Question

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

help please ​

Answer 1

Given :

• Sum of Reciprocals of Rehman's ages 3 years ago and after 5 years from now is 1/3 .

To Find :

• Present Age of Rehman .

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 = -3

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

Answer 2

Given:

• Sum of reciprocals of Rehman's ages 4 years ago and 5 years now is ⅓.

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To find:

• Present age of Rehman.

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Solution:

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• Let the present age of Rehman be x.

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➲ As given in the question that sum of ages 3 years ago and 5 years from now is 1/3.

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☆ 3 years ago

$\tt\dashrightarrow{Rehman's age = x - 3}$

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☆ After 5 years

$\tt\dashrightarrow{Rehman's age = x + 5}$

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According to the question

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

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

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

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$\tt:\implies{\dfrac{2x + 2}{x^2 + 2x - 15} = \dfrac{1}{3}}$

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$\tt:\implies{3(2x + 2) = x^2 + 2x - 15}$

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

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

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$\tt:\implies{x^2 - 4x - 21 = 0}$

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Now, we can see that, it is in the form of a quadratic equation.

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Splitting the middle term

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

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

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

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

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We get,

⠀⠀⠀⠀⠀⠀⇛ x = 7

⠀⠀⠀⠀⠀⠀⇛ x = -3

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Since, ages can't be negative. Therefore, we should take the value of x = 7.

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Hence,

• Present age of Rehman is 7 years.

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