Mastering Ratio and Proportion from Page 154 | 第154页比与比例精讲

📚 Mastering Ratio and Proportion from Page 154 | 第154页比与比例精讲

Welcome to your focused revision guide for the ratio and proportion questions on page 154. This article breaks down every key concept step by step, using clear examples and paired explanations, so you can tackle these problems with confidence and precision. Whether you are preparing for a Checkpoint test or simply mastering the topic, the following sections will give you the tools you need.

欢迎阅读第154页比与比例问题的专项复习指南。本文通过清晰的示例和中英对照讲解，逐步拆解每一个核心概念，帮助你自信、准确地解决这些题目。无论你是在准备Checkpoint考试还是单纯想扎实掌握这一主题，接下来的小节都将为你提供所需的工具。

1. Understanding Ratio Notation | 理解比的表示法

A ratio compares two or more quantities. It can be written in several equivalent forms: using a colon (a:b), as a fraction a/b, or with the phrase ‘a to b’. The order of a ratio matters greatly — 3:5 is not the same as 5:3.

比用于比较两个或更多的数量。它可以用几种等价的形式表示：用冒号（a:b）、作为分数 a/b，或用短语 “a 比 b”。比的顺序非常重要——3:5 与 5:3 并不相同。

For example, if a fruit bowl contains 4 apples and 6 bananas, the ratio of apples to bananas is 4:6. This ratio can also be expressed as 4/6 or ‘4 to 6’. The ratio of bananas to apples would be 6:4.

例如，如果一个水果碗里有4个苹果和6根香蕉，苹果与香蕉的比是 4:6。这个比也可以写成 4/6 或 “4 比 6”。而香蕉与苹果的比则是 6:4。

When dealing with three or more items, extend the notation naturally: a:b:c. A recipe might require flour, butter and sugar in the ratio 3:2:1. This tells you the relative amounts of each ingredient.

当涉及三个或更多项目时，自然地扩展表示法：a:b:c。一份食谱可能需要面粉、黄油和糖的比为 3:2:1。这告诉了你每种配料的相对用量。

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2. Simplifying Ratios | 化简比

Simplifying a ratio makes it easier to understand and compare. To simplify, divide all parts of the ratio by their greatest common divisor (GCD). The simplified ratio should contain the smallest possible whole numbers that keep the same relationship.

化简比可以让它更容易理解和比较。化简时，用比的各个部分除以它们的最大公约数 (GCD)。化简后的比应包含保持相同关系的最小可能整数。

Consider the ratio 15:25. The GCD of 15 and 25 is 5. Divide both numbers by 5 to get 3:5. Check that 15/25 = 3/5, so the proportion is unchanged. If the ratio includes different units, convert to the same unit first.

考虑比 15:25。15 和 25 的最大公约数是 5。将两个数都除以 5 得到 3:5。验证 15/25 = 3/5，所以比例保持不变。如果比中包含不同的单位，要先转换为相同的单位。

Example: simplify 1.5 m : 30 cm. Convert 1.5 m to 150 cm. The ratio becomes 150:30, which simplifies by dividing both by 30 to give 5:1. Always aim for integer-only ratios in simplest form.

示例：化简 1.5 米 : 30 厘米。将 1.5 米转换为 150 厘米。比变为 150:30，两边都除以 30 化简得到 5:1。始终以求最简整数比为目标。

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3. Sharing Quantities in a Given Ratio | 按给定比例分配数量

To share an amount in a given ratio, first find the total number of parts by adding all the ratio numbers. Then divide the total amount by the total number of parts to find the value of one part. Multiply each ratio number by this value to obtain the individual shares.

要按给定比例分配一个总量，首先把比的所有数字相加，得到总份数。然后用总量除以总份数，求出一份的值。再用每个比数乘以这个值，从而得到各个份额。

For instance, share £120 between Alex and Ben in the ratio 2:3. Total parts = 2 + 3 = 5. One part is £120 ÷ 5 = £24. Alex gets 2 × £24 = £48, and Ben gets 3 × £24 = £72. Always check that the sum of shares equals the original total.

