Question

The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is 1/3. Find their present age
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Answer 1

Let the present age of Rehman be x years.

Case Ī

Age of Rehman before 3 years = (x - 3) yrs

Case ĪĪ

Age of Rehman after 5 years = (x + 5) yrs

With the situation of this query, we have,

$\implies \tt \frac{1}{x - 3} + \frac{1}{x + 5} = \frac{1}{3} \\ \\ \implies \tt \frac{x + 5 + x - 3 }{( x- 3)(x + 5)} = \frac{1}{3} \\ \\ \implies \tt \frac{2x + 2}{ {x}^{2} + 2x - 15 } = \frac{1}{3} \\ \tiny{ \text{performing \: cross - multiplication}}\\ \implies \tt 6x + 6 = x^{2} + 2x - 15 \\ \tiny{ \text{switching \: the \: sides}}\\ \implies \tt x^{2} + 2x - 15 = 6x + 6 \\ \tiny{ \text{transposing \: RHS \: to \: LHS}} \\ \implies \tt {x}^{2} + 2x - 6x - 15 - 6 = 0 \\ \\ \implies \tt {x}^{2} - 4x - 21 = 0 \\ \tiny{ \text{applying \: completing \: the \:square \: method }} \\ \implies \tt {(x)}^{2} - 2(x)(2) + {(2)}^{2} - {(2)}^{2} - 21 = 0 \\ \\ \implies \tt {(x - 2)}^{2} - 4 - 21 = 0 \\ \\ \implies \tt {(x - 2)}^{2} - 25 = 0 \\ \\ \implies \tt {(x - 2)}^{2} = 25 \\ \tiny{ \text{taking \: square \: roots \: of \: both \: side}} \\ \implies \tt \sqrt{ {(x - 2)}^{2} } = \sqrt{25} \\ \\ \implies \tt(x - 2) = \pm5$

Now,

• x = (5 + 2) = 7
• x = (- 5 + 2) = - 3

Since, Ages can't be negative. Negative values are omitted.

Thus,

• Present age of Rehman is 7 years.

Answer 2

$\large\bf{\underline{\underline{Let \:us\: say}}}$

$\textbf\color{red}{present age of Rahman is x years.}$

$\large\bf{\underline{\underline{Three\: years \:ago,}}}$

$\textbf\color{red}{Rehman’s age was (x – 3) years.}$

$\large\bf{\underline{\underline{Five\: years \:after,}}}$

$\textbf\color{red}{his age will be (x + 5) years}$

$\large\bf{\underline{\underline{Given}}}$

The sum of the reciprocals of Rehman’s ages 3 years ago and after 5 years is equal to 1/3.

➦ $\sf{1/x-3 + 1/x-5 = 1/3}$

➦ $\sf{(x+5+x-3)/(x-3)(x+5) = 1/3}$

➦$\sf{(2x+2)/(x-3)(x+5) = 1/3}$

➦ $\sf{3(2x + 2) = (x-3)(x+5)}$

➦ $\sf{6x + 6 = x2 + 2x – 15}$

➦b$\sf{x2 – 4x – 21 = 0}$

➦ $\sf{x2 – 7x + 3x – 21 = 0}$

➦ $\sf{x(x – 7) + 3(x – 7) = 0}$

➦ $\sf{(x – 7)(x + 3) = 0}$

$\underline{\boxed{\sf\purple{x\: =\: 7, -3}}}$

$\underline{\bf{As\: we\: know, \:age \:cannot\: be\: negative.}}$

$\underline{\bf{Therefore, \:Rahman’s \:present \:age \:is \:7\: years.}}$

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Thankyou :)