Question

Rakesh is 19 years younger than his father. After 5 years, their ages will be in the ratio $ 2:3 $ . Find their present ages.

Answer 1

Hint: In order to solve this problem, we will assume the present age of Rakesh and his father as $ x $ and $ y $ . As it is given in the question that Rakesh is 19 years younger than his father, so by using this condition we will express the relation between $ x $ and $ y $ . It is also given in the question that after 5 years their ages will have a certain ratio. Hence, we will express the relation for the ages after 5 years. Now by using both the expressions we find the ages of Rakesh and his father.

Complete step-by-step answer:
 Assume Rakesh’s age as $ x $ and his father’s age as $ y $ .
Since it is given in the question, Rakesh is 19 years younger than his father. Therefore, it can be expressed as
 $ x = y - 19 $ ……(i)
After years Rakesh’s age will be $ x + 5 $ and his father’s age as $ y + 5 $ .
Since, it is given that after 5 years Rakesh and his father’s age will be the ratio $ 2:3 $ and hence
it can be expressed as
 $ \dfrac{{x + 5}}{{y + 5}} = \dfrac{2}{3} $
We will further resolve the above expression to get the relation between $ x $ and $ y $ .
 $ \begin{array}{c}
3\left( {x + 5} \right) = 2\left( {y + 5} \right)\\
3x + 15 = 2y + 10\\
3x - 2y = - 5
\end{array} $ ……(ii)
Now we will substitute the value of $ x $ from equation (i) in the above equation (ii).
 $ 3\left( {y - 19} \right) - 2y = - 5 $
On further resolving the above expression we get
 $ \begin{array}{c}
3y - 57 - 2y = - 5\\
y - 57 = - 5\\
y = - 5 + 57\\
y = 52
\end{array} $
Now we will substitute 52 for $ y $ in the equation (i) to find the value of x.
 $ \begin{array}{l}
x = 52 - 19\\
x = 33
\end{array} $
Therefore, Rakesh’s age is 33 and his father’s age is 52.

Note: When we find the age after 5 years, we will add 5 in the present age of both Rakesh and his father. We can also solve this problem by assuming his father’s age as x and Rakesh’s age as $ x - 19 $ according to the condition given. Then after 5 years their age will become $ x + 5 $ and $ x - 14 $ respectively. Now we can find their ages by using the ratio given in the question.