📚 Ratios and Proportions | 比与比例
Ratios and proportions are fundamental tools in mathematics that help us compare quantities and understand relationships between numbers. Whether you are sharing sweets, mixing paint, or reading a map, ratios allow you to see how much of one thing there is compared to another. Proportions build on this idea, showing that two ratios are equal and helping us solve problems where quantities change together. In this article, we will explore how to write, simplify, and use ratios, and we will learn about direct and inverse proportion. These skills are essential for KS3 and everyday life.
比和比例是数学中的重要工具，帮助我们比较数量、理解数字之间的关系。无论是分享糖果、混合颜料还是阅读地图，比都可以让你看到一种事物相对于另一种事物的多少。比例在此基础上进一步展示两个比相等的情况，并帮助我们解决数量共同变化的问题。在这篇文章中，我们将学习如何书写、简化和使用比，并了解正比例和反比例。这些技能对KS3阶段和日常生活都至关重要。
1. Understanding Ratios | 理解比
A ratio compares two or more quantities of the same kind, showing the relative size of one quantity to another. It can be written in several ways: using a colon, as ‘3 to 4’, or as the fraction 3/4. For example, if there are 3 apples and 4 oranges in a bowl, the ratio of apples to oranges is 3 : 4. The order is very important; the ratio 4 : 3 would mean something completely different.
比用于比较两个或多个同类数量，显示一个数量相对于另一个数量的大小。它可以用多种方式书写：使用冒号，如“3比4”，或写成分数 3/4。例如，如果碗里有3个苹果和4个橙子，苹果与橙子的比是 3 : 4。顺序非常重要；比 4 : 3 则表示完全不同的关系。
Ratios describe only the relationship, not the actual numbers. So if the ratio of boys to girls in a class is 2 : 3, it could mean there are 10 boys and 15 girls, or 20 boys and 30 girls. The quantities are multiplied by the same factor, but the ratio stays the same. This idea is called equivalent ratios.
比只描述关系，而不是实际数量。因此，如果一个班级里男孩与女孩的比是 2 : 3，它可能意味着有10个男孩和15个女孩，或者20个男孩和30个女孩。数量乘以相同的倍数，但比保持不变。这个概念称为等价比。
2. Simplifying Ratios | 简化比
Like fractions, ratios should be simplified by dividing all parts by their greatest common divisor (GCD). To simplify 12 : 16, find the GCD of 12 and 16, which is 4. Divide both numbers by 4: 12 ÷ 4 = 3, 16 ÷ 4 = 4, so the simplified ratio is 3 : 4. If there are more than two parts, divide all of them by the same number.
与分数类似，比应该通过将所有部分除以它们的最大公约数来化简。要化简 12 : 16，先找出12和16的最大公约数，为4。将两个数都除以4：12 ÷ 4 = 3，16 ÷ 4 = 4，所以化简后的比是 3 : 4。如果有两个以上的部分，所有部分都除以同一个数。
If the ratio contains a decimal or a fraction, first multiply to make all parts whole numbers. For instance, 1.5 : 2.5 can be multiplied by 2 to give 3 : 5. For a ratio like ½ : ⅓, find a common denominator (6) and multiply both parts: ½ × 6 = 3, ⅓ × 6 = 2, giving 3 : 2.
如果比中含有小数或分数，先通过乘法将所有部分化为整数。例如，1.5 : 2.5 可以乘以2得到 3 : 5。对于 ½ : ⅓ 这样的比，找到公分母6并乘以两部分：½ × 6 = 3，⅓ × 6 = 2，得到 3 : 2。
3. Ratios in Different Units | 不同单位的比
When quantities are expressed in different units, you must convert them to the same unit before writing the ratio. For example, to compare 50 cm to 1.2 m, convert 1.2 m to 120 cm. Then the ratio is 50 : 120, which simplifies to 5 : 12. Always use the smaller common unit to avoid decimals.
当数量以不同单位表示时，书写比之前必须将它们转换为相同单位。例如，要比较50厘米和1.2米，将1.2米转换为120厘米。然后比是 50 : 120，可以化简为 5 : 12。始终使用更小的共同单位以避免小数。
Common conversions you should remember include 1 km = 1000 m, 1 m = 100 cm, 1 kg = 1000 g, and 1 hour = 60 minutes. This skill is very useful in map scales and recipes.
