Question

Sanjay is now \[\dfrac{1}{2}\] as old as his brother. In 6 more years he will be \[\dfrac{3}{5}\] as old as his brother then. What is the present age of each boy?

Answer 1

Hint: Here we assume the present age of brother as a variable and use the relation given in the question to form equations for present age and age after six years. Cross multiply the values from both sides of the equations and find the value of the variable.

Complete step by step answer:
Let us assume the present age of Sanjay’s brother \[ = x\] …...… (1)
Then from the statement in question we have the present age of Sanjay is \[\dfrac{1}{2}\]times the age of his brother
\[ \Rightarrow \]Present age of Sanjay \[ = \dfrac{x}{2}\] …...… (2)
Now we have to find the equation of age of Sanjay and his brother after 6 years.
After six years, the age of Sanjay’s brother will be the present age of Sanjay’s brother plus 6 years.
From equation (1):
\[ \Rightarrow \]Age of Sanjay’s brother after 6 years \[ = x + 6\] ………..… (3)
Age of Sanjay after six years will be the present age of Sanjay plus 6 years.
From equation (2):
\[ \Rightarrow \]Age of Sanjay after 6 years \[ = \dfrac{x}{2} + 6\] ………..… (4)
We know from the statement of the question that after six years, the age of Sanjay is \[\dfrac{3}{5}\]times the age of his brother.
We use equations (3) and (4) to write this relation, i.e.
\[ \Rightarrow \left( {\dfrac{x}{2} + 6} \right) = \dfrac{3}{5}\left( {x + 6} \right)\]
Take LCM on LHS of the equation and multiply the fraction to bracket in RHS of the equation
\[ \Rightarrow \dfrac{{x + 12}}{2} = \dfrac{{3x + 18}}{5}\]
Cross multiply the equations on both sides
\[ \Rightarrow 5(x + 12) = 2(3x + 18)\]
Multiply the terms inside the bracket
\[ \Rightarrow 5x + 60 = 6x + 36\]
Shift all constant values to one side of the equation
\[ \Rightarrow 60 - 36 = 6x - 5x\]
\[ \Rightarrow 24 = x\]
So, the present age of Sanjay’s brother is 24 years.
From equation (2), present age of Sanjay \[ = \dfrac{x}{2}\]
Substituting the value of $x$ as 24,
\[ \Rightarrow \]Present age of Sanjay \[ = \dfrac{{24}}{2}\]
\[ \Rightarrow \]Present age of Sanjay \[ = 12\]
So, the present age of Sanjay is 12 years.

\[\therefore \] Present ages of Sanjay and his brother are 12 years and 24 years respectively.

Note:
Students may get confused in forming the equation of ages after six years as they tend to add value 6 to the age of Sanjay and not to Sanjay’s brother. Keep in mind after a period of time, there will be change in both Sanjay’s and Sanjay’s brother’s age. Also, change the sign of the value when shifting the value from one side of the equation to the other side of the equation.