Ratios and Proportions | 比率与比例

📚 Ratios and Proportions | 比率与比例

A strong understanding of ratios and proportions is essential for everyday problem solving, from scaling recipes to interpreting maps. In this article we break down the key methods you need to handle ratio questions with confidence, fully aligned with the Cambridge KS3 Mathematics curriculum.

扎实掌握比率与比例是解决日常问题的关键，无论是调整食谱用量还是解读地图比例尺都离不开它。本文拆解了处理比率题目所需的核心方法，完全对标剑桥 KS3 数学课程，助你充满信心地应对各类题型。

1. What is a Ratio? | 什么是比率？

A ratio compares quantities of the same kind, showing how much of one thing there is compared to another. We write ratios using a colon, for example 3 : 2, which means for every 3 of the first quantity there are 2 of the second.

比率用来比较同一类量的多少，表示一个量相对于另一个量的大小。我们用冒号来书写比率，例如 3 : 2，表示每有 3 份第一个量，就对应第二个量有 2 份。

Ratios do not tell us the actual amounts, only the relative sizes. If a class has boys and girls in the ratio 3 : 2, it might have 15 boys and 10 girls, or 30 and 20 — the relationship stays the same.

比率不显示具体的数量，只反映相对大小。如果一个班级男生和女生的比例是 3 : 2，那可能是 15 个男生和 10 个女生，也可能是 30 个和 20 个——其关系保持不变。

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2. Simplifying Ratios | 约简比率

To simplify a ratio, divide all parts by their greatest common factor (GCF). For instance, 8 : 12 has a GCF of 4, so dividing both terms by 4 gives 2 : 3. Always express ratios in their simplest integer form unless told otherwise.

要化简一个比率，用各项的最大公因数（GCF）去除。例如，8 : 12 的最大公因数是 4，所以两项同时除以 4 得到 2 : 3。除非另有说明，始终将比率表示为最简整数形式。

Ratios with three or more terms, such as 20 : 35 : 15, can be simplified by finding the GCF of all numbers — here the GCF is 5, giving 4 : 7 : 3. If the ratio contains decimals, multiply every term by a power of 10 to obtain whole numbers first.

含三项或更多项的比率，例如 20 : 35 : 15，可以通过找出所有数的最大公因数来化简——这里最大公因数是 5，得到 4 : 7 : 3。如果比率中含有小数，先给每一项乘以 10 的幂化为整数。

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

When quantities are given in different units, you must convert them to the same unit before writing the ratio. For example, to express 50 pence and £2 as a ratio, write both in pence: 50 : 200, then simplify to 1 : 4.

当给出的量使用不同单位时，必须先转换为相同单位再写出比率。例如，要将 50 便士和 2 英镑表示为比率，都写成便士：50 : 200，再化简为 1 : 4。

Similarly, for lengths 1.5 m and 60 cm, convert 1.5 m = 150 cm, so the ratio is 150 : 60 = 5 : 2. Always double‑check that your conversion is correct before simplifying.

同理，对于长度 1.5 米和 60 厘米，将 1.5 米转换为 150 厘米，则比率为 150 : 60 = 5 : 2。化简前务必再次检查换算是否正确。

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4. Equivalent Ratios | 等价比率

Equivalent ratios are formed by multiplying or dividing all terms by the same non‑zero number. The ratios 2 : 3, 4 : 6, and 10 : 15 are all equivalent. To find a missing value in equivalent ratios, set up a proportion and use the scaling factor.

将一项比率的所有项同时乘以或除以同一个非零数，就得到等价比率。2 : 3、4 : 6 以及 10 : 15 都是等价比率。要求出等价比率中的缺失值，可以列出比例关系并利用缩放因子进行计算。

For example, to find x in 3 : 8 = x : 24, multiply both terms of the left ratio by 3 (since 8 × 3 = 24), so x = 3 × 3 = 9. Another method is cross‑multiplication: 3 × 24 = 8 × x → 72 = 8x → x = 9.

例如，求 3 : 8 = x : 24 中的 x，将左侧比率的两项乘以 3（因为 8 × 3 = 24），所以 x = 3 × 3 = 9。另一种方法是交叉相乘：3 × 24 = 8 × x → 72 = 8x → x = 9。

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5. Sharing in a Given Ratio | 按给定比率分配

To divide a quantity according to a ratio, first find the total number of parts by adding the terms. Then calculate the value of one part by dividing the total amount by the total number of parts. Finally, multiply the value of one part by each term in the ratio.