你应该记住的常见换算有：1千米 = 1000米，1米 = 100厘米，1千克 = 1000克，1小时 = 60分钟。这项技能在地图比例尺和食谱中非常有用。
4. Dividing a Quantity in a Given Ratio | 按给定比分配数量
To divide a total amount in a given ratio, first add the parts of the ratio to find the total number of shares. For example, to divide £60 between two people in the ratio 2 : 3, the total number of shares is 2 + 3 = 5. The value of one share is £60 ÷ 5 = £12. The first person gets 2 shares = £24, and the second gets 3 shares = £36.
要按给定比分配总量，首先将比的各部分相加，求出总份额数。例如，要将60英镑按 2 : 3 分给两个人，总份额数是 2 + 3 = 5。每份的价值是 £60 ÷ 5 = £12。第一个人得到2份 = £24，第二个人得到3份 = £36。
This method works for any ratio with more than two parts. If paint is mixed in the ratio red : blue : white = 2 : 1 : 3 and you need 12 litres in total, the total shares are 2+1+3=6, one share is 12 ÷ 6 = 2 L. So you need 4 L red, 2 L blue, and 6 L white.
这种方法适用于具有两个以上部分的任何比。如果涂料按红 : 蓝 : 白 = 2 : 1 : 3 混合，并且总共需要12升，总份额数为2+1+3=6，每份为12 ÷ 6 = 2升。因此你需要4升红色，2升蓝色，6升白色。
5. Introduction to Proportion | 比例简介
Proportion describes the equality of two ratios. If two quantities are ‘in proportion’, their relationship stays the same when one changes. We can write this as a : b = c : d, which means a/b = c/d. In many problems, you can find an unknown value by using cross-multiplication.
比例描述了两个比的相等关系。如果两个量“成比例”，当一个量变化时，它们的关系保持不变。我们可以将其写为 a : b = c : d，即 a/b = c/d。在许多问题中，你可以通过交叉相乘来找到未知值。
For example, if 5 pens cost £2, how much would 8 pens cost? Set up the proportion: 5/2 = 8/x. Cross-multiply: 5x = 16, so x = 3.20 (£3.20). Always check that the units on each side correspond (pens/money on one side, pens/money on the other).
例如，如果5支笔售价2英镑，8支笔多少钱？列出比例式：5/2 = 8/x。交叉相乘：5x = 16，所以 x = 3.20（3.20英镑）。始终检查等式两边的单位对应（一边是笔/钱，另一边也应是笔/钱）。
6. Direct Proportion | 正比例
Two quantities are in direct proportion when they increase or decrease at the same rate. If y is directly proportional to x, we write y ∝ x, and the relationship is y = kx, where k is the constant of proportionality. This means if x doubles, y doubles; if x is halved, y is halved. The graph of y against x is a straight line through the origin.
当两个量以相同速率增加或减少时，它们成正比例关系。如果 y 与 x 成正比，我们写作 y ∝ x，其关系为 y = kx，其中 k 是比例常数。这意味着如果 x 加倍，y 也加倍；如果 x 减半，y 也减半。y 对 x 的图像是一条过原点的直线。
y = kx
You can find k by substituting any pair of corresponding values. For instance, if a car travels 150 km using 10 litres of fuel, the distance is directly proportional to fuel. k = 150/10 = 15 km/L. With 25 L, distance = 15 × 25 = 375 km.
你可以通过代入任意一对对应值来求 k。例如，一辆车用10升燃油行驶150公里，距离与燃油成正比。k = 150/10 = 15 公里/升。用25升时，距离 = 15 × 25 = 375 公里。
7. Inverse Proportion | 反比例
Two quantities are in inverse proportion if their product is constant. When one increases, the other decreases proportionally. We write y ∝ 1/x, and the relationship is y = k/x. A typical example is time taken to complete a job: if more people work, it takes less time, assuming they all work at the same rate.
如果两个量的乘积恒定，则它们成反比例。当一个量增加时，另一个量按比例减少。我们写作 y ∝ 1/x，关系式为 y = k/x。一个典型的例子是完成工作所需的时间：如果更多人工作，所需时间减少，前提是所有人工作效率相同。
y = k/x
If 3 builders can build a wall in 8 days, how long would 6 builders take? k = 3 × 8 = 24. For 6 builders, days = 24/6 = 4 days. Notice the product of workers × days is always 24. Graphs of inverse proportion are curves that never touch the axes.
