Solving Problems Using Ratio | 用比例解决问题

📚 Solving Problems Using Ratio | 用比例解决问题

A ratio is a way to compare two or more quantities. It tells us how much of one thing there is compared to another. Whether you are mixing paint, scaling a recipe, or sharing sweets among friends, ratio helps you keep the relationship between amounts fair and accurate. In KS3 Mathematics, mastering ratio is essential for tackling proportion problems, map scales, and for understanding topics that appear later in the Cambridge curriculum.

比例是比较两个或多个量的一种方法。它告诉我们一种东西相对于另一种东西有多少。无论是混合颜料、按比例调整食谱，还是在朋友之间分享糖果，比例都能帮助你保持数量之间关系的公平与准确。在 KS3 数学中，掌握比例对于解决比例分配问题、理解地图比例尺，以及后续剑桥课程中出现的许多主题都至关重要。

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1. Understanding Ratio | 理解比例

A ratio compares quantities in the order they are mentioned. For example, if a class has 12 boys and 15 girls, the ratio of boys to girls is written as 12 : 15. We read this as ‘twelve to fifteen’. The order is important: swapping the numbers changes the meaning completely.

比例按照所提及的顺序比较数量。例如，如果一个班级有 12 名男生和 15 名女生，那么男生与女生的比例就写作 12 : 15。我们把它读作“12 比 15”。顺序很重要：一旦把数字交换，含义就完全改变了。

Ratios do not tell us the actual values, only the relative sizes. A ratio of 3 : 1 could mean 3 litres of water to 1 litre of juice, or 30 ml to 10 ml. The scale can vary, but the relationship stays the same as long as we multiply or divide both sides by the same number.

比例不告诉我们具体的数值，只显示相对大小。3 : 1 的比例可以表示 3 升水与 1 升果汁，也可以表示 30 毫升与 10 毫升。具体的数量可以缩放，但只要我们将两边同时乘以或除以相同的数，它们之间的相对关系就保持不变。

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2. Writing Ratios | 书写比例

Ratios can be written in three common forms: using a colon, as a fraction, or with the word ‘to’. For instance, 2 to 5 can be expressed as 2 : 5, 2/5, or ‘2 to 5’. In KS3, we mostly use the colon form, but it is helpful to understand the link to fractions.

比例可以用三种常见形式书写：使用冒号、用分数表示，或者用“比”字。例如，“2 比 5”可以写作 2 : 5、2/5 或者“2 to 5”。在 KS3 阶段，我们大多使用冒号形式，但理解比例与分数的联系也很有帮助。

When writing a ratio from a word problem, always identify which quantity is first and which is second. For a fruit bowl containing 6 apples and 4 bananas, the ratio of apples to bananas is 6 : 4. The ratio of bananas to apples would be 4 : 6.

从文字题中写出比例时，始终要先确定哪个量在前，哪个量在后。对于一个装有 6 个苹果和 4 根香蕉的水果碗，苹果与香蕉的比例是 6 : 4。而香蕉与苹果的比例则是 4 : 6。

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3. Simplifying Ratios | 化简比例

Like fractions, ratios can be simplified by dividing all parts by their highest common factor (HCF). The ratio 6 : 4 can be simplified: the HCF of 6 and 4 is 2, so we divide both numbers by 2 to get 3 : 2. This keeps the same relationship in its simplest form.

和分数一样，比例可以通过将所有部分除以它们的最大公因数 (HCF) 来化简。比例 6 : 4 可以化简：6 和 4 的最大公因数是 2，所以我们将两个数都除以 2，得到 3 : 2。这以最简形式保持相同的关系。

A ratio is in its simplest form when the numbers are whole numbers with no common factor greater than 1. The ratio 9 : 3 simplifies to 3 : 1, and 20 : 25 simplifies to 4 : 5. Always check whether the simplification step makes the ratio easier to interpret.

当比例中的数字均为整数，且没有大于 1 的公因数时，它就是最简形式。比例 9 : 3 化简为 3 : 1，20 : 25 化简为 4 : 5。务必检查化简步骤是否让比例更易于理解。

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4. Equivalent Ratios | 等效比例

Equivalent ratios are formed by multiplying or dividing each part of a ratio by the same non-zero number. For example, 1 : 2 is equivalent to 2 : 4, 3 : 6, and 10 : 20. You can think of these as different ways of expressing the same proportional relationship.

等效比例是通过将比例中的每一部分同时乘以或除以同一个非零数字得到的。例如，1 : 2 与 2 : 4、3 : 6 以及 10 : 20 都是等效的。你可以把它们看作表达同一比例关系的不同方式。

To find a missing value in an equivalent ratio, set up a proportion. If 3 : 5 is equivalent to x : 20, notice that the second part has been multiplied by 4 (since 5 × 4 = 20). Do the same to the first part: 3 × 4 = 12, so x = 12. This method is very useful for solving for unknowns.

