The sum of the reciprocals of Rehman's ages (in years) 3 years ago

Q. The sum of the reciprocals of Rehman’s ages (in years) 3 years ago and 5 years from now is 1/3 . Find his present age.

Ans: Let’s consider Rehman’s present age be X

Hence, his age 3 years ago = X – 3 and his age 5 years from now = X + 5

by given condition: sum of reciprocals of his age 3 years ago & 5 years from now is 1/3

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

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

$\therefore 3 (2 X + 2) = (X - 3)(X + 5)$

$\therefore 6 X + 6 = X^2 + 2 X - 15$

$\therefore X^2 - 4 X - 21 = 0$

By applying “splitting the middle term” method,

$\therefore X^2 - 7 X +3 X - 21 = 0$

$\therefore (X - 7) (X + 3) = 0$

Therefore, X = 7 and X = – 3

Here, we reject X = – 3, because age can not be negative, and accept only X = 7

Therefore, Rehman’s present age is 7 years.

Check:

If Rehman’s present age is 7 years, his age 3 years ago was 7 – 3 = 4 years and 5 years later will be 7 + 5 = 12 years

Sum of reciprocals of 4 and 12, we get: $\frac{1}{4} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3}$

Since it meets our given condition, hence our answer is correct.

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