Ratio and Proportion Made Easy | 轻松掌握比和比例

📚 Ratio and Proportion Made Easy | 轻松掌握比和比例

Welcome to this KS3 Cambridge Mathematics revision guide based on p112_1.pdf. In this article, we break down the key concepts of ratio and proportion, including simplifying ratios, dividing quantities, and solving proportion problems. Whether you are preparing for a Checkpoint test or strengthening your foundation, these explanations and examples will help you master the topic with confidence.

欢迎学习这篇基于 p112_1.pdf 的 KS3 剑桥数学复习指南。本文将深入解析比和比例的核心概念，包括化简比、按比例分配以及解决比例问题。无论你是在为 Checkpoint 测试做准备，还是在夯实基础，这些讲解与例题都将帮助你自信地掌握该主题。

1. Understanding Ratio | 理解比的概念

A ratio is a way of comparing two or more quantities. It shows 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 very important: the ratio 3 : 2 is different from 2 : 3.

比是比较两个或多个量的一种方式，它表示一个量与另一个量之间的相对大小。例如，一个果盘中有 3 个苹果和 2 个橙子，那么苹果与橙子的比写作 3 : 2。顺序非常重要：3 : 2 与 2 : 3 所表示的关系是不同的。

Ratios can also compare more than two quantities, such as 2 : 4 : 6 for red, blue, and green marbles. Ratios have no units and simply express a relationship. They are widely used in recipes, maps, and real-life scaling.

比还可以比较两个以上的量，例如红、蓝、绿弹珠的比可以写作 2 : 4 : 6。比没有单位，仅表示一种关系。它们广泛用于食谱、地图和实际生活中的缩放问题。

2. Simplifying Ratios | 化简比

To simplify a ratio, divide all parts by their highest common factor (HCF). For instance, the ratio 10 : 15 can be simplified by dividing both numbers by 5, giving 2 : 3. Always express a ratio in its simplest integer form, just like simplifying a fraction.

要化简一个比，用各项的最大公因数（HCF）去除每一项。例如，比 10 : 15 可以同时除以 5，得到 2 : 3。应始终将比化为最简整数形式，这与化简分数类似。

If the ratio contains decimals or fractions, multiply all terms by the same power of 10 or by the lowest common denominator to obtain whole numbers first. For example, 0.5 : 1.5 becomes 5 : 15 after multiplying by 10, which simplifies to 1 : 3.

如果比中含有小数或分数，首先将所有项乘以 10 的相同次幂或最小公分母，得到整数。例如，0.5 : 1.5 乘以 10 后变成 5 : 15，再化简得 1 : 3。

3. Equivalent Ratios | 等值比

Equivalent ratios are obtained by multiplying or dividing each term of a ratio by the same non‑zero number. For example, 1 : 2 is equivalent to 2 : 4, 3 : 6, and 10 : 20. Finding equivalent ratios is useful when scaling recipes or comparing quantities with a common term.

等值比是通过将比的每一项同时乘以或除以同一个非零数得到的。例如，1 : 2 与 2 : 4、3 : 6 和 10 : 20 等值。在调整食谱分量或需统一某项进行比较时，使用等值比非常方便。

To check whether two ratios are equivalent, write them as fractions and simplify, or cross‑multiply. For ratios a : b and c : d, they are equivalent if a × d = b × c. This technique also helps in solving missing‑value problems.

要检验两个比是否等值，可以把它们写成分数再化简，或者采用交叉相乘的方法。对于比 a : b 和 c : d，如果 a × d = b × c，则它们等值。这一技巧也可用于求比中的未知项。

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

To divide a quantity into parts according to a ratio, first add the parts of the ratio to find the total number of shares. Next, divide the total quantity by this number of shares to find the value of one share. Then multiply each ratio part by this share value.

要按照给定的比将一个数量分配成若干部分，首先将比的各项相加，得出总份数。然后，用总量除以总份数，得到每一份的值。最后将比的每一项乘以该份值即可。

For example, divide £60 between Alex and Ben in the ratio 2 : 3. Total parts = 2 + 3 = 5. One share = £60 ÷ 5 = £12. Alex gets 2 × £12 = £24, and Ben gets 3 × £12 = £36. Always check by adding the amounts: £24 + £36 = £60.

