### Theory:

Consider, you went to a shop with your friend. You bought $$7$$ chocolates for both of you. You and your friends ate three chocolates and decided to split the last one chocolate equally.

So each one ate $$3$$ chocolates and $$\frac{1}{2}$$ chocolate this can be written as $3+\frac{1}{2}=3\frac{1}{2}$

In terms of improper fraction, we can say that you both had $\frac{7}{2}$ chocolates, and in terms of mixed fraction we can say that, you both had eaten $3\frac{1}{2}$ chocolates.
Mixed fractions contain a whole number and a proper fraction. They can also be represented as an improper fraction.
Example:
$4\frac{10}{5}$; where $$4$$ is a whole number and $$\frac{10}{5}$$ is a proper fraction.
Let us see how to convert the mixed fraction to improper fraction and vice versa.

Converting mixed fraction to improper fraction:
Apply the following steps to convert a mixed fraction to improper fraction.

Step 1: Multiply the denominator of the proper fraction and whole number and add it with the numerator of the fraction to get the numerator of the improper fraction.

Step 2: Denominator of an improper fraction is the same as the denominator of a mixed fraction.
Example:
Convert $4\frac{10}{5}$ to an improper fraction.

Step 1: Numerator of improper fraction $$=$$ (denominator of the proper fraction $$×$$ whole number) $$+$$ numerator of a proper fraction.

$$= 5 × 4 = 20 + 10 = 30$$

Step 2: Denominator of improper fraction $$=$$ denominator of the mixed fraction.

The denominator of improper fraction $$= 5$$.

Improper fraction $$= \frac{30}{5}$$.

Thus, $4\frac{10}{5}$ $$= \frac{30}{5}$$ $$= 6$$.

Converting improper fraction to mixed fraction:
We can express an improper fraction as a mixed fraction by dividing the numerator by denominator to obtain the quotient and remainder.

Divide the given fraction and find the quotient and remainder.

The quotient of the division $$=$$ The whole number in mixed fraction.

The remainder of a division $$=$$ The numerator of mixed fraction.

The denominator of the improper fraction $$=$$ The denominator of mixed fraction.

$$\text{Quotient}$$ $$+$$ $$\frac{\text{Remainder}}{\text{Divisor}}$$ $$=$$ $$\text{Quotient}$$ $$\frac{\text{Remainder}}{\text{Divisor}}$$
Example:
Let us convert the fraction $$\frac{11}{5}$$ to a mixed fraction.

The remainder is $$1$$, the quotient is $$2$$.

The whole number in mixed fraction $$=$$ quotient of division $$= 2$$

The numerator of mixed fraction $$=$$ reminder of division $$= 1$$

The denominator of mixed fraction $$=$$ denominator of improper fraction $$= 5$$

Thus, $$\frac{11}{5} =$$ $2\frac{1}{5}$.