Question

How do you simplify $ \dfrac{\left( \dfrac{1}{x} \right)-\left( \dfrac{1}{5} \right)}{\left( \dfrac{1}{{{x}^{2}}} \right)-\left( \dfrac{1}{25} \right)} $ ?

Answer 1

Hint: In this question, we have to simplify a fractional term. So, we will use the algebraic identities to solve the same. The term given to us is in the form of a fraction, so we will use the algebraic identity $ (a+b)(a-b)={{a}^{2}}-{{b}^{2}} $ in the denominator and then make the necessary changes. Then, we will take the LCM in the denominator and multiply and divide 5x on the equation. After the necessary calculations, we get the required solution to the problem.

Complete step by step answer:
According to the question, we have to simplify the fractional number. Since the fraction contains both the variable and the constant, so after we simplify the term, we get the answer in the variable term as well.
So, to solve this problem, we will use algebraic identities.
The fractional term given to us is $ \dfrac{\left( \dfrac{1}{x} \right)-\left( \dfrac{1}{5} \right)}{\left( \dfrac{1}{{{x}^{2}}} \right)-\left( \dfrac{1}{25} \right)} $ ---------- (1)
So, now we will solve the denominator of the equation (1), we get
 $ \Rightarrow \dfrac{\left( \dfrac{1}{x} \right)-\left( \dfrac{1}{5} \right)}{{{\left( \dfrac{1}{x} \right)}^{2}}-{{\left( \dfrac{1}{5} \right)}^{2}}} $
Now, we will apply the algebraic identity $ (a+b)(a-b)={{a}^{2}}-{{b}^{2}} $ on the denominator in the above equation, we get
 $ \Rightarrow \dfrac{\left( \dfrac{1}{x} \right)-\left( \dfrac{1}{5} \right)}{\left( \left( \dfrac{1}{x} \right)-\left( \dfrac{1}{5} \right) \right)\left( \left( \dfrac{1}{x} \right)+\left( \dfrac{1}{5} \right) \right)} $
So, we know the same terms will cancel out in the division, as it gives quotient equal to 1, therefore we get
\[\Rightarrow \dfrac{1}{\left( \left( \dfrac{1}{x} \right)+\left( \dfrac{1}{5} \right) \right)}\]
Now, we will take LCM in the denominator, we get
\[\Rightarrow \dfrac{1}{\dfrac{5+x}{5x}}\]
Also, we will multiply and divide 5x in the above equation, we get
\[\Rightarrow \dfrac{1}{\dfrac{5+x}{5x}}.\dfrac{5x}{5x}\]
On further solving, we get
\[\Rightarrow \dfrac{5x}{5+x}\]
Therefore, for the equation $ \dfrac{\left( \dfrac{1}{x} \right)-\left( \dfrac{1}{5} \right)}{\left( \dfrac{1}{{{x}^{2}}} \right)-\left( \dfrac{1}{25} \right)} $ , after simplifications we get the value \[\dfrac{5x}{5+x}\] , which is our required solution.

Note:
 While solving this problem, do mention the algebraic identity to avoid confusion and mathematical mistakes. One of the alternative methods to solve the problem is to take LCM in both the numerator and the denominator, and make necessary calculations, to get the required solution to the problem.