Ratio and Proportion: A Complete KS3 Guide — 比与比例：KS3 完整指南

一、什么是比：两个量之间的比较关系 | What Is a Ratio? Comparing Two Quantities

比（ratio）是数学中用来比较两个或两个以上数量大小关系的一种方法。它告诉我们一个量相对于另一个量有多少份。例如，一个班级里有 12 名男生和 16 名女生，我们就说男生与女生的比是 12 比 16，记作 12 : 16，读作”12 比 16″。比的顺序非常重要：12 : 16 和 16 : 12 表示的是完全不同的关系，前者表示男生与女生的人数比，后者表示女生与男生的人数比。

A ratio is a way of comparing two or more quantities. It tells us how many parts one quantity has for every part of another. For example, if a class has 12 boys and 16 girls, we say the ratio of boys to girls is 12 to 16, written 12 : 16 and read as “12 to 16”. The order of a ratio matters a great deal: 12 : 16 and 16 : 12 describe completely different relationships. The first compares boys to girls, while the second compares girls to boys.

比的每一部分叫做”项”（term）。在比 12 : 16 中，12 是第一项，16 是第二项。比可以用三种等价的方式来表示：用冒号（12 : 16）、用”比”字（12 比 16），或者写成分数（12/16）。虽然写成分数看起来和分数一样，但它们的含义略有不同：分数通常表示”整体中的一部分”，而比强调的是”两个量之间的相对大小”。理解这一点是学好本章的关键。

Each part of a ratio is called a “term”. In the ratio 12 : 16, the first term is 12 and the second term is 16. A ratio can be written in three equivalent ways: with a colon (12 : 16), with the word “to” (12 to 16), or as a fraction (12/16). Although the fraction form looks identical to a fraction, the meaning is slightly different: a fraction usually represents a part of a whole, whereas a ratio emphasises the relative size of two quantities. Understanding this distinction is the key to mastering this topic.

在日常生活和科学中，比无处不在。烹饪时面粉和水的比例、调配饮料时果汁与水的比例、地图上的比例尺、以及化学中元素的配比，都是比的实际应用。正因为比如此常见，掌握它不仅能帮助你在考试中得分，更能让你真正理解身边世界中的数量关系。

Ratios appear everywhere in daily life and science. The ratio of flour to water in a recipe, the ratio of juice to water in a mixed drink, the scale on a map, and the proportion of elements in a chemical formula are all real applications of ratios. Because ratios are so common, mastering them not only helps you score well in exams but also lets you genuinely understand the quantitative relationships in the world around you.

二、化简比：约去最大公因数 | Simplifying Ratios: Cancelling the Highest Common Factor

化简比就是把比的两项同时除以它们的最大公因数（HCF，Highest Common Factor），使比变成最简单、最易读的形式。化简的过程和约分分数几乎完全一样。例如，比 12 : 16，12 和 16 的最大公因数是 4，两边同时除以 4，就得到 3 : 4。我们称 3 : 4 为 12 : 16 的最简形式（simplest form）。

Simplifying a ratio means dividing both terms by their highest common factor (HCF), so that the ratio becomes as simple and readable as possible. The process is almost identical to cancelling down a fraction. For example, in the ratio 12 : 16, the highest common factor of 12 and 16 is 4. Dividing both terms by 4 gives 3 : 4, which we call the simplest form of 12 : 16.

化简比的步骤可以总结为三步：第一步，找出两项的公因数；第二步，用最大公因数同时去除两项；第三步，检查结果是否还能继续化简。以 24 : 36 为例，24 和 36 的公因数有 1、2、3、4、6、12，其中最大的是 12，所以 24 : 36 = 2 : 3。如果你一开始只想到除以 6，会得到 4 : 6，这时还能再除以 2，最终仍然是 2 : 3。无论分几步除，只要每一步都正确，最终结果一定相同。

Simplifying a ratio can be summarised in three steps. First, find a common factor of the two terms. Second, divide both terms by the highest common factor. Third, check whether the result can be simplified further. Take 24 : 36 as an example: the common factors of 24 and 36 are 1, 2, 3, 4, 6 and 12, of which the largest is 12, so 24 : 36 = 2 : 3. If you had only thought of dividing by 6 at first, you would get 4 : 6, which can be divided by 2 again to reach 2 : 3. No matter how many steps you take, as long as each step is correct, the final result is always the same.