例如，将 £120 按 2:3 的比例分给 Alex 和 Ben。总份数 = 2 + 3 = 5。一份是 £120 ÷ 5 = £24。Alex 得 2 × £24 = £48，Ben 得 3 × £24 = £72。始终检查全部份额之和是否等于原总量。

When sharing among three or more people, the same method applies. If a ratio includes a fraction, first multiply to clear denominators. A ratio of 1/2 : 2/3 can be transformed by multiplying both by 6 (the LCM of 2 and 3) to get 3:4, then share as usual.

在三个人或更多人之间分配，方法相同。如果比中包含分数，先乘以公分母以消除分母。比 1/2 : 2/3 可以通过两边同乘 6（2 和 3 的最小公倍数）转化为 3:4，然后照常分配。

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4. Working with Unitary Ratios and the Form 1:n | 运用归一比与 1:n 形式

Sometimes it is useful to express a ratio so that one part equals 1. This is called a unitary ratio, often written in the form 1:n. To convert a ratio a:b to 1:n, divide both parts by a. The result is 1 : b/a.

有时将比表示成某一部分为 1 的形式很有用，这被称为归一比，通常写成 1:n 的形式。要把比 a:b 转换成 1:n，将两部分都除以 a，结果就是 1 : b/a。

For example, express 8:20 in the form 1:n. Divide both sides by 8: 8÷8 = 1, and 20÷8 = 2.5. So 8:20 is equivalent to 1:2.5. This is particularly handy when comparing prices or speeds.

例如，将 8:20 表示成 1:n 的形式。两边同时除以 8：8÷8=1，20÷8=2.5。所以 8:20 等价于 1:2.5。在比较价格或速度时这特别方便。

The unitary form helps answer questions like ‘Which is better value?’ If 350 g of cereal costs £1.40 and 500 g costs £1.90, find the price per 1 g. The ratios are 1.40:350 → 1:250 for the first (dividing 1.40/350 = 0.004, so 1 g costs £0.004) and 1.90:500 → 1:263.16, but easier to compare directly the unit cost per gram.

归一形式有助于回答 “哪种更划算？” 这样的问题。如果 350 克麦片售价 £1.40，500 克售价 £1.90，求出每克的价格。第一个比 1.40:350 → 1:250（1.40/350 = 0.004，所以每克 £0.004），第二个 1.90:500 → 约 1:263.16，但直接比较每克的成本更简单。

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5. Direct Proportion and the Constant of Proportionality | 正比例与比例常数

Two quantities are in direct proportion if they increase or decrease together at the same rate. This relationship can be written as y = kx, where k is the constant of proportionality. If y is directly proportional to x, the graph is a straight line through the origin.

如果两个量以相同的速率同时增加或减少，它们就成正比例关系。这种关系可以写作 y = kx，其中 k 是比例常数。如果 y 与 x 成正比，其图像是一条通过原点的直线。

For instance, if 5 notebooks cost £7.50, the cost is directly proportional to the number of notebooks. Here k = £7.50/5 = £1.50 per notebook. The equation is Cost = 1.50 × number. Use this to find the cost of any number of notebooks.

例如，如果 5 个笔记本售价 £7.50，总价就与笔记本的数量成正比。这里 k = £7.50/5 = £1.50 每个。方程为 总价 = 1.50 × 数量。可以用它求出任意数量笔记本的总价。

To solve proportion problems, identify k first. If y = 24 when x = 6, then k = y/x = 24/6 = 4. So the equation is y = 4x. Then if x = 9, y = 4 × 9 = 36. Always look for a constant multiplier between the two quantities.

解决比例问题要先确定 k。如果 x=6 时 y=24，那么 k = y/x = 24/6 = 4。所以方程是 y=4x。那么当 x=9 时，y=4×9=36。始终要在两个量之间寻找一个固定的乘数。

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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 general form is y = k/x, or xy = k. This is common in time and speed problems.

反比例描述的是当一个量增加时另一个量减少、且它们的乘积保持恒定的关系。一般形式是 y = k/x，或 xy = k。这在时间与速度的问题中很常见。

Example: if 3 workers take 8 hours to complete a task, then the time taken is inversely proportional to the number of workers. The constant k = number of workers × time = 3 × 8 = 24. So if 4 workers do the same job, the time t follows 4 × t = 24, giving t = 6 hours.

