# Ex.4.3 Q4 Quadratic Equations Solutions - NCERT Maths Class 10

## Question

The sum of the reciprocals of Rehman’s age (in years) \(3\) years ago and \(5\) years from now is \(\begin{align}\frac{1}{3} \end{align}.\) Find his present age.

## Text Solution

**What is Known?**

i) Sum of reciprocals of Rehman’s age (in years) \(3\) years ago and \(5\) years from now is \(\begin{align}\frac{1}{3} \end{align}.\)

**What is Unknown?**

Rehman’s age.

**Reasoning:**

Let the present age of Rehman be *\(x\)* years.

\(3\) years ago, Rehman’s age was \(= x - 3\)

\(5\) years from now age will be \(= x + 5\)

Using this information and the given condition, we can form the following equation:

\[\begin{align}\frac{1}{{x - 3}} + \frac{1}{{x + 5}} = \frac{1}{3} \end{align}\]

**Steps:**

\[\begin{align}\frac{1}{{x - 3}} + \frac{1}{{x + 5}} = \frac{1}{3} \end{align}\]

By cross multiplying we get:

\[\begin{align}\frac{{(x + 5) + (x - 3)}}{{(x - 3)(x + 5)}} &= \frac{1}{3}\\\frac{{2x + 2}}{{{x^2} + 2x - 15}} &= \frac{1}{3}\\(2x + 2)(3) &= {x^2} + 2x - 15\\6x + 6 &= {x^2} + 2x - 15\\{x^2} + 2x - 15 &= 6x + 6\\{x^2} + 2x - 15 - 6x - 6 &= 0\\{x^2} - 4x - 21 &= 0\end{align}\]

Finding roots by factorization:

\[\begin{align}{x^2} - 7x + 3x - 21& = 0\\x(x - 7) + 3(x - 7) &= 0\\

(x - 7)(x + 3)& = 0\\x - 7 &= 0 \quad x + 3 = 0\\x &= 7 \quad x = - 3\end{align}\]

Age can’t be a negative value.

\(\therefore \;\) Rehman’s present age is \(7\) year.