如果3名建筑工人砌一堵墙需要8天，6名工人需要多少天？k = 3 × 8 = 24。6名工人时，天数 = 24/6 = 4天。注意工人数 × 天数的乘积始终是24。反比例函数图像是永不相交于坐标轴的曲线。
8. Scale Drawings and Maps | 比例尺与地图
Maps and scale drawings use ratios to represent real distances. A scale such as 1 : 50 000 means 1 cm on the map represents 50 000 cm (or 0.5 km) in reality. To find the actual distance, multiply the map distance by the scale factor. If a road is 4 cm on the map, the real length is 4 × 50 000 = 200 000 cm = 2 km.
地图和比例图使用比来表示实际距离。像 1 : 50 000 这样的比例尺表示地图上的1厘米代表实际中的50 000厘米（或0.5公里）。要找到实际距离，将地图距离乘以比例因子。如果一条道路在地图上长4厘米，实际长度就是 4 × 50 000 = 200 000厘米 = 2公里。
| Scale | 1 cm represents |
| 1 : 10 000 | 100 m |
| 1 : 50 000 | 500 m |
| 1 : 100 000 | 1 km |
You can also find the scale if you know a real distance and its map length. If a 3 km trail is shown as 6 cm, the scale is 6 cm : 3 km = 6 cm : 300 000 cm = 1 : 50 000.
如果你知道实际距离及其地图上的长度，也可以求出比例尺。如果一条3公里的小径在地图上显示为6厘米，比例尺就是 6 cm : 3 km = 6 cm : 300 000 cm = 1 : 50 000。
9. Ratio and Proportion in Recipes | 食谱中的比与比例
Recipes are practical examples of ratios. A recipe may call for flour and sugar in the ratio 4 : 1. If you need to make more or less, you keep the ratio the same. To make half the recipe, multiply all quantities by 1/2; to make three times as much, multiply by 3. This is direct proportion.
食谱是比的实际例子。一份食谱可能要求面粉和糖的比为 4 : 1。如果你需要制作更多或更少，应保持比不变。制作食谱的一半分量时，所有数量乘以 1/2；制作三倍分量时则乘以3。这就是正比例。
For example, if a cake recipe serves 8 and uses 200 g of flour, how much flour is needed for 12 people? The ratio of servings to flour is 8 : 200, which simplifies to 1 : 25. So 12 people need 12 × 25 = 300 g. Alternatively, set up 8/200 = 12/x and solve for x.
例如，如果一份蛋糕食谱供8人食用，使用200克面粉，那么12人需要多少面粉？份数与面粉的比是 8 : 200，化简为 1 : 25。因此12人需要 12 × 25 = 300克。也可以列出 8/200 = 12/x 并求解 x。
10. Practice Problems and Applications | 练习题与应用
To master ratios and proportions, regular practice is essential. Start with simplifying ratios like 24 : 36 to 2 : 3, then move on to sharing £90 in 4 : 5. Set up proportion equations for real-world problems: if 7 apples cost £2.10, find the cost of 12 apples (answer: £3.60). Check inverse proportion by solving: 5 machines take 8 hours, how long for 4 machines? (Answer: 10 hours).
要掌握比和比例，定期练习是必要的。从化简 24 : 36 为 2 : 3 开始，然后练习按 4 : 5 分配90英镑。为实际问题建立比例方程：如果7个苹果售价2.10英镑，求12个苹果的价格（答案：3.60英镑）。通过求解反比例来检验：5台机器需要8小时，4台机器需要多长时间？（答案：10小时）。
Always show your steps: write the given ratio, find the total parts or constant k, and then calculate the answer. Use clear units and simplify where possible. With these techniques, you can solve problems involving speed, density, recipes, maps, and many other KS3 topics.
始终写出解题步骤：写出给定比，求出总份数或常数 k，然后计算答案。使用清晰的单位并尽可能化简。掌握这些技巧，你就可以解决涉及速度、密度、食谱、地图以及许多其他KS3主题的问题。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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