例如，将 60 英镑按照 2 : 3 的比例分给 Alex 和 Ben。总份数 = 2 + 3 = 5。每份 = 60 ÷ 5 = 12 英镑。Alex 得 2 × 12 = 24 英镑，Ben 得 3 × 12 = 36 英镑。务必通过加总来检验：24 + 36 = 60。

5. The Unitary Method for Proportions | 比例中的归一法

The unitary method involves finding the value of one unit first, and then scaling up or down to find the required amount. This is especially helpful in proportion problems such as ‘8 pens cost £5.20; how much for 3 pens?’ Find the cost of 1 pen, then multiply by 3.

归一法指的是先求出一个单位的量，再以此为基础放大或缩小得到所需的数量。这在诸如 “8 支笔售价 5.20 英镑，3 支笔多少钱？” 的比例问题中尤为实用。先算出 1 支笔的价格，再乘以 3。

Using the unitary method makes it easy to compare best buys. By reducing different pack sizes to a cost per unit (e.g. per 100 g or per litre), you can directly see which product offers better value for money. It is a powerful real‑life application of ratio thinking.

运用归一法可以轻松比较最优购买方案。通过将不同包装规格化为单位成本（如每 100 克或每升的价格），就能直接看出哪种产品更划算。这是比的实际应用，是生活中强有力的数学工具。

6. Proportion and Best Buys | 比例与最优购买

Proportional reasoning helps identify best buys by comparing prices for the same quantity. For instance, a 300 g pack of cereal costs £2.10, and a 500 g pack costs £3.25. Calculate the price per 100 g: £2.10 ÷ 3 = £0.70 per 100 g, and £3.25 ÷ 5 = £0.65 per 100 g. The larger pack is cheaper per 100 g.

比例推理能够通过比较同等数量的价格帮助找到最优购买方案。例如，一盒 300 克的麦片售价 2.10 英镑，另一盒 500 克售价 3.25 英镑。计算每 100 克的价格：2.10 ÷ 3 = 0.70 英镑/100 克；3.25 ÷ 5 = 0.65 英镑/100 克。因此大包装每 100 克更便宜。

Sometimes offers like ‘3 for 2’ or ‘20% extra free’ need to be interpreted using ratios. Convert the offer into a unit price and compare with the regular option. Always read the question carefully and identify which quantity you are fixing to make a fair comparison.

有时像“买三付二”或“加量 20% 不加价”这类优惠，也需要用比来解读。将优惠形式转化为单位价格，再与普通装进行比较。仔细读题，确定将哪一个量固定下来，以做出公平的比较。

7. Map Scales and Scale Drawings | 地图比例尺与比例图

A map scale is a ratio that relates a distance on the map to the actual distance on the ground. For example, a scale of 1 : 50 000 means 1 cm on the map represents 50 000 cm (or 0.5 km) in real life. You can use the scale to find real distances by multiplying the map length by the scale factor.

地图比例尺是一个比，它表示图上距离与实际地面距离的关系。例如，1 : 50 000 的比例尺意味着图上 1 厘米代表实际中的 50 000 厘米（即 0.5 千米）。你可以用比例尺通过地图上的长度乘以比例因子来求出实际距离。

When converting between map and real measurements, make sure all units are consistent. It is often easier to convert the scale to a more convenient form, such as 1 cm : 0.5 km. Similarly, scale drawings reduce or enlarge objects while keeping their proportions intact.

在地图尺寸与实际尺寸之间进行换算时，务必确保单位一致。通常将比例尺转化为更方便的形式会更简单，例如 1 cm : 0.5 km。同样地，比例图通过缩小或放大物体来保持其形状的比例不变。

8. Ratio and Fractions | 比与分数的关系

Ratios and fractions are closely connected. If a ratio of boys to girls is 3 : 5, you can say that boys make up 3/8 of the total, and girls make up 5/8. The denominator is the sum of the ratio parts. This link helps solve many word problems and probability questions.

比和分数密切相关。如果男生与女生的比是 3 : 5，那么男生占总人数的 3/8，女生占总人数的 5/8。分母是比中各项之和。这种联系有助于解决许多文字题和概率问题。

Be careful: the ratio 3 : 5 does not mean girls are 5/3 of the boys; it means for every 3 boys there are 5 girls. To express one part as a fraction of the other, you can say girls are 5/3 of the boys, but the fraction of the total is what commonly appears in questions.

需要注意：比 3 : 5 并不意味着女生是男生的 5/3；它表示每 3 个男生对应 5 个女生。要将一部分表示为另一部分的分数，可以说女生是男生的 5/3，但题目更常问的是各占总体的几分之几。

9. Direct Proportion Problems | 正比例问题

Two quantities are in direct proportion if they increase or decrease in the same ratio. For example, the cost of apples is directly proportional to the mass of apples bought. If 4 kg cost £6, then 12 kg cost £18, because the mass is multiplied by 3, and the cost is also multiplied by 3.

