Ratios and Proportions (p254) | 比率与比例 (第254页)

📚 Ratios and Proportions (p254) | 比率与比例 (第254页)

In this article, based on page 254 of the Cambridge KS3 Mathematics coursebook, we explore the core concepts of ratios and proportions. You will learn how to write, simplify and apply ratios, understand direct and inverse proportion, and use proportional reasoning to solve real‑world problems. Each section pairs an English explanation with its Chinese equivalent, ensuring bilingual clarity and exam readiness.

本文基于剑桥 KS3 数学教材第254页内容，深入探讨比率与比例的核心概念。你将学习如何书写、化简和应用比率，理解正比例与反比例，并运用比例推理解决实际问题。每个部分都提供中英双语对照讲解，帮助你清晰理解，为考试做好准备。

1. Understanding Ratios | 理解比率

A ratio compares the sizes of two or more quantities. It shows how many times one quantity contains another. For example, if we have 3 red balls and 5 blue balls, the ratio of red to blue is written as 3 : 5. Ratios can also be written in the form of fractions, such as 3/5, but the colon notation is standard when comparing parts to parts.

比率用于比较两个或多个量的大小。它表示一个量包含另一个量的倍数。例如，如果有3个红球和5个蓝球，红球与蓝球的比率写作 3 : 5。比率也可以用分数形式表示，如 3/5，但在进行部分与部分的比较时，冒号记法是标准形式。

A ratio has no units – it is a pure number comparison.
比率没有单位——它是纯数字的比较。
The order of the terms is important: 3 : 5 is not the same as 5 : 3.
各项的顺序很重要：3 : 5 与 5 : 3 并不相同。

2. Simplifying Ratios | 化简比

To simplify a ratio, divide all of its terms by their greatest common divisor (GCD). For instance, the ratio 12 : 8 can be simplified by dividing both numbers by 4, giving 3 : 2. A ratio is in its simplest form when all terms are whole numbers with no common factor other than 1.

化简比率时，用各项的最大公约数（GCD）去除每一项。例如，比率 12 : 8 可以通过同时除以4化简为 3 : 2。当比值的各项都是整数且除1以外没有其他公因数时，该比率就是最简形式。

12 : 8 = 3 : 2 (after dividing by 4)
12 : 8 = 3 : 2（除以4之后）

If the ratio contains fractions or decimals, multiply by a common denominator to clear them.
如果比率包含分数或小数，可以乘以公分母来消去它们。
For example, 0.5 : 1.5 becomes 1 : 3 after multiplying by 2.
例如，0.5 : 1.5 乘以2后变为 1 : 3。

3. Ratios in Different Forms | 比率的不同形式

Ratios can be expressed in several equivalent ways. Apart from the colon form a : b, we often use the language of ‘for every’ or write the ratio as a fraction of the total. Knowing how to move between these forms is crucial for solving problems.

比率可以用多种等价形式表示。除了冒号形式 a : b 外，我们经常使用“每……对应……”的说法，或者把比率写成占总数几分之几的形式。能在这些形式之间灵活转换是解决问题的关键。

Ratio (part : part)	Fraction of one part	Verbal statement
3 : 2	3/5 of the total is the first part	For every 3 of A, there are 2 of B

比率 3 : 2 意味着总份数为5，第一部分占3/5，第二部分占2/5。

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

When dividing an amount in a ratio a : b, first find the total number of parts (a + b). Then divide the total quantity by that sum to find the value of one part. Finally multiply by a and b to obtain the individual shares. For example, divide £60 in the ratio 3 : 2: total parts = 5, one part = £12, so shares are 3 x £12 = £36 and 2 x £12 = £24.

按比率 a : b 分配数量时，先求出总份数 (a + b)。然后将总量除以总份数，得到一份的值。最后分别乘以 a 和 b，就得到各自应占的数量。例如，将 £60 按 3 : 2 分配：总份数5，一份£12，因此份额为 3×£12 = £36 和 2×£12 = £24。

Always check that the sum of the individual shares equals the original total.
务必检查各份额之和是否等于原来的总量。
This method works for lengths, masses, money and any measurable quantity.
该方法适用于长度、质量、金额以及任何可度量的量。

5. Direct Proportion | 正比例

Two quantities are in direct proportion if they increase or decrease at the same rate. The relationship can be expressed as y = k x, where k is the constant of proportionality. For example, if the cost of apples is directly proportional to the number bought, then doubling the number doubles the cost.

两个量如果以相同的速率增加或减少，它们就成正比例。这种关系可以表示为 y = k x，其中 k 是比例常数。例如，若苹果的总价与购买的数量成正比例，那么数量加倍，总价也加倍。

y ∝ x means y = k x

If x = 2 and y = 10, then k = 10 / 2 = 5, so y = 5x.
如果 x = 2，y = 10，则 k = 10 / 2 = 5，因此 y = 5x。
The graph of y = k x is a straight line through the origin.
y = k x 的图像是一条经过原点的直线。

6. Inverse Proportion | 反比例

Inverse proportion describes a relationship where one quantity increases as the other decreases, such that their product remains constant. The equation is y = k / x or x y = k. Examples include the time taken to travel a fixed distance at different speeds.

反比例描述的是一种一个量增大时另一个量反而减小的关系，且它们的乘积保持不变。其方程式为 y = k / x 或 x y = k。常见的例子是以不同速度行驶固定距离所需的时间。

y ∝ 1/x means x y = k

If x = 4 gives y = 3, then k = 4 × 3 = 12, so y = 12 / x.
如果 x = 4 时 y = 3，则 k = 4 × 3 = 12，因此 y = 12 / x。
Inverse proportion graphs are hyperbolas and never touch the axes.
反比例图像是双曲线，且永远不会与坐标轴相交。

7. Proportional Reasoning with Graphs | 用图像进行比例推理

Graphs are a powerful tool for understanding proportional relationships. For direct proportion, the graph is a straight line passing through (0,0) with slope k. For inverse proportion, the product of x and y is constant, giving a curved shape. Interpreting coordinates on these graphs helps solve problems without algebraic manipulation.

