Ratio and Proportion: Sharing Quantities | 比与比例：数量分配

📚 Ratio and Proportion: Sharing Quantities | 比与比例：数量分配

In the Cambridge KS3 mathematics syllabus, ratios form a bridge between simple fractions and the proportional reasoning needed for algebra and real‑world problem solving. The exercises often found on page 141 of the coursebook challenge you to share quantities in a given ratio, compare parts, and recognise when a ratio is in its simplest form. This article unpacks those ideas step by step, giving you the tools to tackle any ratio question with confidence.

在剑桥KS3数学大纲中，比是连接简单分数与比例推理的桥梁，后者是代数及实际问题解决的基础。教材第141页常见的那类练习，要求你按给定比分配数量、比较各部分、识别最简比。本文将一步步拆解这些概念，为你提供信心十足地解决任何比的问题的工具。

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1. Understanding Ratios | 理解比

A ratio is a way of comparing two or more quantities. It tells you how much of one thing there is compared to another. For example, if a fruit bowl contains 3 apples and 2 oranges, the ratio of apples to oranges is written as 3 : 2. The order is crucial – 3 : 2 is not the same as 2 : 3. Ratios do not give the actual amounts, but the relative sizes. In this bowl, the real numbers could be 6 apples and 4 oranges, or 30 apples and 20 oranges – the ratio stays 3 : 2.

比是比较两个或多个数量的一种方式。它告诉你一种东西相对于另一种有多少。例如，如果一个水果碗里有3个苹果和2个橙子，苹果与橙子的比写成 3 : 2。顺序很重要——3 : 2 与 2 : 3 不同。比不给出实际数量，而是相对大小。在这个碗里，实际数量可能是6个苹果和4个橙子，或者30个苹果和20个橙子——比保持 3 : 2。

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2. Simplifying Ratios | 化简比

Just like fractions, ratios should be simplified by dividing all parts by their highest common factor. Take the ratio 12 : 8. The highest common factor of 12 and 8 is 4, so divide both by 4: 12 ÷ 4 = 3 and 8 ÷ 4 = 2, giving 3 : 2. A ratio is in its simplest form when the numbers are whole numbers with no common factor other than 1. If the ratio contains decimals or fractions, multiply through to clear them. For instance, 0.5 : 1.5 can be multiplied by 2 to get 1 : 3.

就像分数一样，比也应该通过除以最大公因数来化简。以 12 : 8 为例。12 和 8 的最大公因数是 4，所以两边都除以 4：12 ÷ 4 = 3，8 ÷ 4 = 2，得到 3 : 2。当一个比中的所有数都是整数且除了1以外没有其他公因数时，它就处于最简形式。如果比含有小数或分数，可以通过乘法消去它们。例如，0.5 : 1.5 可以乘以 2 得到 1 : 3。

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3. Ratios with Different Units | 不同单位的比

When comparing quantities with different units, convert them to the same unit first. A ratio of 2 m to 50 cm must be expressed as 200 cm : 50 cm, which simplifies to 4 : 1. Mixing metres and centimetres directly would give 2 : 50, a completely wrong ratio. Always ensure both parts are in the same unit before simplifying.

比较不同单位的数量时，首先要转换成相同单位。2 米与 50 厘米的比必须表示为 200 厘米 : 50 厘米，化简得 4 : 1。如果直接用米和厘米写 2 : 50，就完全错了。化简前一定要确保两个部分使用相同单位。

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4. Sharing in a Given Ratio | 按给定比例分配

One of the most common KS3 tasks is sharing an amount into a given ratio. Imagine three friends, Amira, Ben and Chloe, who win £180 and agree to split it in the ratio 2 : 3 : 5. Follow these steps: add the parts of the ratio to find the total number of parts. 2 + 3 + 5 = 10 parts. Then work out the value of one part: £180 ÷ 10 = £18. Finally, multiply the value of one part by each ratio number: Amira gets 2 × £18 = £36, Ben gets 3 × £18 = £54, and Chloe gets 5 × £18 = £90. Always check the sum: £36 + £54 + £90 = £180.

