What are ratios?

The ratio of a set of numbers is a tool used in comparing two or more values. Sometimes we need to compare two quantities of the same type. So, the comparison by division method makes calculation easier than comparing by finding out the difference. This important method is used while dealing with ratios of two quantities is division. Hence, the knowledge of dividend and divisor is also important while performing calculations of the ratios. The sign used to represent ratios is a colon or division. Suppose there are four candies and three chocolates. The ratio of these two numbers is represented as $$4:3$$

So, from the above explanation, it can be observed that the ratio indicates how many times one number contains another number. In ratios, the quantity of one thing is compared to the number of something else.

Consider a bag where the ratio of apples and oranges is given. So, to compare the quantity or number of the apples to the quantity of the oranges, the concept of ratio will be used. Suppose there are $$6$$ apples and $$3$$ oranges. So the ratio of the apples to the oranges is $$6:3$$. A ratio can be simplified. Here, $$3$$ is the common factor. Thus, the ratio of apple to oranges can also be written as $$2:1$$. 

E1.11: Demonstrate an understanding of ratio.

The comparison of two quantities ratios talks about how two numbers are related to each other rather than to decide if a number is greater, lesser, or equal to another number. The ratio can be written in a word form as $$x$$ is to $$y$$; it can also be written in fraction form $$\frac{x}{y}$$. 

Equivalent ratios are another way to show the equivalent fractions. The equivalent ratios are the ratios that make the same comparison of numbers. 

For example, if there are $$2$$ girls and $$1$$ boy in a school, the ratio of boys to girls is $$1:2$$. Now, to get the equivalents of the ratios, multiply both quantities by any number. Try to multiply it by $$3$$, so the ratio will become $$3:6$$. It means the ratio $$1:2$$ is equivalent to $$3:6$$. Now, try to multiply it by 4, so the ratio will become $$4:8$$. It means the ratio $$1:2$$ is also equivalent to $$4:8$$. 

So, to make equivalent ratios, multiply the numerator and denominator of the fractional form of ratio by the same number. For example: $$2:5$$ is equivalent to $$4:10$$. Here, $$2$$ is multiplied by both parts of the ratio.

Another way to make equivalent ratios is to divide the given fraction by the same number. For example, the fraction is $$\frac{4}{12}$$. Now, divide both the numerator and the denominator of the fraction by $$4$$. Therefore, the equivalent ratio will be $$\frac{1}{3}$$.

Worked examples

Example 1: There are $$5$$ dogs for every $$15$$ cows on a farm. Find out the ratio of the cows to the total number of animals. 

Step 1: Find the total number of animals.

Add the number of cows and the number of dogs.

$$5+15=20$$

Step 2: Find the ratio of the cows to the total number of animals. 

$$5:20$$

Step 3: Simplify the ratio. 

$$1:4$$

Example 2: A person has $$3$$ dogs and $$5$$ cats. Compare them using a ratio and write it in fraction form.

Step 1: Find the ratio.

$$3:5$$

Step 2: Convert the ratio in fraction form.

$$\frac{3}{8} : \frac{5}{8}$$