如果比的两项带有单位，化简前必须先把它们换成相同的单位。例如 2 m : 40 cm，需要先把 2 m 换成 200 cm，得到 200 : 40，化简为 5 : 1。一个常见的错误是直接写 2 : 40，这样会得到完全错误的结果。所以遇到带单位的比，务必先统一单位再化简。

If the two terms of a ratio carry units, you must first convert them to the same unit before simplifying. For example, in 2 m : 40 cm, you should convert 2 m into 200 cm to get 200 : 40, which simplifies to 5 : 1. A very common mistake is to write 2 : 40 directly, which leads to a completely wrong answer. Whenever a ratio involves units, always make the units the same before simplifying.

三、等价比与单位比（1:n）| Equivalent Ratios and the Unitary Form (1:n)

等价比（equivalent ratios）是指表示相同关系的不同比。就像 1/2 和 2/4 表示同一个分数一样，1 : 2 和 2 : 4 也表示同一个比。把一个比的两项同时乘以或除以同一个非零的数，就能得到等价比。例如 3 : 5 两边同时乘以 2 得到 6 : 10，同时乘以 3 得到 9 : 15，它们都等价于 3 : 5。

Equivalent ratios are different ratios that represent the same relationship. Just as 1/2 and 2/4 represent the same fraction, 1 : 2 and 2 : 4 represent the same ratio. You can produce an equivalent ratio by multiplying or dividing both terms by the same non-zero number. For example, multiplying both terms of 3 : 5 by 2 gives 6 : 10, and multiplying by 3 gives 9 : 15; both are equivalent to 3 : 5.

判断两个比是否等价，最可靠的方法是化简它们。如果两个比化简后完全相同，它们就是等价的。例如 6 : 9 化简为 2 : 3，10 : 15 也化简为 2 : 3，因此 6 : 9 和 10 : 15 等价。在考试中，”找出等价比”这类题目通常会给出一个比和几个选项，你只需把每个选项化简后与目标比比较即可。

The most reliable way to test whether two ratios are equivalent is to simplify them. If two ratios simplify to the same form, they are equivalent. For example, 6 : 9 simplifies to 2 : 3, and 10 : 15 also simplifies to 2 : 3, so 6 : 9 and 10 : 15 are equivalent. In exams, questions of the type “find the equivalent ratio” usually give one ratio and several options; you simply simplify each option and compare it with the target ratio.

单位比（unitary form）是把比写成 1 : n 或 n : 1 的形式，其中一项为 1。这种形式在比较两个比例时特别有用。例如，A 店的苹果 5 个卖 3 元，B 店的苹果 4 个卖 2.4 元，要判断哪家更便宜，可以统一为”1 个苹果多少钱”：A 店每个 0.6 元，B 店每个 0.6 元，价格相同。把比 5 : 3 写成 1 : 0.6，就是单位比的形式。单位比让”每个单位”或”每份”的成本一目了然。

The unitary form writes a ratio as 1 : n or n : 1, with one of the terms equal to 1. This form is especially useful when comparing two proportions. For example, shop A sells 5 apples for 3 yuan, and shop B sells 4 apples for 2.4 yuan. To decide which is cheaper, we can work out “how much for one apple”: each apple costs 0.6 yuan at both shops, so the prices are the same. Writing the ratio 5 : 3 as 1 : 0.6 is the unitary form, which makes the cost “per unit” or “per part” immediately clear.

四、按比例分配：把一个量分成若干份 | Sharing a Quantity in a Given Ratio

按比例分配是把一个总量按照给定的比分成若干份。这是比这一章最重要的应用之一，也是最常考的题型。方法可以概括为三步：第一，把比的所有项加起来，得到”总份数”；第二，用总量除以总份数，得到”每一份”的值；第三，用每一份的值分别乘以比的各项，得到各部分的数量。

Sharing a quantity in a given ratio means dividing a total amount into parts according to a given ratio. This is one of the most important applications of ratios and one of the most frequently tested question types. The method can be summarised in three steps: first, add all the terms of the ratio to find the total number of parts; second, divide the total amount by the total number of parts to find the value of one part; third, multiply the value of one part by each term of the ratio to find each share.

