(Q22)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.

Let the present age of Rehman be x years.

3 years ago Rehman's age = x - 3 and its reciprocal is
1
x-3
Rehman's age 5 years from now = x + 5 and its reciprocal is
1
x+5

The sum of reciprocals

=
1
x-3
 +
1
x+5
 =
(x+5)+(x-3)
(x-3)(x+5)

=
x+5+x-3
x(x+5)-3(x+5)

=
2x+2
x2 + 5x - 3x - 15

=
2x + 2
x2 + 2x - 15
By problem
2(x+1)
x2 + 2x - 15
 =
1
3

x2 + 2x - 15 = 3(2x+2)

x2 + 2x - 15 = 6x + 6

x2 + 2x - 15 - 6x - 6 = 0

x2 - 4x - 21 = 0

x2 - x + 3x - 21 = 0

x(x - ) + 3(x - )=0

(x - )(x + 3) = 0

x - = 0 or x + 3 = 0

x = or x = -3

But x can't be negative, x =

Present age of Rehman = years.