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# The sum of reciprocals of Rehman’s ages (in years) 3 years ago and 5 years from now is 13. Find his present age.

Let present age of Rehman = x years

Age of Rehman 3 years ago = (x - 3) years.
Age of Rehman after 5 years = (x + 5) years

According to the given condition:

$$\Rightarrow$$ $${{1} \over {x - 3}} + {{1} \over {x + 5}} = {{1} \over {3}}$$
$$\Rightarrow$$$${{(x + 5) + (x - 3)} \over {(x - 3)(x + 5)}} = {{1} \over {3}}$$
$$\Rightarrow$$$$3 (2x + 2) = (x - 3) (x + 5)$$
$$\Rightarrow$$$$6x + 6 = x^2 - 3x + 5x -15$$
$$\Rightarrow$$$$x^2 - 4x - 15 - 6 = 0$$
$$\Rightarrow$$$$x^2 - 4x -21 = 0$$

Comparing quadratic equation $$x^2 - 4x -21 = 0$$ with general form $$ax^2 + bx + c = 0$$,

We get a = 1, b = -4 and c = -21

Using quadratic formula = $$x = {{-b ± \sqrt{b^2 - 4ac}} \over {2a}}$$

$$\Rightarrow$$$$x = {{4 ± \sqrt{(4)^2 - 4(1)(-21)}} \over {(2)(1)}}$$
$$\Rightarrow$$$$x = {{4 ± \sqrt{16 + 84}} \over {2}}$$
$$\Rightarrow$$$$x = {{4 + 10} \over {2}} , {{4 - 10} \over {2}}$$
$$\Rightarrow$$$$x = 7, -3$$

We discard x=-3. Since age cannot be in negative.

Therefore, present age of Rehman is 7 years.