Question

The sum of the reciprocals of Rehman's ages, (in years) 3 years ago and 5 years from now is $\frac{1}{3}$. Find his present age.

• A. 8 years
• B. 6 years
• C. 7 years
• D. 3 years

Answer 1

The correct option is C 7 years

Let the present age of Rehman be x years.
Three years ago, his age was (x - 3) years.
Five years hence, his age will be (x + 5) years.
It is given that the sum of the reciprocals of Rehman's ages 3 years ago and 5 years from now is $\frac{1}{3}$.
$\therefore \frac{1}{x-3}+\frac{1}{x+5}=\frac{1}{3}$

$\frac{x+5+x-3}{(x-3)(x+5)}=\frac{1}{3}$

$\frac{2x+2}{(x-3)(x+5)}=\frac{1}{3}$

$\Rightarrow 3(2x+2)=(x-3)(x+5)$

$\Rightarrow 6x+6=x^2+2x-15$

$\Rightarrow x^2-4x-21=0$

$\Rightarrow x^2-4x=21$

Adding the square of one-half of the coefficient of x [i.e., $(\frac{1}{2}\times -4{)}^2$] to both sides, we get,

$\Rightarrow x^2-4x+(-2{)}^2=21+(-2{)}^2$

$\Rightarrow x^2-2\times x\times 2+2^2=25$

$\Rightarrow (x-2{)}^2=25$

$\Rightarrow (x-2)=\pm 5$

$\Rightarrow x=\pm 5+2$

$\Rightarrow x=5+2,x=-5+2$

$\Rightarrow x=7,-3$

However, age cannot be negative.

Therefore, Rehman's present age is 7 years.