# Ratio

A ratio is a simple comparison between two quantities. It says how much one thing is there compared to another.

For example, in a group of 7 students, 4 are men and 3 are women. Then, the ratio of men to women is 4:3. In words, it is spelled as for every 3 women, there are 4 men.

## Formula

We use the ratio formula to compare the relationship between two numbers or quantities. Ratios can be expressed in 3 different ways:

1. Using a â€˜:â€™ sign to separate the values, such as 4:3
2. Using the word â€˜toâ€™, for example, 4 to 3
3. Writing as a fraction like ${\dfrac{4}{3}}$

However, the most commonly used way of expressing the ratio is:

The numerator (the first term) is called the antecedent and the denominator (the second term) is the consequent.

If â€˜xâ€™ be the antecedent and â€˜yâ€™ be the consequent:

• When x = y, a:b = 1
• When a > b, a:b > 1
• When a < b, a:b < 1

## Part to Whole Ratio

The examples considered so far are all part-to-part ratios. It compares one part of a whole to the other part. However, a ratio can also show a part in comparison to a whole.

For example, if there are 2 men, 4 women, and 3 children in a car, the number of men to the whole is 2:9, women to the whole is 4:9, and children to the whole is 3:9.

## Simplifying Ratios

Sometimes ratios need to be in their lowest form. Such as the fraction ${\dfrac{20}{12}}$

In such cases, the ratios are further reduced or simplified, similar to what we do in case of fractions.

Here, we find the greatest common factor of the numbers in antecedent and consequent and divide them with the GCF.

Let us simplify the fraction ${\dfrac{24}{16}}$

Step 1: The GCF of 16 and 24 is 8

Step 2: Dividing antecedent and consequent by 8

${\dfrac{24\div 8}{16\div 8}}$

Step 3: The result is 3:2

Simplify the ratio 21:49

Solution:

Step 1: The GCF of 21 and 49 is 49
Step 2: Dividing antecedent and consequent by 7
${\dfrac{21\div 7}{49\div 7}}$
Step 3: The result is 3:7

## Scaling a Ratio

A ratio is often scaled up or down by multiplying or dividing the antecedent and consequent by the same number.

For example, in the above-given ratio of 4:3, if we multiply antecedent and consequent by 2, the new ratio is 8:6, which is equal to the ratio 4:3.

The recipe for a 20g pancake uses 5 cups of flour and 3 cups of milk. However, a party needs 80g of pancake. Find the new ratio of flour to milk required to prepare 80g of cake.

Solution:

Making an 80g pancake will require 4 times what is required to make a 20g pancake.
The given ratio of flour to milk is 5:3. Multiplying both antecedent and consequent by 4, we get the near ratio as:
${\dfrac{5\times 4}{3\times 4}=\dfrac{20}{12}}$
Thus preparing 80g of cake will require 20 cups of flour and 12 cups of milk.
Note that the ratio ${\dfrac{20}{12}}$ is same as ${\dfrac{5}{3}}$

## Equivalent Ratios

The ratio obtained by scaling a current ratio through multiplication or division is called an equivalent ratio.

For example, the ratios ${\dfrac{20}{12}}$ and ${\dfrac{5}{3}}$ are equivalent ratios. We can obtain an infinite number of equivalent ratios for any given ratio through the multiplication of the antecedent and the consequent by a positive number.

## Ratio Table

A ratio table is a list containing the equivalent ratios of any given ratio in an ordered form. It is prepared by finding the equivalent ratios of any given ratio.

For example, the given ratio table gives the relation between the ratio 1:3 and four of its equivalent ratios.