Question

The sum of reciprocals of rehman's ages 3 years ago and 5 years now is 1 by 3 . find his present age.

A sort of fre e points. But solve the answer properly or else I'll re port it.

$\large \bold{@WildCat7083}$
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Answer 1

Question

The sum of reciprocals of rehman's ages 3 years ago and 5 years now is 1 by 3 . find his present age.

Solution :

$\longmapsto\tt{Let\:present\:age\:of\:Rehman=x}$

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

$\longmapsto\tt{After\:5\:years=x+5}$

A.T.Q :

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

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

$\longmapsto\tt{\dfrac{2x+2}{{x}^{2}+5x-3x-15}=\dfrac{1}{3}}$

Cross Multipy :

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

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

$\longmapsto\tt{{x}^{2}+5x-6x-3x-15-6=0}$

$\longmapsto\tt{{x}^{2}-1x-3x-21=0}$

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

By Splitting Middle Term :

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

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

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

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

• x = 7
• x = 7x = -3

Age can't be negative.

So , The Present Age of Rehman is 7 years .

Answer 2

Answer:

$\huge\fcolorbox{blue}{pink}{☃αղsաҽɾ☃}$

$\bold{⇒ \: let \: rehman's \: current \: age = x} \\ \\ \bold{⇒ \: rehman's \: age \: 3 \: years \: ago = x - 3} \\ \\ \bold{⇒ \: rehman's \: age \: 5 \: years \: from \: now = x + 5}$

$\huge \bold \red{Given \: that : - }$

$\bold{The \: sum \: of \: reciprocals \: of \: rehman's} \\ \bold { ages \: 3 \: years \: ago \: and \: 5 years \: now \: is \frac{1}{3} }$

$⇒ \frac{1}{x - 3} + \frac{1}{x + 5} = \frac{1}{3} \\ \\ ⇒ \: \frac{ x + 5 \: + x - 3 }{( x - 3)(x + 5)} = \frac{1}{3} \\ \\ ⇒ \: \frac{2x + 2}{(x - 3)(x + 5)} = \frac{1}{3} \\ \\ ⇒ \: (2x + 2) \times 3 = 1 \times (x - 3)(x + 5) \\ \\ ⇒ \: 6x + 6 = x(x + 5) - 3(x + 5) \\ \\ ⇒ \: 6x + 6 = {x}^{2} + 5x - 3x - 15 \\ \\ ⇒ \: 0 = {x }^{2} + 5x - 3x - 15 - 6x - 6 \\ \\ ⇒ \: 0 = {x}^{2} + 5x - 3x - 6x - 15 \\ \\ ⇒ \: 0 = {x}^{2} - 4x - 21 \\ \\ ⇒ \: {x}^{2} - 4x - 21 = 0$

$\bold{we \: factorize \: by \: splitting \: the \: middle \: term}$

$⇒ \: {x}^{2} + 3x - 7x - 21 = 0 \\ \\⇒ \: x(x + 3) - 7(x + 3) = 0 \\ \\ ⇒ \: (x + 3)(x - 7) = 0$

$⇒ \: x - 7 = 0 \\ \\ ⇒ \: x = 7$

$⇒ \: x + 3 = 0 \\ \\ ⇒ \: x = - 3$

$\bold{so \: x = 7 \: and \: x = - 3} \\ \\ \bold{but \: x \: cannot \: be \: in \: negative}$

$\bold{so \: rehaman's \: current \: age \: is \: 7 \: years}$