Question

The sum of the reciprocals of Rahul’s ages 3 years ago and 5 years from now is 1/3. The present age of Rahul is:

Answer 1

Given :

• The sum of the reciprocals of Rahul’s ages 3 years ago and 5 years from now is 1/3.

To Find :

• Present Age of Rahul .

Solution :

$\longmapsto\tt{Let\:Present\:age\:of\:Rahul\:be=x\:yrs}$

$\longmapsto\tt{Before\:3\:yrs=x-3\:yrs}$

After 5 years :

$\longmapsto\tt{Age\:of\:Rahul=x+5}$

A.T.Q :

$\longmapsto\tt{\dfrac{1}{x-3}+\dfrac{1}{x+5}=\dfrac{1}{3}}$

$\longmapsto\tt{\dfrac{(x+5)+(x-3)}{(x-3)\:(x+5)}=\dfrac{1}{3}}$

$\longmapsto\tt{\dfrac{2x+2}{(x-3)\:(x+5)}=\dfrac{1}{3}}$

$\longmapsto\tt{3(2x+2)=(x-3)\:(x+5)}$

$\longmapsto\tt{6x+6={x}^{2}+5x-3x-15}$

$\longmapsto\tt{6x+6={x}^{2}+2x-15}$

$\longmapsto\tt{0={x}^{2}+2x-15-6x-6}$

$\longmapsto\tt\bf{{x}^{2}-4x-21=0}$

By Splitting Middle Term :

$\longmapsto\tt{{x}^{2}-(7x-3x)-21=0}$

$\longmapsto\tt{{x}^{2}-7x+3x-21=0}$

$\longmapsto\tt{x(x-7)+3(x-7)=0}$

$\longmapsto\tt{(x+3)\:\:(x-7)=0}$

• x = -3
• x = 7

Age can't be negative . So , The Present Age of Rahul is 7 years .

Answer 2

Answer :-

$\quad\pink{\bigstar}\:\sf Present\:age\:of\:Rahul\:\leadsto\:{\boxed{\tt{\blue{7\:years}}}}$

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Given

• The sum of the reciprocals of Rahul’s ages 3 years ago and 5 years from now is ⅓.

To Find

• Present age of Rahul = ?

Solution

• Let present age of Rahul be m years.

• So, before 3 years Rahul's age = (x - 3) years

• And, after 5 years Rahul's age = (x + 5) years

$\tiny\underline{\bigstar\:{\mathcal{\pink{ACCORDING\:\:TO\:\:THE\:\:QUESTION\::-}}}}$

➪ $\sf \dfrac{1}{x - 3} + \dfrac{1}{x + 4} = \dfrac{1}{3}$

➪ $\sf \dfrac{(x + 5) + (x - 3)}{(x - 3)(x + 5)} = \dfrac{1}{3}$

➪ $\sf \dfrac{x + 5 + x - 3}{x(x + 5) - 3(x + 5)} = \dfrac{1}{3}$

➪ $\sf \dfrac{x + x + 5 - 3}{x^2 + 5x - 3x - 15)} = \dfrac{1}{3}$

➪ $\sf \dfrac{2x + 2}{x^2 + 2x - 15)} = \dfrac{1}{3}$

$\tiny\underline{\bigstar\:{\mathcal{\green{BY\:\:CROSS\:\:MULTIPLICATION\::-}}}}$

➪ $\sf 3(2x + 2) = 1(x^2 + 2x - 15)$

➪ $\sf 6x + 6 = x^2 + 2x - 15$

➪ $\sf 0 = x^2 + 2x - 6x - 15 - 6$

➪ $\sf 0 = x^2 - 4x - 21$

➪ $\sf x^2 - 4x - 21 = 0$

$\tiny\underline{\bigstar\:{\mathcal{\purple{SPLITTING\:\:THE\:\:MIDDLE\:\:TERM\::-}}}}$

➪ $\sf x^2 - 7x + 3x - 21 = 0$

➪ $\sf x(x - 7) + 3(x - 7) = 0$

➪ $\sf (x + 3)(x - 7) = 0$

➪ $\sf x + 3 = 0\:,\:x - 7 = 0$

➪ $\sf x = 0 - 3\:,\:x = 0 + 7$

➪ $\bf { \red{ x = -3}\:,\:\red{x = 7}}$

☯ 〘 Age can't be negative !! 〙

• So, we will reject x = -3

∴ Hence, present age of Rahul is 7 years

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