Question

Rekha is now 15 years older than Akaash but in 3 more years she will be 8 times as old as Akaash was 3 years ago. How old are they now?

Answer 1

Hint: We will begin by letting the present age of Rekha be $x$ years and the present age of Akaash be $y$ years. Then, use the given condition, that Rekha is now 15 years older than Akaash to form equation (1) . Next, write the age of Rekha 3 years from now and age of Akaash 3 years ago and form the equation using the given condition. Solve both the equations to find the value of $x$ and $y$

Complete step-by-step answer:
Let the present age of Rekha be $x$ years and the present age of Akaash be $y$ years.
Then, we are given that Rekha is now 15 years older than Akaash.
Which implies,
$x - y = 15$ eqn. (1)
After 3 years, the age of Rekha will be \[x + 3\]
and the age of Akaash three 3 ago will be \[y - 3\].
Also, we are given that in 3 more years Rekha will be 8 times as old as Akaash was 3 years ago
That is, $x + 3 = 8\left( {y - 3} \right)$
On simplifying , we will get,
$x + 3 = 8y - 24$
$ \Rightarrow x - 8y = - 27$ eqn. (2)
We will now solve equation (1) and (2).
Subtract equation (2) from (1) to eliminate the terms of $x$ and then solve for $y$.
$
  x - x - y + 8y = 15 + 27 \\
   \Rightarrow 7y = 42 \\
$
Divide the equation throughout by 7,
$y = 6$
Substitute the value of $y$ in equation (1) to find the value of $x$
$
  x - 6 = 15 \\
  x = 21 \\
$
Therefore, the present age of Rekha is 21 years and the present age of Akaash is 6 years.

Note: While using elimination, the coefficient of term that has to be eliminated must be the same in both the equations. Here, we have first used elimination and then substitution to solve the equations. We can also use only elimination or only substitution to solve both the equations.