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Question
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.
Solution
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 1/3.
`∴ 1/(x-3) + 1/(x-5) = 1/3`
`(x+5+x-3)/((x-3)(x+5)) = 1/3`
`(2x+2)/((x-3)(x+5)) = 1/3`
⇒ 3(2x + 2) = (x-3)(x+5)
⇒ 6x + 6 = x2 + 2x - 15
⇒ x2 - 4x - 21 = 0
⇒ x2 - 7x + 3x - 21 = 0
⇒ x(x - 7) + 3(x - 7) = 0
⇒ (x - 7)(x + 3) = 0
⇒ x = 7, -3
However, age cannot be negative.
Therefore, Rehman's present age is 7 years.
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