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.​

Answer 1

Let Rehman's age be x years

⌬ Rehman's age 3 year ago = (x - 3) years

⌬ Rehman's age 5 years from now = (x + 5) years

According to the Question:

The sum of the reciprocals of Rehman's ages (in years) 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 + 5x - 3x - 15

⇏ x² + 5x - 3x - 15 - 6x - 6 = 0

⇏ x² + 5x - 3x - 6x - 15 - 6 = 0

⇏ x² - 4x - 21 = 0

Now, Splitting middle term,

⇏ x² + 3x - 7x - 21 = 0

⇏ x(x + 3) - 7(x + 3) = 0

⇏ (x + 3)(x - 7) = 0

⇏ (x + 3) = 0 or (x - 7) = 0

⇏ x = - 3 or x = 7

⇏ x = - 3 and 7

Since, x can't be negative, as x is Rehman's current age.

So, x = 7

∴ Rehman's current age is 7 years.

Answer 2

$\large\sf{Let,the\:age\:of\:rehman=x\:years}$

$\large\sf{3\:years\:ago,}$

$\large\sf{age=(x-3)}$

━━━━━━━━━━━━━━━

$\large\sf{5\:years\:after=(x+5)}$

━━━━━━━━━━━━━━━

$\huge\sf{A/C,}$

$\large\sf{\frac{1}{x - 3} + \frac{1}{x + 5} = \frac{1}{3}}$

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

$\large\sf{\frac{x + 5 + x - 3}{ {x}^{2} + 5x - 3x - 15} = \frac{1}{3}}$

$\large\sf{\frac{2x + 2}{ {x}^{2} + 2x - 15 } = \frac{1}{3}}$

━━━━━━━━━━━━━━━

$\large\sf{{x}^{2}+2x-15=3(2x+2)}$

$\large\sf{{x}^{2}+2x-15=6x+6}$

$\large\sf{{x}^{2}+2x-6x-15-6=0}$

$\large\sf{{x}^{2}-4x-21=0}$

$\large\sf{{x}^{2}-7x+3x-21=0}$

$\large\sf{x(x-7)+3(x-7=0}$

$\large\sf{(x+3)(x-7=0}$

━━━━━━━━━━━━━━━

$\large\sf{x+3=0}$

$\large\sf{x=-3}$

$\large\sf{x-7=0}$

$\large\sf{x=7}$

━━━━━━━━━━━━━━━

But the age cannot be negative,

THEREFORE, the age of Rehman is 7 years.