## How to Solve Word Problems Involving Ratio Part 3

In the previous two posts, we have learned how to solve word problems involving ratio with two and three quantities. In posts, we are going to learn how to solve a slightly different problem where both numbers are increased.

Problem

The ratio of two numbers is 3:5 and their sum is 48. What must be added to both numbers so that the ratio becomes 3:4?

Solution and Explanation

First, let us solve the first sentence. We need to find the two numbers whose ratio is 3:5 and whose sum is 48.

Now, let x be the number of sets of 3 and 5.

3x + 5x = 48
8x = 48
x = 6

Now, this means that the numbers are 3(6) = 18 and 5(6) = 30.

Now if the same number is added to both numbers, then the ratio becomes 3:4.

Recall that in the previous posts, we have discussed that ratio can also be represented by fraction. So, we can represent 18:30 as $\frac{18}{30}$. Now, if we add the same number to both numbers (the numerator and the denominator), we get $\frac{3}{4}$. If we let that number y, then $\dfrac{18 + y}{30 + y} = \dfrac{3}{4}$.

Cross multiplying, we have $4(18 + y) = 3(30 + y)$.

By the distributive property, $72 + 4y = 90 + 3y$ $4y - 3y = 90 - 72$ $y = 18$.

So, we add 18 to both the numerator and denominator of $\frac{18}{30}$. That is, $\dfrac{18 + 18}{30 + 18} = \dfrac{36}{48}$.

Now, to check, is $\dfrac{36}{48} = \frac{3}{4}$? Yes, it is. Divide both the numerator and the denominator by 12 to reduce the fraction to lowest terms.