示例：如果 3 名工人完成一项任务需要 8 小时，那么所用时间与工人数量成反比。常数 k = 工人数 × 时间 = 3 × 8 = 24。因此，如果 4 名工人做同样的工作，时间 t 满足 4 × t = 24，解得 t = 6 小时。

Note the difference between direct and inverse proportion. In direct proportion, doubling one quantity doubles the other. In inverse proportion, doubling one quantity halves the other. Always check whether the total amount stays fixed or whether the quantities vary together in the same direction.

注意正比例与反比例的区别。在正比例中，一个量翻倍，另一个量也翻倍。在反比例中，一个量翻倍，另一个量则减半。一定要检查总量是否保持不变，还是两个量沿同一方向变化。

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7. Scale Factors, Maps and Models | 比例因子、地图与模型

Ratios are essential for reading maps and building scale models. A map scale such as 1:50 000 means that 1 cm on the map represents 50 000 cm (or 500 m) in real life. This is a typical unitary ratio form, making conversions straightforward.

比是查看地图与制作比例模型的关键。比例尺如 1:50 000 表示地图上的 1 厘米代表现实中的 50 000 厘米（即 500 米）。这是一种典型的归一比形式，使得转换非常直接。

To find the actual distance between two points on a map, measure the map distance and multiply by the scale factor. If two towns are 8 cm apart on a 1:25 000 map, the real distance is 8 × 25 000 = 200 000 cm, which is 2 km.

要计算地图上两点间的实际距离，先量出图上距离，然后乘以比例因子。在 1:25 000 的地图上，如果两镇相距 8 厘米，实际距离就是 8 × 25 000 = 200 000 厘米，即 2 公里。

When scaling up a model, use the scale factor as a multiplier. For a model car at scale 1:24, the length of the real car is 24 times the model’s length. Conversely, to make a scale drawing of a room, divide real measurements by the scale factor. Mixed units are common, so always convert to consistent units before multiplying.

当要把模型放大时，把比例因子当作乘数。对于一个 1:24 的汽车模型，真车长度是模型的 24 倍。反之，要画一个房间的比例图，则将实际尺寸除以比例因子。混合单位很常见，所以乘之前一定要转换成一致的单位。

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8. Rate, Speed and Compound Ratios | 速率、速度与复合比

A rate is a special ratio that compares two quantities with different units. Speed is a rate comparing distance and time, often expressed in km/h or m/s. The relationship is Speed = Distance ÷ Time, which is a direct proportion if speed is constant.

速率是一种特殊的比，它比较两个不同单位下的量。速度就是一种比较距离和时间的速率，通常以 km/h 或 m/s 表示。关系式为 速度 = 距离 ÷ 时间，在速度恒定时这是一个正比例关系。

Example: a cyclist travels 45 km in 3 hours. The average speed is 45 ÷ 3 = 15 km/h. If the cyclist keeps this rate, the distance travelled in 5 hours is 15 × 5 = 75 km. Ratios involving time require careful unit conversion: 1 hour = 60 minutes, 1 minute = 60 seconds.

示例：一名自行车手 3 小时骑行 45 公里。平均速度为 45 ÷ 3 = 15 km/h。如果他保持这个速率，5 小时骑行的距离就是 15 × 5 = 75 公里。涉及时间的比需要仔细转换单位：1 小时 = 60 分钟，1 分钟 = 60 秒。

Compound ratios combine more than two pieces of information, such as fuel efficiency (miles per gallon) or population density (people per km²). Treat them like a simple ratio: write the quantity before ‘per’ in the numerator and the quantity after in the denominator.

复合比融合了两个以上的信息，比如燃油效率（英里每加仑）或人口密度（人每平方公里）。把它们当作简单的比来处理：把 “per” 前面的量放在分子上，后面的量放在分母上。

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9. Solving Ratio and Proportion Word Problems from Page 154 | 解答第154页的比与比例文字题

Page 154 features typical word problems that blend ratio reasoning with everyday situations. The key is to identify whether the problem involves a part-part relationship or a part-whole relationship. Then model the information using a diagram or an equation.

