and 90 cm, write 150 cm : 90 cm, then simplify to 5 : 3.

化简时，要始终检查所有项是否均为整数，并且除了 1 之外没有其他公因数。同时，要注意混合单位的问题：在书写比率之前，一定要先转换为相同的单位。例如，比较 1.5 米和 90 厘米时，应写作 150 cm : 90 cm，再化简为 5 : 3。

Finally, always ask yourself whether your answer makes sense in the context. A cake recipe for 10 people should not require less flour than the recipe for 4 people. Proportional reasoning is about keeping the multiplicative relationship the same.

最后，要时常反问自己，所得答案在相关情境中是否合理。一份供 10 人食用的蛋糕食谱所需的免费不应该比供 4 人食用的更少。比例推理的关键在于保持乘法关系不变。

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Ratio and Proportion | 比率与比例