KS3最常见的任务之一是按给定比例分配一个总数。设想三位朋友，Amira、Ben 和 Chloe，赢了180英镑并同意按 2 : 3 : 5 的比例分配。遵循以下步骤：将比的所有项相加得到总份数。2 + 3 + 5 = 10 份。然后算出一份的价值：£180 ÷ 10 = £18。最后，用一份的价值乘以各项：Amira 得 2 × £18 = £36，Ben 得 3 × £18 = £54，Chloe 得 5 × £18 = £90。务必检查总和：£36 + £54 + £90 = £180。

Total parts = 2 + 3 + 5 = 10;   One part = £180 ÷ 10 = £18;   Shares = 2×£18, 3×£18, 5×£18

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5. The Unitary Method for Ratios | 比例的归一法

The unitary method is a powerful way to find missing values in ratio problems. If a recipe uses 300 g of flour for 4 servings, how much flour is needed for 10 servings? Set up the ratio 4 : 10, but simpler: find the flour per serving first. 300 g ÷ 4 = 75 g per serving. Then multiply by 10: 75 g × 10 = 750 g. This method works for any ‘how much for…?’ question involving ratios and proportions.

归一法是解决比例问题中寻找缺失值的强大工具。如果一份食谱用 300 克面粉做 4 份，那么做 10 份需要多少面粉？设立比例 4 : 10，但更简单的做法是：先求出每份面粉量。300 ÷ 4 = 75 克/份。再乘以 10：75 × 10 = 750 克。这种方法适用于任何涉及比例关系的“多少对应多少？”问题。

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6. Comparing Ratios | 比较比

To compare ratios, you can write them as fractions or scale them to have the same total or one common term. Suppose a class has a ratio of boys to girls of 7 : 8, and another class has 13 : 14. Which class has the greater proportion of boys? Convert each ratio to a fraction: 7/(7+8) = 7/15 ≈ 0.467, and 13/(13+14) = 13/27 ≈ 0.481. The second class has a slightly higher proportion of boys. Equally, you could find a common term – scaling the first ratio by 27 and the second by 15 – but fraction comparison is quicker.

要比较两个比，可以把它们写成分数，或者通分到相同总数或有一个公共项。假设一个班级男女比为 7 : 8，另一个班级为 13 : 14。哪个班级男生比例更高？把每个比转化为分数：7/(7+8) = 7/15 ≈ 0.467，13/(13+14) = 13/27 ≈ 0.481。第二个班级男生比例略高。同样地，也可以找一个公共项——将第一个比乘以27，第二个比乘以15——但分数比较更快捷。

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7. Ratios and Fractions | 比与分数

A ratio links directly to fractions. In a ratio a : b, the fraction of the whole that is a is a / (a+b), and the fraction that is b is b / (a+b). For a 3 : 4 ratio, the total parts are 7, so the first quantity represents 3/7 of the whole, the second 4/7. This conversion is essential when solving problems where you know a fraction of an amount and need to find the ratio, or vice versa. Always remember: ratio parts become the numerators, while the sum of the parts is the denominator.

比与分数直接关联。在 a : b 的比中，表示 a 占总体的分数为 a/(a+b)，表示 b 的分数为 b/(a+b)。对于 3 : 4 的比，总份数为 7，因此第一个量占整体的 3/7，第二个占 4/7。当你知道总量的一个分数而需要找出比时，或者反过来，这种转换至关重要。始终记住：比的各项变成分子，而各项之和成为分母。

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8. Scaling Up and Down | 扩大与缩小

You can multiply or divide all terms of a ratio by the same non‑zero number without changing the relationship. This is scaling. A map scale of 1 : 50000 means 1 cm on the map represents 50000 cm in reality. If a road is 3 cm on the map, the real length is 3 × 50000 = 150000 cm, or 1.5 km. Scaling up works for model trains, architectural plans, and even photography. The key principle: whatever you multiply or divide one part by, do exactly the same to all other parts.

