Ratio and Proportion: Mastering Key Concepts from p109.pdf | 比和比例：掌握p109.pdf中的核心概念

📚 Ratio and Proportion: Mastering Key Concepts from p109.pdf | 比和比例：掌握p109.pdf中的核心概念

Welcome to this comprehensive revision guide on ratio and proportion, adapted from the essential exercises found in p109.pdf of the Cambridge KS3 Mathematics course. Whether you are preparing for a checkpoint test or simply strengthening your skills, understanding ratios is crucial for solving real‑world problems. In this article, we will break down the fundamentals, tackle common questions from the worksheet, and equip you with robust problem‑solving strategies.

欢迎阅读这份关于比和比例的全面复习指南，内容改编自剑桥KS3数学课程中p109.pdf的重要练习。无论你是在准备Checkpoint测试还是巩固技能，理解比对于解决实际问题至关重要。本文将分解基础知识，攻克练习册中的常见题型，并为你提供强大的解题策略。

1. Understanding Ratios | 理解比

A ratio compares two or more quantities, showing the relative size of one quantity to another. For example, in a class of 25 students, if there are 10 boys and 15 girls, the ratio of boys to girls is 10 : 15. In p109.pdf, Question 1 asks you to write ratios from given scenarios, such as “the number of red apples to green apples”.

比用来比较两个或多个数量，显示一个量相对于另一个量的大小。例如，一个班有25名学生，其中10名男生和15名女生，男生与女生的比是10 : 15。在p109.pdf的第1题中，要求你根据场景写出比，比如“红苹果与青苹果的数量比”。

The order in a ratio is important; the ratio of boys to girls is not the same as girls to boys. A ratio can be written using a colon ( : ), as a fraction (10/15) or in words ’10 to 15′.

比中的顺序很重要；男生与女生的比不同于女生与男生的比。比可以用冒号( : )表示，也可以写成分数(10/15)或用词语’10比15’表示。

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2. Simplifying Ratios | 简化比

To simplify a ratio, divide both terms by their greatest common factor (GCF). Q2 in p109.pdf presents the ratio 18:24. The GCF of 18 and 24 is 6, so 18 ÷ 6 : 24 ÷ 6 = 3 : 4.

简化比时，将两项同时除以它们的最大公因数(GCF)。p109.pdf中的第2题给出了比18:24。18和24的最大公因数是6，因此18 ÷ 6 : 24 ÷ 6 = 3 : 4。

When a ratio contains fractions or decimals, multiply through by a common denominator to obtain whole numbers. For instance, 0.5 : 1.5 becomes 1 : 3 after multiplying by 2.

当比中包含分数或小数时，乘以一个公分母以得到整数。例如，0.5 : 1.5在乘以2后变为1 : 3。

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3. Ratios and Fractions | 比与分数

Ratios can be expressed as fractions of a whole. For a part‑to‑part ratio of 2:3, the total parts = 2+3 = 5. The first quantity represents 2/5 of the whole, and the second represents 3/5. Question 4 on p109.pdf asks what fraction of the class are boys given the ratio of boys to girls is 2:3.

比可以表示为整体的分数。对于部分与部分之比2:3，总份数=2+3=5。第一个量占整体的2/5，第二个量占3/5。p109.pdf第4题问，在男生与女生比是2:3的情况下，男生占整个班级的几分之几。

It is important to note that the fraction from a part‑to‑part ratio is different from a part‑to‑whole fraction. Always identify whether the question asks for a part:whole fraction.

需要注意的是，从部分与部分之比得出的分数，与部分与整体之比不同。一定要看清题目是求部分:整体分数。

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

To divide a quantity in a given ratio, first find the total number of parts. For example, divide £240 in the ratio 3:5. Total parts = 3+5=8. One part = £240 ÷ 8 = £30. So the amounts are 3 × £30 = £90 and 5 × £30 = £150. This corresponds to Exercise 5 on p109.pdf.

按给定比例分配数量时，先求出总份数。例如，将£240按3:5分配。总份数=3+5=8。一份=£240÷8=£30。因此金额分别为3×£30=£90和5×£30=£150。这对应p109.pdf的第5题。

Always check your answer by adding the shares to ensure they sum to the original total. If you get 90 + 150 = 240, the work is correct.

务必检查你的答案，将各部分相加看是否等于原来的总量。如果90 + 150 = 240，则解题正确。

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5. Finding Missing Values in Proportions | 在比例中寻找缺失值

When two ratios are equivalent, you can find a missing term by scaling. If 3:4 = x:24, note that 4 has been multiplied by 6 to get 24, so multiply 3 by 6 to get x = 18. In p109.pdf, question 6 features similar proportion problems.

当两个比等价时，可以通过缩放找到缺失项。如果3:4 = x:24，可以注意到4乘以6得到24，因此将3乘以6得到x=18。p109.pdf第6题就是类似的比例问题。

Alternatively, use cross‑multiplication: a/b = c/d implies ad = bc. So 3/4 = x/24 → 3×24 = 4x → 72 = 4x → x = 18.

或者使用交叉相乘：a/b = c/d 推出 ad = bc。因此3/4 = x/24 → 3×24 = 4x → 72=4x → x=18。

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6. Scale Drawings and Maps | 比例图和地图

Scale drawings use ratios to represent real sizes. A map scale of 1 : 50,000 means 1 cm on the map represents 50,000 cm in reality. Question 8 in p109.pdf asks to calculate the actual distance between two towns given the map distance and scale.

