Question

The sum of the ages of P and Q is 40 years. Five years ago, twice the age of P added to three times the age of Q was 70 years. Find their present ages.

Answer 1

Hint:
In this question, we need to determine the present age of P and Q such that the sum of the ages of P and Q is 40 years and five years ago, the sum of twice the age of P and thrice the age of Q was 70 years. For this, we will prepare the equations for the two variables, i.e., the present age of P and the present are of Q and then, solve both the equations to get the result.

Complete step by step solution:
Let the present age of P and Q be $x$ and $y$ respectively.
According to the question, the sum of the ages of P and Q is 40 years. So,
$x + y = 40 - - - - (i)$
Also, five years ago, the sum of twice the age of P and thrice the age of Q is 70 years. So,
\[2\left( {x - 5} \right) + 3\left( {y - 5} \right) = 70 - - - - (ii)\]
Solving equations (i) and (ii) to determine the present age of P and Q.
From the equation (i), we get
$
  x + y = 40 \\ \Rightarrow
  x = 40 - y - - - - (iii) \\
 $
Substituting the values obtained in the equation (iii) in the equation (ii), we get
$
  2\left( {x - 5} \right) + 3\left( {y - 5} \right) = 70 \\ \Rightarrow
  2\left( {40 - y - 5} \right) + 3y - 15 = 70 \\ \Rightarrow
  2\left( {35 - y} \right) + 3y - 15 = 70 - - - - (iv) \\
 $
Further simplifying the equation (iv) to determine the value of y, which is the present age of Q.
$
  2\left( {35 - y} \right) + 3y - 15 = 70 \\
  70 - 2y + 3y - 15 = 70 \\ \Rightarrow
   - 2y + 3y = 70 - 70 + 15 \\ \Rightarrow
  y = 15{\text{ years}} \\
 $
Now, substituting the value of y as 15 years in the equation (iii), we get
$
  x = 40 - y \\
   = 40 - 15 \\
   = 25{\text{ years}} \\
 $

Hence, the present age of P and Q is 25 years and 15 years, respectively.

Note:
It is worth noting down here that, we have taken the variables as the present age of P and Q and so, during the formation of the equation for 5 years ago, we have subtracted the present ages by 5 for both P and Q.