Question

If twice the age of a son is added to the age of a mother, the sum is 56. But if twice the age of the mother is added to the age of the son, the sum is 82. Find the age of mother and son?

Answer 1

Hint: Let us assume that the age of son and mother is x years and y years respectively. It is given that when twice the age of son is added to the age of the mother then the sum is 56 which can be written as $2x+y=56$. Mark this equation as 1. It is also given that when twice the age of the mother is added to the age of the son then the sum is 82 which can be written as $2y+x=82$. Mark this equation as 2. Now, solve these two simultaneous equations by elimination method.

Complete step-by-step answer:
Let us assume that the age of the son is x years.
And let us assume that the age of the mother is y years.
It is given that when twice the age of son is added to the age of the mother then the sum is 56.
$2x+y=56$…….. Eq. (1)
It is also given that when twice the age of mother is added to the age of the son then the sum is 82.
 $2y+x=82$………. Eq. (2)
Now, we have two simultaneous equations in x and y that is;
 $2x+y=56$……… Eq. (1)
$2y+x=82$………… Eq. (2)
We are going to solve the above simultaneous equations by elimination method in which we multiply eq. (1) by 2 then subtract eq. (2) from eq. (1).
Multiplying equation (1) by 2 we get,
$4x+2y=112$………. Eq. (3)
Now, subtracting eq. (2) from eq. (3) we get,
$\begin{align}
  & 4x+2y=112 \\
 & \dfrac{-x-2y=-82}{3x+0=30} \\
\end{align}$
Solving the above equation we get,
$\begin{align}
  & 3x=30 \\
 & \Rightarrow x=10 \\
\end{align}$
Substituting the above value of x in eq. (1) we get,
$\begin{align}
  & 2y+x=82 \\
 & \Rightarrow 2y+10=82 \\
 & \Rightarrow 2y=72 \\
 & \Rightarrow y=36 \\
\end{align}$
From the above solution, we have found the age of son is 10 years and age of mother is 36 years.

Note: You can cross – check whether the values of x and y that you have solved by substituting in the one of the equations.
The solutions of x and y that we have got is:
$x=10;y=36$
Substituting these values in eq. (1) we get,
$\begin{align}
  & 2x+y=56 \\
 & \Rightarrow 2\left( 10 \right)+36=56 \\
 & \Rightarrow 20+36=56 \\
 & \Rightarrow 56=56 \\
\end{align}$
As L.H.S is equal to R.H.S in the above equation so we have proved that the values of x and y that we have obtained in the solution is correct.