如果两个量按照相同的比增大或减小，它们就成正比例。例如，苹果的总价与购买的苹果质量成正比。如果 4 千克苹果 6 英镑，那么 12 千克就是 18 英镑，因为质量乘以 3，价格也乘以 3。

To solve direct proportion, set up equivalent ratios. Using the form y = kx, the constant of proportionality k can be found from a given pair of values. Once k is known, any other value can be calculated. This method underpins many science and financial calculations.

解正比例问题时，可以建立等值比。利用公式 y = kx，比例常数 k 可以由已知的一组对应值求出。一旦求出 k，就可以计算任何其他值。这种方法为许多科学和金融计算奠定了基础。

10. Inverse Proportion Basics | 反比例基础

In inverse proportion, as one quantity increases, the other decreases in such a way that their product remains constant. For example, if 4 workers complete a task in 6 hours, 8 workers (double the number) would complete it in 3 hours (half the time), assuming they work at the same rate.

在反比例关系中，当一个量增大时，另一个量随之减小，且它们的乘积保持不变。例如，如果 4 名工人需要 6 小时完成一项任务，那么 8 名工人（人数翻倍）只需 3 小时（时间减半），前提是工作效率相同。

The key equation for inverse proportion is x × y = constant. To solve problems, find the constant using a known pair, then divide by the new value of one quantity. Inverse proportion often appears in contexts of sharing work, speed and time, or volume and pressure.

反比例的关键等式是 x × y = 常数。解题时，先用已知的一对值求出常数，再用常数除以一个量的新值。反比例常出现在协作完成工作、速度与时间、或体积与压强等情景中。

11. Common Misconceptions and Tips | 常见误解与提示

A typical mistake is to confuse the order of a ratio. Remember that ‘ratio of A to B’ means A : B. Another error is forgetting to simplify ratios fully, or cancelling incorrectly when only one term is a multiple. Always look for the highest common factor of all terms to simplify completely.

一个常见错误是混淆比的顺序。记住“A 与 B 的比”意味着 A : B。另一个错误是忘记将比彻底化简，或者在只有某一项可被整除时进行错误约分。始终应找出所有项的最大公因数来进行完整化简。

When dividing a quantity, many students forget to add the parts of the ratio first. Do not split the total directly according to the numbers in the ratio. Instead, calculate total shares. For proportion, confirm whether the relationship is direct or inverse before applying a formula. Drawing a table often helps.

在分配数量时，许多学生忘记先求出比的总份数。不能直接用比中的数字去分割总量。正确做法是先求总份数。对于比例关系，在套用公式前应先确认是正比还是反比关系。画表格常常会有帮助。

12. Practice Questions and Summary | 练习题与总结

To consolidate your learning, try these quick questions. 1) Simplify the ratio 24 : 36 : 60. 2) Divide £90 in the ratio 1 : 2 : 3. 3) If 5 lemons cost £1.25, how much do 8 lemons cost? 4) A map scale is 1 : 250 000; two towns are 8 cm apart on the map; find the real distance in km. Answers: 1) 2 : 3 : 5; 2) £15, £30, £45; 3) £2.00; 4) 20 km.

为了巩固所学，尝试解答以下题目。1) 化简比 24 : 36 : 60。2) 将 90 英镑按 1 : 2 : 3 分配。3) 5 个柠檬售价 1.25 英镑，8 个柠檬需多少钱？4) 地图比例尺为 1 : 250 000，两镇图上距离 8 cm，实际距离是多少千米？答案：1) 2 : 3 : 5；2) 15、30 和 45 英镑；3) 2.00 英镑；4) 20 km。

Ratio and proportion are fundamental topics that appear throughout the KS3 curriculum and in everyday life. By mastering equivalent ratios, the unitary method, and direct/inverse proportion, you will be well‑equipped for Checkpoint assessments and beyond. Practice regularly, check your working, and keep asking ‘what does this ratio actually tell me?’—that reflective question is the key to deep understanding.

比和比例是贯穿 KS3 课程及日常生活的基础主题。通过掌握等值比、归一法以及正反比例，你将能为 Checkpoint 评估及更高层次的学习做好充分准备。坚持练习，检查你的解题过程，并时常自问“这个比实际上告诉我什么？”——这个反思性问题是通向深层理解的关键。

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Ratio and Proportion Made Easy | 轻松掌握比和比例