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The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is $13$ . What is his current age?

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answer is B.

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Detailed Solution

Given that the sum of the reciprocals of Rehman’s ages (in years) 3 years ago and 5 years from now is

13

.
Let Rehman's present age be a.
Rehman's age 3 years ago is

a − 3

years.
Rehman's age after 5 years

= a + 5

years.
The sum of the reciprocals of Rehman’s ages 3 years ago and 5 years from now will be,

⇒ 1 a − 3 + 1 a + 5 = 13

⇒ a + 5 + a − 3 (a − 3) (a + 5) = 13 ⇒ 2 a + 2 (a − 3) (a + 5) = 13

Cross-multiply the above equation, we get

⇒ 3 (2 a + 2) = (a − 3) (a + 5) ⇒ 6 a + 6 = a 2 − 3 a + 5 a − 15 ⇒ a 2 + 2 a − 15 − 6 a − 6 = 0 ⇒ a 2 − 4 a − 21 = 0

We know that the standard form of a quadratic equation is

a x 2 + b x + c = 0, a ≠ 0

.
Here we have,
a=1, b=-4, c=-21.
Product of a and c,

a × c = 1 × (− 21) = − 21

.
Factors of 21

= 7 × 3

⇒ a 2 − 7 a + 3 a − 21 = 0

Take common terms out, we get

⇒ a 2 − 7 a + 3 a − 21 = 0 ⇒ a (a − 7) + 3 (a − 7) = 0 ⇒ (a − 7) (a + 3) = 0 ⇒ a + 3 = 0 or a − 7 = 0 ⇒ a = − 3 or a = 7

As the age can never be negative, so we get the present age of Rehman to be 7 years, i.e., a=7.
Present age of Rehman is 7 years.
Therefore, the correct option is 2.

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