要找出等效比例中的缺失值，可以建立一个比例关系。如果 3 : 5 与 x : 20 等效，注意到后项乘以了 4（因为 5 × 4 = 20）。前项也做同样的操作：3 × 4 = 12，所以 x = 12。这种方法在求解未知数时非常有用。

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

One of the most practical uses of ratio is sharing an amount into unequal parts. To divide 60 sweets between two children in the ratio 3 : 2, first add the parts to find the total: 3 + 2 = 5 parts. Each part is worth 60 ÷ 5 = 12 sweets. The first child gets 3 parts (3 × 12 = 36) and the second gets 2 parts (2 × 12 = 24).

比例最实际的用途之一是按照不等的份额分配一个总量。要按照 3 : 2 的比例把 60 颗糖果分给两个孩子，首先将两部分相加得出总份数：3 + 2 = 5 份。每份价值 60 ÷ 5 = 12 颗糖果。第一个孩子得到 3 份 (3 × 12 = 36)，第二个得到 2 份 (2 × 12 = 24)。

When three or more terms appear, such as sharing a prize of £100 among Ana, Ben, and Cara in the ratio 2 : 3 : 5, still add all parts: 2 + 3 + 5 = 10. Each part is £100 ÷ 10 = £10. So Ana gets £20, Ben gets £30, and Cara gets £50. Always check that the individual amounts add back to the original total.

当出现三项或更多项时，例如将 100 英镑的奖金按照 2 : 3 : 5 的比例分给 Ana、Ben 和 Cara，依然要把所有份数加起来：2 + 3 + 5 = 10。每份是 £100 ÷ 10 = £10。因此 Ana 得到 £20，Ben 得到 £30，Cara 得到 £50。务必检查各人所得相加后是否等于原来的总量。

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6. Ratio and Fractions | 比例与分数

A ratio and a fraction are closely connected. In a ratio a : b, the fraction of the whole that is a is a/(a+b). For example, if the ratio of boys to girls is 1 : 3, the fraction of the class that are boys is 1/4, and girls make up 3/4 of the class. Understanding this link makes many problems simpler.

比例和分数是紧密相连的。在比例 a : b 中，a 占整体的分数是 a/(a+b)。例如，如果男生与女生的比例是 1 : 3，那么男生占全班的 1/4，女生占全班的 3/4。理解这种联系会让很多问题变得更简单。

When a problem asks ‘what fraction of the total are green marbles?’ and gives a ratio of green to red as 5 : 7, the total number of parts is 12. The green marbles represent 5/12 of the collection. This is a key skill for converting between ratios and fractions.

当一道题问“绿色弹珠占总数的几分之几？”并给出绿色与红色的比例是 5 : 7 时，总份数就是 12。绿色弹珠占总数的 5/12。这是实现比例与分数相互转换的一项关键技能。

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7. Solving Word Problems | 解文字题

Word problems often require careful reading to identify the correct ratio. A common type: ‘In a bag, red and blue counters are in the ratio 4 : 7. There are 35 blue counters. How many red counters are there?’ Since 7 parts represent 35, one part is 35 ÷ 7 = 5. Therefore, red counters = 4 × 5 = 20.

文字题通常需要仔细阅读，才能识别出正确的比例。常见的类型是：“一个袋子里红色与蓝色计数片之比为 4 : 7。蓝色计数片有 35 个。红色计数片有多少？”7 份代表 35，那么每份是 35 ÷ 7 = 5。因此，红色计数片有 4 × 5 = 20。

Another pattern involves a difference: ‘The ratio of adults to children at a show is 2 : 5. There are 24 more children than adults. Find the total audience.’ The difference in parts is 5 – 2 = 3 parts. These 3 parts equal 24 people, so 1 part = 8. Total parts = 7, giving 56 people.

另一种模式涉及差值：“一场表演中成人与儿童的比例是 2 : 5。儿童比成人多 24 人。求观众总人数。”份数差为 5 – 2 = 3 份。这 3 份等于 24 人，因此 1 份 = 8 人。总份数为 7，总人数为 56 人。

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8. Using Scale Factors | 使用比例因子

A scale factor is a number that tells you how much to enlarge or reduce a ratio. If a recipe uses 3 cups of flour for 2 cups of sugar, you can scale it up for a larger batch: multiply both quantities by the same scale factor. Multiplying by 4 gives 12 cups of flour and 8 cups of sugar.

比例因子是一个告诉你将比例放大或缩小多少的数字。如果一个食谱用 3 杯面粉搭配 2 杯糖，你可以按比例因子把它放大，制作更多分量：将两者同时乘以相同的比例因子。乘以 4 就得到 12 杯面粉和 8 杯糖。

Scale factors also appear in maps and drawings. A map scale of 1 : 50000 means that 1 cm on the map represents 50000 cm (or 500 m) in real life. To find a real distance, multiply the map distance by the scale factor. To find a map distance, divide the real distance by the scale factor.

