Question

What would be the reciprocal of the sum of the reciprocal of the numbers \[\dfrac{3}{5}\]and \[\dfrac{7}{3}\]?
A. \[\dfrac{1}{42}\]
B. \[\dfrac{21}{41}\]
C. \[\dfrac{4}{5}\]
D. \[\dfrac{36}{55}\]

Answer 1

Hint: In order to solve this problem, first of all we need to take the reciprocal of two numbers. Then, we have to add those two numbers. And then, after adding two reciprocal numbers, we must take the reciprocal of the obtained answer.

Complete step-by-step solution:
Before solving this problem,
First of all, we need to understand the concept of reciprocal of a number that means
The reciprocal of a number is one divided by that number. For example, reciprocal of x is \[\dfrac{1}{x}\]
Now, we have to understand the concept of reciprocal of a fraction that means
The reciprocal of a fraction is a fraction obtained by swapping or interchanging the values in the numerator and the denominator of the given fraction, just turn it upside down.
For example: Reciprocal of \[\dfrac{3}{5}\]is \[\dfrac{5}{3}\]and Reciprocal of \[\dfrac{7}{3}\] is \[\dfrac{3}{7}\]
Now, we come to the problem, here we have asked to find reciprocals of sum of reciprocal of a numbers which is given as \[\dfrac{3}{5}\]and \[\dfrac{7}{3}\]
Here, first we will find the reciprocal of each number that is
Reciprocal of \[\dfrac{3}{5}=\dfrac{5}{3}--(1)\]
Reciprocal of \[\dfrac{7}{3}=\dfrac{3}{7}--(2)\].
By adding the equation (1) and (2) we get:
\[\Rightarrow \dfrac{5}{3}+\dfrac{3}{7}\]
By taking LCM on above step we get:
\[\Rightarrow \dfrac{\left( 5\times 7 \right)+\left( 3\times 3 \right)}{21}\]
By simplifying further, we get:
\[\Rightarrow \dfrac{35+9}{21}=\dfrac{41}{21}\]
Now, according to the definition of the reciprocal of a fraction that is
Reciprocal of \[\dfrac{41}{21}=\dfrac{21}{41}\].
So, the correct option is “option B”.

Note: We should know that the terminology reciprocal stands for the inverse of the number. So, we just need to interchange the numerator and the denominator. Here, students might make a mistake by adding the given number directly. That is wrong because in the question it is written that we have to take reciprocal of each number then we have to add it up and take reciprocal of the simplified answer. Hence, read the question carefully, what is asked and what is given, then solve the problem.