Solve the fraction \[4\dfrac{2}{3} + 3\dfrac{1}{4}\]?

Hint: Fractions are a big part of our daily life and the mathematical world, so we must understand how to perform mathematical operations on two different fractions. In the given question, we have to add the two given fractions. Here we have a mixed fraction. First, we need to convert the mixed fraction into an improper fraction and then we need to apply the addition operation.

Complete step-by-step solution:
We have, \[4\dfrac{2}{3} + 3\dfrac{1}{4}\].
let consider \[4\dfrac{2}{3}\],
\[4\dfrac{2}{3} = 4 + \dfrac{2}{3}\]
Now taking Lcm and solving we have,
\[4\dfrac{2}{3} = \dfrac{{12 + 2}}{3}\]
\[ \Rightarrow 4\dfrac{2}{3} = \dfrac{{14}}{3}\]
Similarly, \[3\dfrac{1}{4}\],
\[3\dfrac{1}{4} = 3 + \dfrac{1}{4}\]
\[3\dfrac{1}{4} = \dfrac{{12 + 1}}{4}\]
\[3\dfrac{1}{4} = \dfrac{{13}}{4}\]
That is,
\[ \Rightarrow \dfrac{{14}}{3} + \dfrac{{13}}{4}\]
To solve this we need to find the LCM of 3 and 4.
We can see that the LCM of 3 and 4 is 12.
Now we need to multiply 12 and divide by 12 to the given problem, we get:
\[ = \dfrac{{\left( {\dfrac{{14}}{3} + \dfrac{{13}}{4}} \right) \times 12}}{{12}}\]
Now multiplying 12 for each fraction in the numerator we get,
\[ = \dfrac{{\left( {\dfrac{{14}}{3} \times 12 + \dfrac{{13}}{4} \times 12} \right)}}{{12}}\]
Simplifying on the numerator we have,
\[ = \dfrac{{\left( {\left( {14 \times 4} \right) + \left( {13 \times 3} \right)} \right)}}{{12}}\]
\[ = \dfrac{{\left( {56 + 39} \right)}}{{12}}\]
\[ = \dfrac{{95}}{{12}}\].
Thus we have \[4\dfrac{2}{3} + 3\dfrac{1}{4} = \dfrac{{95}}{{12}}\].

Note: The fractions having the same denominator can be added easily but when the denominators are different, then we first find the LCM of the terms in the denominator and then add the fractions. Using this approach, we can find out the correct answer.