Question

Ten years ago the ratio of the ages of Ramesh and Suresh was $ 1:5 $ . Ten years hence the ratio of their ages will be $ 3:5 $ then
a) Find the present age of Ramesh?
 $
  i)14\,\,years \\
  ii)\,10\,\,years \\
  iii)\,30\,\,years \\
  iv)\,24\,\,years \\
  $
b) How old was Suresh ten years ago?
 $
  i)\,9\,\,years \\
  ii)\,20\,\,years \\
  iii)\,40\,\,years \\
  iv)\,30\,\,years \\
  $

Answer 1

Hint: For age problem first let consider the present age of Ramesh and Suresh as x and y, then using it find their ages ten years ago and ten years hence and using given conditions to form equations. In solving equations hence, formed we will have values of ‘x’ and ‘y’ and so the required solution is asked in the problem.

Complete step-by-step answer:
Let present age of Ramesh is = x
And present age of Suresh is = y
Then age of Ramesh ten years ago = $ (x - 10) $
Age of Suresh ten years ago = $ (y - 10) $
Then according to statement we have ratio of their ages ten years ago was given as: $ 1:5 $
 $ \Rightarrow \dfrac{{x - 10}}{{y - 10}} = \dfrac{1}{5} $
Cross multiplying above equation we have,
 $
  5\left( {x - 10} \right) = \left( {y - 10} \right) \\
   \Rightarrow 5x - 50 = y - 10 \\
   \Rightarrow 5x - y = 40.............(i) \\
  $
Now, age of Ramesh ten years hence = $ (x + 10) $
Age of Suresh ten years hence = $ (y + 10) $
Also, it is given in statement ratio of their ages after ten years = $ 3:5 $
 $ \Rightarrow \dfrac{{x + 10}}{{y + 10}} = \dfrac{3}{5} $

Cross multiplying above equation we have,
 $
   \Rightarrow 5(x + 10) = 3(y + 10) \\
   \Rightarrow 5x + 50 = 3y + 30 \\
   \Rightarrow 5x - 3y = - 20............(ii) \\
  $
Solving (i) and (ii) by using the elimination method.
(i) - (ii)
 $
  \left( {5x - y} \right) - \left( {5x - 3y} \right) = 40 - ( - 20) \\
   \Rightarrow - y + 3y = 40 + 20 \\
   \Rightarrow 2y = 60 \\
   \Rightarrow y = 30 \\
  $

Substituting value of y = $ 30 $ in (i) to get value of x.
 $
  5x - 30 = 40 \\
   \Rightarrow 5x = 70 \\
   \Rightarrow x = \dfrac{{70}}{5} \\
   \Rightarrow x = 14 \\
  $
Therefore, from above we see that the present age of Ramesh is $ 14 $ years and that of Suresh is $ 30 $ years.
Hence, for the first part (a) the correct option is (i) as the present age of Ramesh is $ 14 $ years.
For part (b) we have to calculate the age of Suresh ten years ago.
For this we will subtract ten to age present age of Suresh which we have calculated above.
Hence, the age of Suresh ten years ago = $ 30 - 10 = 20 $ years.
Therefore the age of Suresh ten years ago = $ 20\,years $ .
Hence, from given options we see that for part (b) option (ii) is the correct option.

So, the correct answer is “Option I FROM PART A , II FROM PART B”.

Note: For age problems one must start with the present age of persons, either it is of one person problem or two person problem. After letting present age one can easily find ages according to past or future of corresponding persons and so to form respective equations and easily get the required solution of given problem.