图像是理解比例关系的有力工具。正比例的图像是一条经过原点 (0,0)、斜率为 k 的直线。反比例的图像由于 x 与 y 的乘积恒定，呈现曲线形状。通过解读图像上的坐标，可以不用代数变形就解决问题。

To check if a set of points represents direct proportion, plot them and see if they lie on a straight line through the origin. If they do, the ratio y/x is the same for all points. For inverse proportion, check whether x × y is constant.

要检查一组数据点是否表示正比例，可以把它们画出来，看是否位于一条经过原点的直线上。如果是，那么所有点的 y/x 比值都相同。对于反比例，则检查 x × y 是否为定值。

8. Scale Drawings and Maps | 比例尺绘图与地图

A scale is a ratio that compares a distance on a drawing or map to the actual distance. Scale is usually written as 1 : n, meaning 1 unit on the drawing represents n units in real life. For example, a map scale of 1 : 50 000 means 1 cm on the map represents 50 000 cm (or 0.5 km) in reality.

比例尺是用来比较图纸或地图上的距离与实际距离的比率。比例尺通常写作 1 : n，表示图上1个单位代表现实中的 n 个单位。例如，地图比例尺 1 : 50 000 表示图上1厘米代表实际 50 000 厘米（即0.5千米）。

To find the real length, multiply the drawing length by the scale factor n.
要求实际长度，用图上长度乘以比例因子 n。
To find the drawing length, divide the real length by n.
要求图上长度，用实际长度除以 n。
Always convert all measurements to the same units before calculating.
计算前务必将所有度量单位统一。

9. Rates and Unit Rates | 率与单位率

A rate compares two quantities with different units, such as speed (km/h) or density (kg/m³). The unit rate is the value for one unit of the first quantity, and it helps us compare different situations easily. Working with unit rates is very similar to ratio reasoning.

率是用来比较两个不同单位量的，如速度（千米/小时）或密度（千克/立方米）。单位率是指第一个量取一个单位时所对应的值，它能帮助我们方便地进行比较。单位率的处理与比率的推理非常相似。

For example, if a car travels 300 km on 25 litres of fuel, the fuel consumption unit rate is 300/25 = 12 km per litre. Conversely, the amount of fuel used per km is 25/300 = 0.0833 litres per km.

例如，一辆汽车用25升燃油行驶300千米，则油耗单位率为 300/25 = 12 千米/升。反过来，每千米的燃油消耗量为 25/300 ≈ 0.0833 升/千米。

10. Converting Between Fractions, Decimals and Ratios | 分数、小数与比率之间的转化

Ratios are closely linked to fractions and decimals. A part‑to‑whole ratio can be written as a fraction of the whole. For instance, if the ratio of boys to girls is 2 : 3, then the fraction of boys is 2/(2+3) = 2/5, which equals 0.4 or 40%. This fluency in conversion is tested frequently at KS3.

比率与分数、小数密切相关。部分与整体的比可以写成一个分数。例如，男生与女生的比率是 2 : 3，则男生所占的比例为 2/(2+3) = 2/5，即 0.4 或 40%。在 KS3 阶段经常会考查这种流利的转换能力。

Ratio (part : part)	Fraction of first part	Decimal / Percentage
1 : 4	1/5	0.2 = 20%
3 : 7	3/10	0.3 = 30%

将比率转换为分数或百分比，便于在统计和概率问题中使用。

11. Problem Solving with Ratios | 比率问题解决

Many real‑life problems involve ratios, from mixing ingredients in cooking to adjusting the dimensions of similar shapes. The key is to identify the total number of parts and the value of one part. Drawing a bar model or using an equation can help visualise the relationship and avoid mistakes.

许多现实生活中的问题都涉及比率，从烹饪中混合配料到调整相似图形的尺寸。关键在于确定总份数和一份的值。画条形模型或使用方程有助于将关系可视化，避免错误。

Example: In a recipe, flour and sugar are mixed in the ratio 5 : 2. If you use 350 g of flour, how much sugar is needed? Set up the proportion: 5 parts = 350 g, so 1 part = 70 g. Then sugar (2 parts) = 2 × 70 = 140 g.

例题：一个食谱中，面粉和糖以 5 : 2 的比例混合。如果使用了350克面粉，需要多少克糖？建立比例关系：5份 = 350克，则1份 = 70克。那么糖（2份）= 2 × 70 = 140克。

12. Summary and Key Points | 总结与要点

Ratios and proportions provide a framework for comparing quantities and scaling values efficiently. Remember to always simplify ratios, identify the constant of proportionality in direct and inverse relationships, and practise converting between different representations. Mastering these skills will give you a strong foundation for algebra, geometry and data handling in later studies.

比率和比例为比较量和等比例缩放提供了有效框架。记住要始终化简比率，识别正、反比例关系中的比例常数，并多加练习不同表示形式之间的转换。掌握这些技能将为后续的代数、几何和数据处理学习打下坚实基础。

Ratio compares parts; fraction compares a part to a whole.
比率比较部分与部分；分数比较部分与整体。
Direct proportion: y = k x; Inverse proportion: x y = k.
正比例：y = k x；反比例：x y = k。
Scale drawings use a ratio to relate map distances to real distances.
比例尺绘图使用比率联系图上距离与实际距离。
Unit rates help compare different scenarios fairly.
单位率有助于公平地比较不同情况。

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Ratios and Proportions (p254) | 比率与比例 (第254页)