要按比率分配一个总量，首先把所有项相加得出总份数。然后用总量除以总份数来计算每一份的值。最后，将每一份的值乘以比率中的每一项。

For example, share £60 between Anna and Ben in the ratio 3 : 2. Total parts = 3 + 2 = 5. One part = £60 ÷ 5 = £12. Anna receives 3 × £12 = £36, Ben receives 2 × £12 = £24.

例如，按 3 : 2 的比例将 60 英镑分给安娜和本。总份数 = 3 + 2 = 5。一份 = 60 英镑 ÷ 5 = 12 英镑。安娜得到 3 × 12 英镑 = 36 英镑，本得到 2 × 12 英镑 = 24 英镑。

Always check your answer by adding the individual amounts: £36 + £24 = £60, which matches the original total. This verification step is quick and helps avoid mistakes.

务必通过把各人所得相加来检查答案：36 英镑 + 24 英镑 = 60 英镑，与原始总量一致。这个验证步骤快捷，有助于避免失误。

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6. The Unitary Method and Ratios | 单位法与比率

The unitary method involves finding the value of one unit first, which links directly to ratio problems. If 5 pens cost £3.50, the cost of 8 pens can be found by calculating the cost per pen: £3.50 ÷ 5 = £0.70, then £0.70 × 8 = £5.60.

单位法涉及首先求出一个单位的值，这与比率问题直接相关。如果 5 支笔花费 3.50 英镑，求 8 支笔的费用可以先算出每支笔的价格：3.50 英镑 ÷ 5 = 0.70 英镑，然后 0.70 英镑 × 8 = 5.60 英镑。

This approach is especially useful when comparing ‘best buys’. For example, 300 g of cereal for £1.80 versus 500 g for £2.80. Find the price per 100 g: £1.80 ÷ 3 = £0.60 and £2.80 ÷ 5 = £0.56, so the larger pack is better value.

这种方法在比较“最优惠购买”方案时特别有用。例如，300 克麦片售价 1.80 英镑与 500 克售价 2.80 英镑。求出每 100 克的价格：1.80 英镑 ÷ 3 = 0.60 英镑，2.80 英镑 ÷ 5 = 0.56 英镑，因此大包装更划算。

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7. Proportions and Direct Proportion | 比例与正比例

Two quantities are in direct proportion when they increase or decrease at the same rate. The ratio between them stays constant. If y is directly proportional to x, we write y ∝ x, meaning y = kx for some constant k.

当两个量以相同速率增大或减小时，它们就成正比关系。它们之间的比值保持不变。如果 y 与 x 成正比，我们写作 y ∝ x，意味着 y = kx，其中 k 是某个常数。

For instance, if 3 exercise books cost £4.50, you can find the cost of 7 books using the constant cost per book: k = £4.50 ÷ 3 = £1.50 per book. For 7 books, cost = 7 × £1.50 = £10.50. The ratio 3 : 4.50 is equivalent to 7 : 10.50.

例如，如果 3 本练习本花费 4.50 英镑，你可以利用每本价格常数求出 7 本的花费：k = 4.50 英镑 ÷ 3 = 1.50 英镑/本。7 本的总花费 = 7 × 1.50 英镑 = 10.50 英镑。比率 3 : 4.50 等价于 7 : 10.50。

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8. Using Ratio Tables | 使用比率表格

Ratio tables help organise equivalent ratios and solve proportion problems systematically. Create a table with the given quantities in the first column and multiply or divide across columns to find unknowns.

比率表格有助于系统地整理等价比率并解决比例问题。建立一张表格，将给定量放在第一列，然后通过各列之间的乘法或除法求未知数。

Number of apples	3	?
Cost (pence)	90	240

Here, 3 apples cost 90p. To find how many apples cost £2.40 (240p), note the multiplier from 90 to 240 is 240 ÷ 90 = 8/3. So number of apples = 3 × (8/3) = 8. Alternatively, use the unitary method: 1 apple costs 30p, so 240 ÷ 30 = 8 apples.