比例因子也出现在地图和图纸中。地图比例尺 1 : 50000 表示地图上的 1 厘米代表实际中的 50000 厘米（即 500 米）。要求实际距离，就将图上距离乘以比例因子。要求图上距离，则将实际距离除以比例因子。

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9. Direct Proportion | 正比例

Two quantities are directly proportional when they increase or decrease at the same rate and their ratio stays constant. If 5 pens cost £2, then 10 pens cost £4. Here the ratio of pens to cost (5 : 2) is equivalent to 10 : 4. The constant of proportionality can be found by dividing one quantity by the other.

当两个量以相同的速率增加或减少，并且它们的比值保持不变时，它们就是成正比例的。如果 5 支笔售价 £2，那么 10 支笔就是 £4。这里笔与价格的比值 (5 : 2) 与 (10 : 4) 是等效的。比例常数可以用一个量除以另一个量求得。

In KS3, direct proportion is often set up using a table. For example, to find the cost of 8 pizzas when 3 pizzas cost £15, find the cost per pizza first: £15 ÷ 3 = £5 each. Then multiply: 8 × £5 = £40. This ‘unitary method’ relies on the idea that the ratio of pizzas to cost is constant.

在 KS3 中，正比例问题通常借助表格来建立。例如，已知 3 个比萨饼售价 £15，求 8 个比萨的价格，可以先求出每个比萨的价格：£15 ÷ 3 = £5 每个，再相乘：8 × £5 = £40。这种“归一法”正是依赖于比萨个数与价格之比保持不变的思想。

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10. Common Mistakes | 常见错误

A frequent error is forgetting to simplify the ratio. Leaving 10 : 15 as the final answer when it should be 2 : 3 can cause later steps to be more difficult. Always take a moment to check if the ratio can be reduced.

一个常见的错误是忘记化简比例。把 10 : 15 作为最终答案，而它本应化简为 2 : 3，这会使后续步骤变得更加困难。请务必花一点时间检查比例是否可以约简。

Another mistake is mixing the order of terms. Writing a ratio of 2 : 3 instead of 3 : 2 completely changes the meaning, especially in word problems. Always identify which item is mentioned first and place it before the colon.

另一个错误是混淆各项的顺序。把比例写成 2 : 3 而不是 3 : 2，会完全改变含义，特别是在文字题中。一定要先确定题目中哪一个对象先被提到，把它放在冒号前面。

Finally, many pupils confuse ratios with fractions when dividing totals. Remember that a ratio of 1 : 2 means the first part is 1/3 of the whole, not 1/2. Always use the sum of the parts as the denominator.

最后，许多学生在分配总量时会把比例与分数混淆。请记住，1 : 2 的比例意味着第一部分占整体的 1/3，而不是 1/2。一定要用所有份数的和作为分母。

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11. Practice Questions | 练习题

Try these questions to test your understanding. (i) Simplify the ratio 18 : 24. (ii) Share £84 between two people in the ratio 3 : 4. (iii) A bottle of cordial says to mix with water in the ratio 1 : 9. How much cordial is needed if you use 2.7 litres of water? (iv) In a school, students and teachers are in the ratio 25 : 2. If there are 60 teachers, how many students are there?

试着回答以下问题，检测你的理解程度。(i) 化简比例 18 : 24。(ii) 按照 3 : 4 的比例把 £84 分给两个人。(iii) 一瓶浓缩饮料标明要以 1 : 9 的比例与水混合。如果使用 2.7 升水，需要多少浓缩饮料？(iv) 一所学校里学生和教师的比例为 25 : 2。如果有 60 名教师，那么学生有多少？

Check your answers using the methods above. (i) 3 : 4. (ii) £36 and £48. (iii) Cordial = 0.3 litres (because water is 9 parts, 1 part = 2.7 ÷ 9 = 0.3). (iv) Students = 750 (2 parts = 60, so 1 part = 30; 25 parts = 750). Practising a variety of problems builds confidence and speed.

用上面学过的方法核对答案。(i) 3 : 4。(ii) £36 和 £48。(iii) 浓缩饮料 = 0.3 升（因为水对应 9 份，每份 = 2.7 ÷ 9 = 0.3）。(iv) 学生 = 750 人（2 份 = 60，所以 1 份 = 30；25 份 = 750）。通过练习不同类型的问题，可以建立信心并提高速度。

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12. Summary | 总结

Ratio is a powerful tool for comparing quantities and solving real-world problems. By learning to write, simplify, and use equivalent ratios, you build a strong foundation for proportion, scale drawing, and algebra topics in the Cambridge syllabus. Remember the rules: keep the order, simplify where possible, and think in terms of total parts. With regular practice, ratio becomes second nature.

比例是比较数量、解决现实问题的一个强大工具。通过学习书写、化简和使用等效比例，你将为剑桥课程中的比例问题、比例图和代数课题打下扎实的基础。请牢记这些规则：保持顺序、尽可能化简，并从总份数的角度思考问题。只要坚持练习，比例就会成为你的第二天性。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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