Question

Four times B’s age exceeds A’s age by twenty years, and one-third of A’s age is less than B’s age by two years; find their ages.

Answer 1

Hint: We here have been given two people A and B and two relationships between their ages. Using them, we have to calculate their ages. For this, we will first assume their ages to be x and y respectively. Then we will form two equations in x and y using the two relationships given between their ages. Then we will solve those two equations by the elimination method and hence we will get our answer.

Complete step-by-step solution
Now, to find their ages let us assume the age of A and B to be x and y respectively.
So, here we have been given that four times B’s age exceeds the age of A by twenty years. Thus, four times y exceeds x by 20.
Hence, we can say that:
\[4y-x=20\] .....(i)
This is our first equation in x and y.
Now, we have also been given that one-third of A’s age is less than B’s age by two years. Thus, one-third of x is less than y by 2 years.
Hence, we can say that:
$y-\dfrac{x}{3}=2$
Multiplying this equation by 3, we get:
$3y-x=6$ .....(ii)
This is the second equation in x and y.
Now, we will solve equations (i) and (ii) by elimination method as a result of which we will get:
$\begin{align}
  & 4y-x=20 \\
 & \underline{-\left( 3y-x=6 \right)} \\
 & \underline{\text{ }y=14\text{ }} \\
\end{align}$
Hence, the value of y is 14.
Now putting y=14 in equation (i) we get:
$\begin{align}
  & 4y-x=20 \\
 & \Rightarrow 4\left( 14 \right)-x=20 \\
 & \Rightarrow 56-x=20 \\
 & \therefore x=36 \\
\end{align}$
Thus, the value of x is 36.
Hence, the age of A is x, i.e. 36 and that of B is y, i.e. 14.

Note: Be very careful while reading the given relationships between the ages of A and B and forming the required equations which will give us the answer as even a minor mistake in reading the question can give us a wrong equation resulting in a wrong answer. Hence, it is very important to be careful while forming the equations.