用一个具体例子来说明。把 60 元按 2 : 3 分给小明和小红。总份数是 2 + 3 = 5 份，每一份是 60 ÷ 5 = 12 元，所以小明得到 2 × 12 = 24 元，小红得到 3 × 12 = 36 元。检验一下：24 + 36 = 60，正好等于总量，说明分配正确。这种”加总检验”是很好的自检习惯，能帮你及时发现计算错误。

Let us look at a concrete example. Share 60 yuan between Xiaoming and Xiaohong in the ratio 2 : 3. The total number of parts is 2 + 3 = 5, so one part is 60 ÷ 5 = 12 yuan. Therefore Xiaoming receives 2 × 12 = 24 yuan and Xiaohong receives 3 × 12 = 36 yuan. We can check the answer: 24 + 36 = 60, which equals the original total, confirming the sharing is correct. This “add-up check” is a good habit that helps you spot calculation errors quickly.

当比有三项或更多项时，方法完全相同。例如把 1200 毫升果汁按 1 : 2 : 3 分成三种口味，总份数是 1 + 2 + 3 = 6 份，每一份是 200 毫升，于是三种口味分别是 200 毫升、400 毫升和 600 毫升。只要记住”先求总份数，再求每份值，最后按项分配”，无论比有多少项都能从容应对。

The method is exactly the same when a ratio has three or more terms. For example, to divide 1200 ml of juice into three flavours in the ratio 1 : 2 : 3, the total number of parts is 1 + 2 + 3 = 6, so one part is 200 ml, and the three flavours are 200 ml, 400 ml and 600 ml respectively. As long as you remember “first find the total parts, then find the value of one part, and finally share according to each term”, you can handle a ratio with any number of terms with confidence.

还有一个常见变体：题目不直接给总量，而是给出”某一项比另一项多多少”。例如小红比小明多得 12 元，且分配比是 2 : 3。这里两项相差 3 – 2 = 1 份，而这一份对应 12 元，所以每份是 12 元，于是小明 24 元、小红 36 元。这种”差对应份数”的题目，关键在于先算出两份之间的份数差。

There is also a common variation: instead of giving the total amount, the question gives “how much more one part receives than another”. For example, Xiaohong receives 12 yuan more than Xiaoming, and the sharing ratio is 2 : 3. Here the two terms differ by 3 – 2 = 1 part, and this one part corresponds to 12 yuan, so one part is 12 yuan, giving Xiaoming 24 yuan and Xiaohong 36 yuan. For this “difference corresponds to parts” type of question, the key is to first work out the difference in parts between the two terms.

五、比与分数的关系 | The Link Between Ratio and Fractions

比和分数之间有着密切的联系，理解这种联系能帮助你灵活地在两者之间转换。如果两个量的比是 3 : 4，那么总份数是 3 + 4 = 7 份，第一个量占整体的 3/7，第二个量占整体的 4/7。也就是说，比 3 : 4 意味着两个量分别是整体的 3/7 和 4/7。

Ratios and fractions are closely related, and understanding this link lets you move flexibly between the two. If two quantities are in the ratio 3 : 4, the total number of parts is 3 + 4 = 7, so the first quantity makes up 3/7 of the whole and the second makes up 4/7. In other words, the ratio 3 : 4 means the two quantities are 3/7 and 4/7 of the whole respectively.

反过来，如果题目告诉你一个量占整体的某个分数，你也能把它写成比。例如，一个班级中 2/5 的学生是男生，那么男生与女生的比是 2 : 3（因为男生占 2 份，女生占 5 – 2 = 3 份）。这里的分母 5 就是总份数，分子 2 就是男生对应的份数，剩下的 3 份就是女生。掌握这种”分数转比”的技巧，可以解决大量混合应用题。

Conversely, if a question tells you what fraction of the whole one quantity represents, you can write it as a ratio. For example, if 2/5 of a class are boys, then the ratio of boys to girls is 2 : 3, because boys take 2 parts and girls take 5 – 2 = 3 parts. Here the denominator 5 is the total number of parts, the numerator 2 is the number of parts for boys, and the remaining 3 parts are the girls. Mastering this “fraction to ratio” conversion lets you solve a wide range of mixed word problems.