第154页呈现的是将比推理与日常情境相结合的典型文字题。关键在于识别问题是涉及部分与部分的关系还是部分与整体的关系。然后用图表或方程来构建信息模型。

One common type: ‘The ratio of boys to girls in a class is 3:4. If there are 28 students altogether, how many boys are there?’ Total parts = 3+4 = 7. One part = 28 ÷ 7 = 4. Boys = 3 × 4 = 12. Another: ‘A rope is cut into three pieces in the ratio 2:3:5. The longest piece is 35 cm. Find the total length.’ Longest part = 5 parts = 35 cm → 1 part = 7 cm. Total = 2+3+5 = 10 parts → 70 cm.

一个常见类型：“一个班级中男生与女生的比是 3:4。如果总共有 28 名学生，男生有多少人？” 总份数 = 3+4=7。一份 = 28÷7=4。男生 = 3×4=12。另一个例子：“一根绳子按 2:3:5 的比例剪成三段。最长一段是 35 厘米。求总长。” 最长部分 = 5 份 = 35 cm → 1 份 = 7 cm。总份数 = 2+3+5=10 → 总长 70 cm。

When a ratio changes after adding or removing items, set up an equation using the original ratio and the new ratio. For example, a mixture has oil and vinegar in ratio 3:1. After adding 60 ml of oil, the ratio becomes 5:1. Find the original amount of vinegar. Let the original amounts be 3x oil and x vinegar. After adding 60 ml oil, oil becomes 3x+60. New ratio (3x+60):x = 5:1 → 3x+60 = 5x → 2x = 60 → x = 30 ml. Vinegar is 30 ml.

当比在添加或移除物品后发生变化时，利用原比和新比建立方程。例如，一种混合物中油和醋的比为 3:1。加入 60 毫升油后，比变为 5:1。求原来醋的量。设原来油量为 3x，醋为 x。加入 60 毫升油后，油变为 3x+60。新比 (3x+60):x = 5:1 → 3x+60 = 5x → 2x = 60 → x = 30 ml。所以醋为 30 毫升。

Always double-check that your answer satisfies all the conditions given in the problem. Drawing a bar model can help you visualise the ratios and avoid common mistakes.

始终复查你的答案是否满足题目中给出的所有条件。画条形模型有助于将比例可视化和避免常见错误。

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10. Common Errors and How to Avoid Them | 常见错误及其避免方法

Mistake 1: Misreading the order of the ratio. If a problem says ‘the ratio of red to blue is 1:3’, red is the first number. Swapping them leads to wrong shares. Always label which quantity corresponds to each number.

错误一：弄错比的顺序。如果题目说 “红与蓝的比是 1:3”，红色就是第一个数。把它们互换会导致份额错误。一定要标注每个数对应哪个量。

Mistake 2: Not simplifying units. A ratio of 50p : £2 must be expressed in the same unit, e.g. 50:200 → 1:4. Leaving mixed units gives an incorrect ratio.

错误二：没有统一单位。比 50便士 : 2英镑 必须用同一单位表示，例如 50:200 → 1:4。保留混合单位会得到错误的比。

Mistake 3: Forgetting that the total number of parts is the sum, not the difference. In sharing £90 in ratio 5:7, total parts = 12, not 2. One part = £90/12 = £7.50.

错误三：忘记总份数是各数之和，而不是差。按 5:7 分配 £90，总份数 = 12，不是 2。一份 = £90/12 = £7.50。

Mistake 4: Ignoring the constant of proportionality in inverse proportion. Always find k first, then use xy = k. Do not assume direct proportion unless the problem states it.

错误四：在反比例中忽视比例常数。始终先求出 k，然后使用 xy = k。除非题目明确指出，否则不要假设为正比例。

Review each step and ask: ‘Does my answer make sense in the context?’ This habit will save you marks on your Checkpoint exam and problem sets like page 154.

检查每一步并问自己：“在这个语境下我的答案合理吗？” 这个习惯能为你在 Checkpoint 考试和像第154页这样的练习中赢得分数。

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