你可以将比的所有项都乘以或除以同一个非零数，而不改变它们之间的关系。这就是缩放。地图比例尺 1 : 50000 意味着地图上 1 厘米代表实际 50000 厘米。如果地图上一条路长 3 厘米，实际长度就是 3 × 50000 = 150000 厘米，即 1.5 千米。放大缩小原理同样适用于火车模型、建筑图纸，甚至摄影。关键原则：你对一项乘以或除以什么数，对所有的其他项也必须进行完全相同的操作。

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9. Real-Life Applications: Recipes and Mixtures | 实际应用：食谱与混合物

Ratios appear everywhere in daily life. In cooking, a cake recipe might list flour, sugar and butter in the ratio 3 : 2 : 1. If you have 600 g of flour, how much sugar and butter do you need? The flour corresponds to 3 parts, so 3 parts = 600 g, giving 1 part = 200 g. Then sugar (2 parts) = 400 g, butter (1 part) = 200 g. Paint mixing, diluting squash, and fuel‑oil mixes all rely on the same ratio‑scaling method.

比在日常生活中无处不在。烹饪中，一个蛋糕食谱可能列出面粉、糖和黄油的比为 3 : 2 : 1。如果你有 600 克面粉，需要多少糖和黄油？面粉对应 3 份，因此 3 份 = 600 克，得 1 份 = 200 克。那么糖 (2 份) = 400 克，黄油 (1 份) = 200 克。油漆调配、浓缩果汁稀释、燃油混合，全都依赖相同的比例缩放方法。

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10. Common Pitfalls and How to Avoid Them | 常见错误及避免方法

Many students make mistakes by writing the ratio in the wrong order, forgetting to simplify, or using the total amount incorrectly. When sharing £50 in the ratio 2 : 3, some mistakenly take £50 as the value of 2 parts or 3 parts. Always add the parts first to find the total number of parts. Another trap is ignoring units: 1 m to 30 cm is not 1 : 30, but 10 : 3 or 100 : 30 simplified. Finally, never round numbers midway through ratio calculations – keep exact values until the end to avoid errors.

许多学生常犯的错误包括：把比的顺序写反、忘记化简、或误用总数。当按 2 : 3 分配 50 英镑时，有人会错把 50 英镑当成 2 份或 3 份的值。一定要先加总份数。另一个陷阱是忽略单位：1 米比 30 厘米不是 1 : 30，而是 10 : 3（或 100 : 30 化简）。最后，在比例计算过程中绝不要中途四舍五入——保留精确值直到最后，以避免误差。

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11. Practice Questions to Consolidate | 巩固练习

Try this typical question similar to the one found on page 141 of the Cambridge coursebook: Jack and Jill share £84 in the ratio 5 : 2. How much does each get? (Answer: Jack £60, Jill £24). Another: A school has 450 students and the ratio of boys to girls is 11 : 7. How many girls are there? (Answer: girls = 450 × 7/18 = 175). Work through these carefully, showing your addition of parts and unitary calculation.

试试这道类似于剑桥教材第141页的典型问题：Jack 和 Jill 按 5 : 2 的比例分享 84 英镑。每人分得多少？（答案：Jack 60 英镑，Jill 24 英镑）。再一题：一所学校有 450 名学生，男女比例为 11 : 7。有多少名女生？（答案：女生 = 450 × 7/18 = 175）。仔细解答，写出份数相加和归一计算的过程。

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12. Linking Ratios to Algebra | 比与代数的衔接

In KS3, ratios often introduce algebraic thinking. If the ratio of two numbers is 3 : 8 and their difference is 25, let the numbers be 3x and 8x. Then 8x – 3x = 25, so 5x = 25 and x = 5. The numbers are 15 and 40. This method prepares you for linear equations and later work on direct and inverse proportion. Recognising that a ratio can be expressed using a variable multiplier x is a skill that will stay with you through IGCSE and beyond.

在KS3阶段，比常常引入代数思维。如果两个数的比是 3 : 8，且它们的差是 25，设这两个数为 3x 和 8x。那么 8x – 3x = 25，所以 5x = 25，x = 5。这两个数就是 15 和 40。这种方法为你学习一次方程以及后续的正比反比内容打下基础。认识到比可以用变量乘数 x 表示，是一项会伴随你走过IGCSE乃至更远阶段的技能。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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