比例图利用比例表现真实尺寸。地图比例尺1:50,000表示地图上1厘米代表实际50,000厘米。p109.pdf第8题要求根据地图距离和比例尺计算两个城镇之间的实际距离。

To work out the real length, multiply the measurement on the drawing by the scale factor. Ensure units are consistent; for example, convert km to cm if needed.

要计算实际长度，将图上的测量值乘以比例因子。确保单位一致；例如，需要时将公里转换为厘米。

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7. Ratio and Proportion Word Problems | 比和比例应用题

Word problems often incorporate ratios in contexts like mixing paint, sharing money, or comparing speeds. Read carefully to identify the total or the relationship. p109.pdf includes problem 10 about mixing cement and sand in the ratio 1:4, needing to find the amount of sand for a given cement.

应用题通常将比融入混合油漆、分钱或比较速度等场景。仔细阅读，找出总量或关系。p109.pdf包含第10题关于按1:4混合水泥和沙子，需要求出给定水泥所需的沙子量。

Use a unitary method: find the value of one part, then scale up. For example, with 1 part cement to 4 parts sand, if you have 5 kg of cement, one part = 5 kg, so sand = 4 × 5 kg = 20 kg.

使用归一法：先找到每一份的值，然后扩展。例如，1份水泥对应4份沙子，如果你有5千克水泥，一份=5千克，因此沙子=4×5千克=20千克。

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

If quantities have different units, convert them to the same unit before simplifying. For instance, 1 m : 30 cm should be converted to 100 cm : 30 cm, then simplified to 10:3. Question 12 in p109.pdf tests this skill.

如果量具有不同单位，在化简前先转换为相同单位。例如，1米:30厘米应转换为100厘米:30厘米，然后再简化为10:3。p109.pdf第12题考查这一技能。

Always express the ratio in its simplest form and without units. After simplifying, 10:3 carries no mention of cm.

始终用最简形式表示比，且不带单位。化简后，10:3不带任何厘米的提及。

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9. Recipe Problems | 食谱问题

Recipe problems involve scaling quantities proportionally. If a recipe for 4 people uses 200 g of flour, for 10 people you multiply by (10/4) = 2.5, so 200 g × 2.5 = 500 g. p109.pdf question 14 shows a similar scenario.

食谱问题涉及按比例调整用量。如果4人份的食谱需要200克面粉，那么10人份需要乘以(10/4)=2.5，即200克×2.5=500克。p109.pdf第14题展示了类似的场景。

Identify the constant of proportionality from the original recipe per person. Here, flour per person = 200 g ÷ 4 = 50 g, then for 10 people use 10 × 50 g = 500 g.

找出原食谱每人份的比例常数。此处，每人面粉量=200克÷4=50克，然后10人份为10×50克=500克。

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10. Best Buy and Value for Money | 最佳购买与性价比

To find the best buy, compare unit prices. A 500 g pack costing £2.50 gives a unit price of £2.50/500 = £0.005 per gram, or 0.5 pence per gram. Compare with another pack. p109.pdf question 16 asks which size of juice offers the best value.

要找到最佳购买，比较单位价格。一包500克售价2.50英镑，单位价格为2.50/500=0.005英镑每克，即0.5便士每克。与另一包装比较。p109.pdf第16题问哪种果汁包装最划算。

Use ratio tables or division to find cost per unit, then decide which option gives the lowest unit cost.

使用比表格或除法求出单位成本，然后决定哪个选项的单位成本最低。

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11. Inverse Proportion | 反比例

While direct proportion means as one quantity increases, the other increases at the same rate, inverse proportion means as one increases, the other decreases. For example, if 4 workers can build a wall in 6 hours, how long would 8 workers take? Assuming inverse proportion, the product is constant: 4 × 6 = 8 × h → h = 3 hours. p109.pdf introduces this concept in question 18.

正比例意味着一个量增加时，另一个以相同速率增加；而反比例则意味着一个增加，另一个减少。例如，如果4个工人6小时可以砌一堵墙，8个工人需要多长时间？假设反比例，乘积恒定：4×6 = 8×h → h = 3小时。p109.pdf在第18题介绍了这一概念。

Always verify that the scenario is indeed inversely proportional before applying the method. Look for words like ‘identical machines’ or ‘same work rate’.

在应用该方法前，务必确认该情景确实为反比例。寻找像’相同的机器’或’相同的工作速率’这样的词语。

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12. Mixed Practice from p109.pdf | p109.pdf 综合练习

Let’s consolidate with a mixed exercise from p109.pdf. Simplify the ratio 24:36:60. Find the HCF of 24, 36, 60 which is 12, so divide each term by 12 → 2:3:5.

我们通过p109.pdf的综合练习来巩固。化简比24:36:60。找到24、36、60的最大公因数12，各项除以12 → 2:3:5。

Another question: If 5 pens cost £3.25, what is the cost of 8 pens? Find unit cost: £3.25 ÷ 5 = £0.65 per pen, then multiply by 8 → £5.20.

另一道题：如果5支笔£3.25，8支笔多少钱？求单价：£3.25÷5=£0.65每支，然后乘以8→£5.20。

A map has a scale 1:25,000. Two villages are 8 cm apart on the map. What is the real distance in km? 8 cm × 25,000 = 200,000 cm = 2,000 m = 2 km.

一张地图比例尺为1:25,000。两个村庄在地图上相距8厘米。实际距离是多少公里？8厘米×25,000=200,000厘米=2,000米=2公里。

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