Question

The father’s age 10 years ago was 35 years more than twice his son’s age. After how many years from now will the father be twice his son’s age.
[a] 15
[b] 20
[c] 25
[d] Cannot be determined

Answer 1

Hint: Assume the age of father to be y, the age of son to be x and let after z years father’s age will be twice his son’s age. From the question above create two linear equations in x,y and z. Solve for z by eliminating the variables x and y. If the elimination is not possible, then there is no definite answer to the question. Otherwise, the value of z which is found by eliminating x and y is the answer.

Complete step-by-step solution -

Let the age of son be x, the age of father be y and let after z years fathers age become twice that of his son.
Since 10 years ago father’s age was 35 years more than twice the age of his son we have
y-10 = 2(x-10)+35
Applying distributive law in RHS, we get
y-10 = 2x -20 + 35
i.e. y-10 = 2x+15
Subtracting 15 from both sides, we get
y-10-15 = 2x+15-15
i.e. y-25 = 2x (i)
Also, since z years from now father’s age is twice that of his son, we have
y+z = 2(x+z)
Applying distributive law in RHS, we get
y+z = 2x+2z
Subtracting z from both sides, we get
y+z-z= 2x+2z-z
i.e. y = 2x+z
Substituting the value of 2x from equation (i) we get
y= y-25+z
Adding 25-y on both sides, we get
y+25-y = y -25+z+25-y
i.e. z = 25.
Hence 25 years from now father’s age will be twice that of his son’s age.

Note: [1] Although the given system of equations has infinitely many solutions, the value of z = 25, for the system to have a solution. Hence in the given system x and y take infinitely many values, but z is fixed. The given system can be visualised as a line in a plane parallel to the x-y plane at a distance of z = 25 from it.
[2] Age of an individual is an integer. Hence x, y and z only take integral values. Hence if the given system of equations had solutions at non-integral values of z, e.g. 25.5, then the age was not deterministic, and in fact, no such situation is possible.