这里，3 个苹果花费 90 便士。求 2.40 英镑（240 便士）可以买多少个苹果，注意到从 90 到 240 的乘数是 240 ÷ 90 = 8/3。因此苹果数量 = 3 × (8/3) = 8。也可以使用单位法：每个苹果 30 便士，所以 240 ÷ 30 = 8 个苹果。

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9. Scale Drawings and Maps | 比例制图与地图

A scale ratio like 1 : 100 means that 1 unit on the drawing represents 100 units in real life. If a map has a scale of 1 : 50 000, then 1 cm on the map represents 50 000 cm (or 500 m) on the ground.

比例尺如 1 : 100 表示图上的 1 个单位代表实际生活中的 100 个单位。如果地图的比例尺是 1 : 50 000，那么图上的 1 厘米代表实际地面的 50 000 厘米（即 500 米）。

To find the real distance between two points that are 3.5 cm apart on the map, calculate: real distance = 3.5 × 50 000 cm = 175 000 cm = 1 750 m = 1.75 km. Always convert to sensible units for the final answer.

求地图上相距 3.5 厘米的两点间的实际距离，计算：实际距离 = 3.5 × 50 000 厘米 = 175 000 厘米 = 1 750 米 = 1.75 千米。最终答案务必转换为合理的单位。

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10. Working with Mixed Units in Scales | 处理比例尺中的混合单位

Sometimes you must convert between metres and centimetres in scale problems. Remember: 1 m = 100 cm. For a plan drawn to scale 1 : 200, a wall of real length 4.6 m is 460 cm, so on the plan it measures 460 ÷ 200 = 2.3 cm.

有时在比例尺问题中需要在米和厘米之间进行转换。记住：1 米 = 100 厘米。对于按 1 : 200 比例绘制的平面图，一段实际长度为 4.6 米的墙即 460 厘米，因此在图上测量为 460 ÷ 200 = 2.3 厘米。

When converting from drawing to real size, multiply the drawing length by the scale factor. A 1.5 cm line on a 1 : 250 plan represents 1.5 × 250 = 375 cm = 3.75 m in reality.

当从图纸尺寸转换为实际尺寸时，将图纸上的长度乘以比例因子。一张 1 : 250 的图纸上 1.5 厘米的线段，实际代表 1.5 × 250 = 375 厘米 = 3.75 米。

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

A frequent mistake is to treat a ratio as a simple additive relationship. If the ratio of boys to girls is 3 : 2, it does NOT mean there are 3 boys and 2 girls — those are just parts. Similarly, do not mix up the order: the ratio 3 : 2 is different from 2 : 3.

一个常见错误是将比率看作是简单的加法关系。如果男女比是 3 : 2，这并不意味着有 3 个男生和 2 个女生——那些只是份数。同样，不要混淆顺序：3 : 2 与 2 : 3 是不同的。

Another tip: when a quantity is divided in a ratio, always add the parts to find the total, then divide the actual amount by that total. And if a ratio includes fractions, find a common denominator and multiply all terms by that denominator to clear the fractions.

另一个提示：当按比例分配数量时，一定要先加总份数求出总份数，再用实际数量除以总份数。如果比率中包含分数，求出公分母，然后所有项乘以该分母以消去分数。

To convert a ratio of fractions like ½ : ⅓, multiply both sides by the LCM of 2 and 3, which is 6: (½ × 6) : (⅓ × 6) = 3 : 2. This gives a simple integer ratio.

要将含有分数的比率如 ½ : ⅓ 化为整数比，两边同时乘以 2 和 3 的最小公倍数 6：(½ × 6) : (⅓ × 6) = 3 : 2。这就得到了一个简单的整数比。

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12. Practice and Real‑Life Applications | 练习与现实生活应用

Ratios and proportions appear in countless everyday situations: mixing concrete, diluting juice, converting currencies, reading nutrition labels, and adjusting photo sizes. Mastering them builds a foundation for advanced topics like trigonometry and coordinate geometry.

比率与比例出现在无数日常情境中：搅拌混凝土、稀释果汁、兑换货币、阅读营养标签以及调整照片尺寸。掌握它为学习三角函数和坐标几何等高级课题打下基础。

When revising, practice problems that require you to identify the correct ratio, simplify it, and then use proportion to find unknown quantities. Draw diagrams or use ratio tables to visualise the relationships. With consistent practice, ratio problems will become second nature.

复习时，多多练习需要你识别正确的比率、化简、然后使用比例求未知量的题目。可以画图或借助比率表格将关系可视化。通过持续练习，比率问题将变得轻松自如。

Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

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