一个容易混淆的地方是：比 3 : 4 并不等于分数 3/4。比 3 : 4 表示第一个量占 3/7、第二个量占 4/7；而分数 3/4 表示一个整体被分成 4 份后取 3 份。两者分母的含义完全不同。很多学生在初学时会把”3 : 4″错误地理解为”3/4 和 4/3″，这就是没有弄清”总份数”这一概念。记住：比的分母（总份数）是各项之和，而分数的分母是整体被分成的份数。

One easily confused point is that the ratio 3 : 4 is not the same as the fraction 3/4. The ratio 3 : 4 means the first quantity is 3/7 and the second is 4/7 of the whole, whereas the fraction 3/4 means taking 3 parts out of a whole divided into 4. The denominators mean completely different things. Many beginners mistakenly treat “3 : 4” as “3/4 and 4/3”, which comes from not understanding the concept of “total parts”. Remember: the denominator of a ratio (the total parts) is the sum of its terms, while the denominator of a fraction is the number of parts the whole is divided into.

六、正比例关系 | Direct Proportion

比例（proportion）描述两个量之间保持固定比值的稳定关系。当两个量成正比例（direct proportion）时，一个量增大为原来的几倍，另一个量也会增大为原来的几倍；一个量减半，另一个量也减半。例如，如果苹果每公斤 6 元，那么 1 公斤 6 元、2 公斤 12 元、3 公斤 18 元，总价与重量成正比例，比值始终是 6。

Proportion describes a stable relationship in which two quantities keep a constant ratio. When two quantities are in direct proportion, if one quantity is multiplied by a certain factor, the other is multiplied by the same factor; if one is halved, the other is halved too. For example, if apples cost 6 yuan per kilogram, then 1 kg costs 6 yuan, 2 kg costs 12 yuan and 3 kg costs 18 yuan. The total price and the weight are in direct proportion, and the constant ratio is always 6.

判断两个量是否成正比例，可以看它们的比值是否恒定。用 y 表示总价、x 表示重量，如果 y 与 x 成正比例，就有 y = kx，其中 k 是固定的常数，叫做比例常数。在上面的例子中，k = 6。判定方法是：取几组对应的 x 和 y，计算 y/x，如果结果始终相同，就说明两个量成正比例。这个”比值恒定”的判定方法在考试中非常重要。

To test whether two quantities are in direct proportion, check whether their ratio stays constant. Let y be the total price and x be the weight. If y is directly proportional to x, then y = kx, where k is a fixed constant called the constant of proportionality. In the example above, k = 6. The test is: take several pairs of corresponding x and y values, compute y/x, and if the result is always the same, the two quantities are in direct proportion. This “constant ratio” test is very important in exams.

比例和比的关系是：比描述的是”两个量某一次的相对大小”，而比例描述的是”两个量持续保持的关系”。很多现实问题可以先用比例关系列出方程，再求解。例如，若 4 本笔记本的价格是 3 本笔记本价格的多倍关系，或”3 支笔卖 4.5 元，那么 8 支笔卖多少元”，都可以通过”先求单价，再乘数量”的单位法（unitary method）解决，也可以设比例方程 4.5/3 = x/8 求解。

The relationship between ratio and proportion is this: a ratio describes the relative size of two quantities at a particular moment, while proportion describes an ongoing relationship that two quantities maintain. Many real problems can be solved by setting up a proportion first and then solving it. For example, “3 pens cost 4.5 yuan, so how much do 8 pens cost?” can be solved by the unitary method (first find the price of one pen, then multiply by the number of pens), or by setting up the proportion 4.5/3 = x/8 and solving for x.

七、比例尺与地图 | Scale and Maps

比例尺（scale）是比在地图、建筑图纸和模型制作中的重要应用。地图上的比例尺通常写成 1 : n 的形式，表示”图上 1 个单位长度对应实际 n 个单位长度”。例如，一张比例尺为 1 : 50000 的地图，图上 1 厘米代表实际的 50000 厘米，也就是 500 米。因此，图上 3 厘米就代表实际 1500 米。

Scale is an important application of ratios in maps, architectural drawings and model-making. A map scale is usually written in the form 1 : n, meaning “1 unit of length on the map corresponds to n units of length in reality”. For example, on a map with scale 1 : 50000, 1 cm on the map represents 50000 cm in reality, which is 500 m. Therefore 3 cm on the map represents 1500 m in reality.

