Question

How do you simplify $\dfrac{{1 + \dfrac{1}{x}}}{{\dfrac{1}{x}}}$ ?

Answer 1

Hint: Here the basic concept that is going to be used is LCM i.e. taking the lowest common factor. So, firstly we will take the lowest common factor (LCM) from the numerator and then will solve separately the numerator and denominator.

Complete Step by Step Solution:
Let us consider $\dfrac{{1 + \dfrac{1}{x}}}{{\dfrac{1}{x}}}$ as y in which let $1 + \dfrac{1}{x}$ as u and $\dfrac{1}{x}$ as v
So, $y = \dfrac{u}{v}$ ……(i)
We have $u = 1 + \dfrac{1}{x}$
Now, take LCM (Lowest Common Factor) on the right side in the above equation
$ \Rightarrow u = \dfrac{{x + 1}}{x}$
We have $v = \dfrac{1}{x}$
Put values of u and v in equation (i)
$ \Rightarrow y = \dfrac{{\dfrac{{x + 1}}{x}}}{{\dfrac{1}{x}}}$
Multiply both numerator and denominator by x
$ \Rightarrow y = \dfrac{{x\left( {\dfrac{{x + 1}}{x}} \right)}}{{x\left( {\dfrac{1}{x}} \right)}}$
$ \Rightarrow y = \dfrac{{x + 1}}{1}$
We can write the above equation as $y = x + 1$

Additional Information:
We should take care that if we are multiplying something with a numerator then we must also multiply it with the denominator. As here in the above solution, we multiplied x with both the numerator and denominator.

Note:
There is an alternative method to solve this problem
Let us consider $\dfrac{{1 + \dfrac{1}{x}}}{{\dfrac{1}{x}}}$ as y
$ \Rightarrow y = \dfrac{{1 + \dfrac{1}{x}}}{{\dfrac{1}{x}}}$
Now, after taking LCM in the numerator, we get
$ \Rightarrow y = \dfrac{{\dfrac{{x + 1}}{x}}}{{\dfrac{1}{x}}}$
Now let us multiply the denominator with the numerator. But for that, firstly, we have to take reciprocal of denominator i.e. $\dfrac{x}{1}$
$ \Rightarrow y = \left( {\dfrac{{x + 1}}{x}} \right)\left( {\dfrac{x}{1}} \right)$
$ \Rightarrow y = x + 1$.