Question

Divide the sum of \[\left( {\dfrac{{65}}{{12}}} \right)\] and \[\left( {\dfrac{8}{3}} \right)\] by their difference.

Answer 1

Hint:
Here we will first take LCM of both the fractions to find the sum of these two fractions. Then we will find the difference of these two fractions by using basic mathematical operations. Then we will divide the sum of these two fractions by their difference to get the required value.

Complete step by step solution:
The given two fractions are \[\left( {\dfrac{{65}}{{12}}} \right)\] and \[\left( {\dfrac{8}{3}} \right)\].
We will first find the sum of these two fractions.
But their denominators are difference, so we will first make the denominators same.
For that, we will find the LCM of the numbers in the denominator of both the fractions i.e. LCM of 12 and 3.
We know that the factors of the number 12 \[ = 2 \times 2 \times 3\].
The factors of the number 3 \[ = 1 \times 3\].
Therefore,
Hence, the LCM of 12 and 3 is equal to \[2 \times 2 \times 3 = 12\].
We need to make the denominator 12. The denominator of first fraction is already equal to 12
Now, we will make the denominator of the second fraction 12. For that, we will multiply the denominator and numerator of the second by the number 4.
 \[\dfrac{{8 \times 4}}{{3 \times 4}} = \dfrac{{32}}{{12}}\]
So, now we have the fractions \[\dfrac{{65}}{{12}}\] and \[\dfrac{{32}}{{12}}\]
Now, we will find the sum of these fractions.
\[\begin{array}{l}{\rm{sum}} = \dfrac{{65}}{{12}} + \dfrac{{32}}{{12}}\\ \Rightarrow {\rm{sum}} = \dfrac{{65 + 32}}{{12}} = \dfrac{{97}}{{12}}\end{array}\]
Now, we will find the difference between these fractions.
\[\begin{array}{l}{\rm{difference}} = \dfrac{{65}}{{12}} - \dfrac{{32}}{{12}}\\ \Rightarrow {\rm{difference}} = \dfrac{{65 - 32}}{{12}} = \dfrac{{33}}{{12}}\end{array}\]
Now, we will divide the sum of the fractions by the difference of the fractions.

\[\begin{array}{l}\dfrac{{{\rm{sum}}}}{{{\rm{difference}}}} = \dfrac{{\dfrac{{97}}{{12}}}}{{\dfrac{{33}}{{12}}}}\\ \Rightarrow \dfrac{{{\rm{sum}}}}{{{\rm{difference}}}} = \dfrac{{97}}{{33}}\end{array}\].
Hence, this is the required answer.

Note:
To find the sum of the fractions and difference between the fractions, we need to make the denominator equal as we can’t find the sum or difference of the fractions having different denominators. Here instead of factoring the denominators for finding LCM we can directly take 12 as LCM because 3 is a factor of 12. LCM or least common multiple is a method of finding the smallest common multiple between two or more numbers.