Question

The sum of the ages of mother and son is 90 year, after 5 years the ratio of their ages will be $ 3:2 $ . What is the present age of the mother?

Answer 1

Hint: We first assume the ages of the mother and her son as variables. We use the given conditions to form the equations and solve them using the process of substitution. We solve the two equations of two unknowns and find the solution.

Complete step by step solution:
We first assume the ages of the mother and her son. We take the variables $ x,y $ as their ages.
The sum of the ages of mother and son is 90 year which gives $ x+y=90 $ .
After 5 years their ages will be $ x+5,y+5 $ respectively.
It’s given that the ratio is $ 3:2 $ which gives $ \dfrac{x+5}{y+5}=\dfrac{3}{2} $ . Simplifying we get $ 2x-3y=5 $ .
The given equations $ x+y=90 $ and $ 2x-3y=5 $ are linear equations of two variables.
We know that the number of equations has to be equal to the number of unknowns to solve them.
We take the equations as $ x+y=90.....(i) $ and $ 2x-3y=5......(ii) $ .
We multiply 3 to the both sides of the first equation and get
 $
   3\times \left( x+y \right)=90\times 3 \\
  \Rightarrow 3x+3y=270 \;
 $
We take the equation as $ 3x+3y=270.....(iii) $ .
Now we add the equation (ii) from equation (iii) and get
 $
   \left( 2x-3y \right)+\left( 3x+3y \right)=5+270 \\
  2x-3y+3x+3y=275 \\
  \Rightarrow 5x=275 \\
  \Rightarrow x=\dfrac{275}{5}=55 \;
 $
The value of $ x $ is 55. Now putting the value in the equation $ x+y=90.....(i) $ , we get
 $
   x+y=90 \\
  \Rightarrow y=90-55=35 \;
 $
Therefore, the ages of the mother and her son are 55 and 35 respectively.
So, the correct answer is “55 and 35”.

Note: We can also find the value of one variable $ y $ with respect to $ x $ based on the equation
 $ x+y=90 $ where $ y=90-x $ . We replace the value of $ y $ in the second equation of
 $ 2x-3y=5 $ and get
\[
   2x-3y=5 \\
  \Rightarrow 2x-3\left( 90-x \right)=5 \\
  \Rightarrow 2x-270+3x=5 \;
\]
We get the equation of $x$ and solve
$
2x-270+3x=5 \\
 \Rightarrow 5x=5+270=275 \\
 \Rightarrow x=\dfrac{275}{5}=55 \\
$
Putting the value of $x$ we get $x+y=90\Rightarrow y=90-55=35$.
Therefore, the ages of the mother and her son are 55 and 35 respectively.