比例尺的计算可以套用公式：实际距离 = 图上距离 × 比例尺的后项。例如比例尺 1 : 20000 的地图上，两地相距 4 厘米，则实际距离为 4 × 20000 = 80000 厘米 = 800 米。反过来，如果已知实际距离，要算图上距离，就用实际距离除以比例尺的后项。计算时务必注意单位换算：1 米 = 100 厘米，1 千米 = 100000 厘米。

Calculating with scale follows the formula: actual distance = map distance × the second term of the scale. For example, on a map with scale 1 : 20000, if two places are 4 cm apart on the map, the actual distance is 4 × 20000 = 80000 cm = 800 m. Conversely, if you know the actual distance and need the map distance, divide the actual distance by the second term of the scale. Be very careful with unit conversion: 1 m = 100 cm, and 1 km = 100000 cm.

还有一类题目是”放大的比例尺”，用于表示放大图。例如昆虫图片按 5 : 1 放大，表示图上 5 厘米对应实际 1 厘米，也就是放大了 5 倍。这时比例尺的前项大于后项。理解比例尺前项与后项的含义（前项是”图上”，后项是”实际”）是正确解题的前提。模型汽车按 1 : 24 制作，表示模型长度是真实汽车的 1/24。

There is also the “enlargement scale”, used to represent magnified drawings. For example, a picture of an insect magnified by 5 : 1 means 5 cm on the drawing corresponds to 1 cm in reality, i.e. it is enlarged 5 times. In this case the first term of the scale is larger than the second. Understanding what the two terms of a scale mean (the first is “on the drawing”, the second is “in reality”) is the prerequisite for solving these problems correctly. A model car built at 1 : 24 means the model length is 1/24 of the real car’s length.

八、生活中的比：配方、汇率与速度 | Ratio in Real Life: Recipes, Exchange Rates and Speed

比在烹饪配方中应用得非常直接。一个蛋糕配方需要 200 克面粉和 100 克糖，面粉与糖的比就是 2 : 1。如果你想做 3 倍量的蛋糕，就需要把两项都乘以 3，即 600 克面粉和 300 克糖，此时比仍然是 2 : 1。这体现了比的一个核心性质：等价比表示相同的”味道”或”配比”，只是总量不同。通过等价比，可以轻松地按任意倍数调整配方。

Ratios apply very directly in cooking recipes. A cake recipe needs 200 g of flour and 100 g of sugar, so the ratio of flour to sugar is 2 : 1. If you want to make three times the amount of cake, you multiply both terms by 3, giving 600 g of flour and 300 g of sugar, and the ratio remains 2 : 1. This demonstrates a core property of ratios: equivalent ratios represent the same “flavour” or “mixture”, just with a different total amount. Using equivalent ratios, you can easily scale a recipe by any factor.

汇率（exchange rate）也是比的实际应用。假设 1 英镑可以兑换 9 元人民币，那么英镑与人民币的比是 1 : 9。用这个比可以换算任何金额：50 英镑可以兑换 50 × 9 = 450 元；反过来，450 元可以兑换 450 ÷ 9 = 50 英镑。汇率的本质就是一个”兑换比”，掌握了比的知识，货币换算就变得非常简单。

Exchange rates are another real application of ratios. Suppose 1 pound can be exchanged for 9 yuan; then the ratio of pounds to yuan is 1 : 9. You can use this ratio to convert any amount: 50 pounds can be exchanged for 50 × 9 = 450 yuan, and conversely 450 yuan can be exchanged for 450 ÷ 9 = 50 pounds. An exchange rate is essentially a “conversion ratio”, so once you understand ratios, currency conversion becomes very simple.

速度、时间与距离之间也有比例关系。速度等于距离除以时间，所以当速度一定时，距离和时间成正比例：时间翻倍，行驶的距离也翻倍。例如汽车以 60 千米/小时行驶，1 小时走 60 千米，2 小时走 120 千米。这其实就是比例常数 k = 60 的正比例关系。理解”速度一定，距离与时间成正比”能帮助你快速解决行程问题。

There is also a proportional relationship among speed, time and distance. Speed equals distance divided by time, so when speed is constant, distance and time are in direct proportion: double the time, and the distance travelled doubles. For example, a car travelling at 60 km/h covers 60 km in 1 hour and 120 km in 2 hours. This is simply a direct proportion with constant k = 60. Understanding that “at constant speed, distance is proportional to time” helps you solve journey problems quickly.

