Question

The ratio between the two numbers is 3:5. If each number is increased by 5, the ratio becomes 2:3. Find the numbers.

Answer 1

Hint: First, assume the two numbers. Then, find the value of one number in terms of other numbers. After that substitute the value in the second equation and solve it to get the value. Then substitute the value to get the value of another number. The numbers obtained are the desired result.

Complete step by step answer:
Given: - The ratio between the two numbers is 3:5. If each number is increased by 5, the ratio becomes 2:3.
Let assume the two numbers are x and y.
The ratio is defined as the quantity of one thing concerning the other thing in simplified form. It can be represented in many ways by using the symbols: / or as a decimal or as a percentage.
The formula of the ratio is,
Ratio= $\dfrac{a}{b}$
Put a= x and b= y and the value of the ratio is $\dfrac{3}{5}$. Then,
$\dfrac{x}{y} = \dfrac{3}{5}$
Now, cross multiply the terms of the equation,
$\Rightarrow 5x = 3y$
Now, divide both sides by 5 of the equation,
$\Rightarrow x = \dfrac{3}{5}y$ …… (1)
If each number is increased by 5, the ratio becomes 2:3. Then,
$\Rightarrow \dfrac{{x + 5}}{{y + 5}} = \dfrac{2}{3}$
Now, cross multiply the terms of the equation,
$\Rightarrow 3x + 15 = 2y + 10$
Substitute the value of x from equation (1),
$\Rightarrow 3 \times \dfrac{3}{5}y + 15 = 2y + 10$
Move the variable part on one side and constant part on the other side,
$\Rightarrow 2y - \dfrac{9}{5}y = 15 - 10$
Take LCM on the left side and subtract the value on the right side,
$\Rightarrow \dfrac{{10y - 9y}}{5} = 5$
Subtract the value in the numerator and multiply both sides by 5,
$\Rightarrow y = 5 \times 5$
Multiply the terms on the right side,
$\Rightarrow y = 25$
Substitute the value of y in equation (1),
$\Rightarrow x = \dfrac{3}{5} \times 25$
Cancel out the common terms and multiply,
$\Rightarrow x = 15$

Hence, the numbers are 15 and 25.

Note:
A ratio is a comparison of two or more numbers that indicates their sizes in relation to each other. A ratio compares two quantities by division, with the dividend or number being divided termed the antecedent, and the divisor or number that is dividing termed the consequent. The ratio is denoted by the symbol :