Question

The sum of the ages of Shiva and his father is 100 years. When Shiva is as old as his father is now, he will be five times as old as his son Shyam is now. Shyam will be eight years older than Shiva is now, when Shiva is as old as his father. What are their ages now?

Answer 1

Hint: Let the present ages of Shiva’ father, Shiva and Shyam be x,y and z years, respectively. Now use the conditions given in the question to form equations in terms of x, y and z and solve the equations to get the value of x, y and z. Remember that the point that the difference between Shiva and his father’s age is (x-y) years.

Complete step-by-step answer:
To start with the question, we let the present ages of Shiva’ father, Shiva and Shyam be x,y and z years, respectively.
Now it is given in the question that the sum of the ages of Shiva and his father is 100 years. So, if we represent this in form of equation, we get
$x+y=100$
$y=100-x.....(i)$
Also, it is given, when Shiva is as old as his father is now, he will be five times as old as his son Shyam is now. In this case, the age of Shiva is x years. So, if we represent this mathematically, we get
$x=5z$
$\dfrac{x}{5}=z.....(ii)$
Also, it is mentioned that Shyam will be eight years older than Shiva is now, when Shiva is as old as his father. So, the age of Shyam for this case would be (y+8) years and the time passed from the present situation would be the difference of age of Shiva and his father, i.e., (x-y) years.
$\therefore z+(x-y)=y+8$ .
$\Rightarrow z+x=2y+8$
Now, we will substitute the value of y and z from equations (i) and (ii). On doing so, we get
$\dfrac{x}{5}+x=2\left( 100-x \right)+8$
$\Rightarrow \dfrac{6x}{5}=208-2x$
$\Rightarrow \dfrac{6x}{5}+2x=208$
$\Rightarrow 16x=208\times 5$
$\Rightarrow x=65years$
 Now if we back substitute x in equation (ii), we get z=13 years. And if we substitute x in equation (i), we get y=35 years.
Therefore, the current age of Shiva’s father, Shiva and Shyam are 65years, 35years and 13 years, respectively.

Note: While interpreting the statements given in the question, be very careful whether ages mentioned are present age or not. The key is to remember that the ages of each of the three will increase by the same constant with passage of time, as time passed is the same for everyone. For instance, when you interpret the statement “Shyam will be eight years older than Shiva is now, when Shiva is as old as his father” you may write the equation as \[y+8=x\] while it should be $z+(x-y)=y+8$ .