QUESTION

# Hema is 24 years older than her daughter Damini. 6 years ago, Hema was thrice as old as Damini. Find their present ages.

Hint: Assign two variables to the age of Hema and the age of her daughter Damini. Then, form the two equations in two variables with the given information and solve them by substitution to obtain their present ages.

It is given that Hema is 24 years older than her daughter Damini and before 6 years, Hema was thrice as old as Damini and we need to determine their present ages.
Let the present age of Hema be x and the present age of Damini be y.
Hema is 24 years older than her daughter Damini, hence, we have:
$x = y + 24..........(1)$
Before 6 years, the age of Hema was x – 6 and the age of Damini was y – 6.
Six years ago, Hema was thrice as old as Damini, hence, we have as follows:
$\Rightarrow$ $x - 6 = 3(y - 6)$
Simplifying the above expression, we have:
$\Rightarrow$ $x - 6 = 3y - 18$
Taking 6 to the right-hand side of the equation, we have:
$\Rightarrow$ $x = 3y - 18 + 6$
$\Rightarrow$ $x = 3y - 12..............(2)$
Substituting equation (2) in equation (1), we have:
$\Rightarrow$ $3y - 12 = y + 24$
Taking all the terms containing y to the left-hand side of the expression, we have:
$\Rightarrow$ $3y - y = 24 + 12$
Simplifying the above equation, we get:
$\Rightarrow$ $2y = 36$
Solving for y, we have:
$\Rightarrow$ $y = \dfrac{{36}}{2}$
$\Rightarrow$ $y = 18$years............(3)
Substituting equation (3) in equation (1), we get:
$\Rightarrow$ $x = 18 + 24$
$\Rightarrow$ $x = 42$ years
The present age of Hema is 42 years and the present age of Damini is 18 years old.

Note: When writing the linear equation for relation between the ages before 6 years, you need to subtract 6 from the present ages and then substitute.