Question

Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?
A. Present age of Jacob: $ 50 $ years and Present age of Jacob’s son: $ 15\,years $
B. Present age of Jacob: $ 50 $ years and Present age of Jacob’s son: $ 10\,years $
C. Present age of Jacob: $ 40 $ years and Present age of Jacob’s son: $ 10\,years $
D. Present age of Jacob: $ 35 $ years and Present age of Jacob’s son: $ 15\,years $

Answer 1

Hint: For this type of age problems we first let age of given persons as ‘x’ and ‘y’ and then finding their age first according to given hence years and forming an equation as per of given condition and then finding their age according to given years ago and forming an equation as per of given condition. Finally solving these two equations to find values of ‘x’ and ‘y’ and hence required solution of the given problem.

Complete step-by-step answer:
Let present age of Jacob is = x years
Present age of Jacob’s son is = y years
Age of Jacob five years hence will be = $ (x + 5)years $
Age of Jacob’s five years hence will be = $ \left( {y + 5} \right)years $
Then, according to the question. We have
Age of Jacob is three times the age of his son, five years hence.
Therefore, we have:
 $
  \left( {x + 5} \right) = 3\left( {y + 5} \right) \\
   \Rightarrow x + 5 = 3y + 15 \\
   \Rightarrow x - 3y = 15 - 5 \\
   \Rightarrow x - 3y = 10.....................(i) \;
  $
Also, age of Jacob five years ago = $ (x - 5)years $
Age of Jacob’s son five years ago = $ (y - 5)years $
But according to the question five years ago the age of Jacob is seven times the age of Jacob’s son.
Therefore, we have
 $
  \left( {x - 5} \right) = 7\left( {y - 5} \right) \\
   \Rightarrow x - 5 = 7y - 35 \\
   \Rightarrow x - 7y = - 35 + 5 \;
   \Rightarrow x - 7y = - 30.....................(ii) \\
  $
Subtracting (ii) from (i) we have
 $
  \left( {x - 3y} \right) - \left( {x - 7y} \right) = 10 - \left( { - 30} \right) \\
   \Rightarrow x - 3y - x + 7y = 10 + 30 \\
   \Rightarrow 4y = 40 \\
   \Rightarrow y = 10 \;
  $
Substituting value of $ y = 10 $ in equation (i) we have:
 $
  x - 3\left( {10} \right) = 10 \\
   \Rightarrow x - 30 = 10 \\
   \Rightarrow x = 10 + 30 \\
   \Rightarrow x = 40 \;
  $
Therefore, from above we see that the present age of Jacob is $ 40\,\,years $ and the present age of Jacob’s son is $ 10\,years $ .
So, the correct answer is “Option C”.

Note: There are different ways of solving age problems questions. But better the best way of doing these problems is considering present age of persons as ‘x’ and ‘y’ and then forming equation as per of given conditions one for hence age and one for ago ages and then solving equations so formed by either method of elimination or by substitution method to find value of ‘x’ and ‘y’ and hence solution of required problem.