九、常见错误与考试技巧 | Common Mistakes and Exam Techniques

学习比的过程中，有几个高频错误需要特别警惕。第一个错误是忘记化简：很多学生算完分配后就直接写答案，却忽略了题目要求”以最简比作答”。第二个错误是混淆比与分数：把 3 : 4 直接当成 3/4 来用。第三个错误是带单位的比没有统一单位：把 2 m : 40 cm 写成 2 : 40。第四个错误是在按比例分配时，只乘了其中一项或漏算了总份数。

When studying ratios, there are several high-frequency mistakes to watch out for. The first is forgetting to simplify: many students write the answer immediately after a sharing calculation, ignoring the requirement to give the answer in its simplest form. The second is confusing ratios with fractions, treating 3 : 4 directly as 3/4. The third is not converting units in a ratio that carries units, writing 2 m : 40 cm as 2 : 40. The fourth is, when sharing in a ratio, multiplying only one term or forgetting to work out the total number of parts.

考试技巧方面，第一，做按比例分配题时，一定要先写出”总份数 = 各项之和”这一步，并把”每份值 = 总量 ÷ 总份数”写清楚，阅卷老师会给步骤分。第二，最后一定要做”加总检验”，把各部分加起来看是否等于原总量。第三，遇到带单位的比，先统一单位。第四，遇到”差对应份数”的题目，先算份数差再求每份值。第五，选择题中判断等价比时，把选项逐一带入化简比较，不要凭感觉猜。

As for exam technique: first, when doing sharing questions, always write down the step “total parts = sum of terms” and clearly show “value of one part = total ÷ total parts”, because examiners award method marks for these steps. Second, always do the “add-up check” at the end to see whether the parts sum to the original total. Third, convert units first whenever a ratio carries units. Fourth, for “difference corresponds to parts” questions, work out the difference in parts before finding the value of one part. Fifth, when judging equivalent ratios in multiple-choice questions, simplify each option and compare, rather than guessing by intuition.

在时间允许的情况下，建议用另一种方法验证答案。例如做完按比例分配的题后，可以用”比值检验”：把得到的两个数量写成比并化简，看是否等于题目给出的比。这种交叉验证能极大降低计算错误的概率，是高分学生普遍采用的习惯。

When time allows, verify your answer using a different method. For example, after a sharing question, use the “ratio check”: write the two resulting quantities as a ratio and simplify it, then see whether it equals the ratio given in the question. This cross-checking greatly reduces the chance of calculation errors and is a habit widely adopted by high-scoring students.

Summary | 总结

本章系统讲解了”比与比例”这一 KS3 数学核心主题。我们首先认识了比的定义和三种写法，学会了用最大公因数化简比，并掌握了单位比（1 : n）的写法与用途。随后，我们重点学习了按比例分配的三步法：先求总份数、再求每份值、最后按项分配，并通过加总检验来验证答案。我们还厘清了比与分数的联系与区别，学习了正比例关系及其判定方法（比值恒定），以及比例尺在地图和模型中的应用。

This chapter systematically covered “Ratio and Proportion”, a core KS3 Mathematics topic. We first learned the definition and three ways of writing a ratio, how to simplify a ratio using the highest common factor, and the unitary form (1 : n) and its uses. We then focused on the three-step method for sharing a quantity in a given ratio: find the total parts, find the value of one part, and share according to each term, verifying the answer with the add-up check. We also clarified the link and difference between ratios and fractions, studied direct proportion and its test (constant ratio), and applied scale in maps and models.

掌握比与比例，不仅是为了应付考试，更是为了理解生活中的数量关系。无论是调整配方、换算货币、阅读地图，还是分析速度与距离，比的思维都无处不在。建议你反复练习化简比、按比例分配和正比例判定这三类核心题型，并在每次练习后都做一次加总检验或比值检验。只要掌握了这些方法，比与比例将成为你数学工具箱中最得心应手的工具之一。

Mastering ratio and proportion is not only about passing exams but also about understanding the quantitative relationships in everyday life. Whether adjusting a recipe, converting currency, reading a map, or analysing speed and distance, the thinking behind ratios is everywhere. We recommend practising the three core question types repeatedly: simplifying ratios, sharing in a ratio, and testing for direct proportion, and doing an add-up check or ratio check after every exercise. Once you master these methods, ratio and proportion will become one of the most useful tools in your mathematical toolkit.

更多咨询请联系16621398022（同微信）