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Improper fractions and mixed numbers

There are three types of fractions: proper, improper, mixed.

Fractions that are greater than $0$ but less than $1$ are called proper fractions. In proper fractions, the numerator is less than the denominator. When a fraction has a numerator that is greater than or equal to the denominator, the fraction is an improper fraction. An improper fraction is always $1$ or greater than $1$. And, finally, a mixed number is a combination of a whole number and a proper fraction.

In a proper fraction, the numerator is always less than the denominator, for example $\displaystyle{\frac{3}{5}}$ and $\displaystyle{\frac{11}{13}}$.

In an improper fraction, the numerator is always greater than or equal to the denominator, for example $\displaystyle{\frac{7}{2}}$ and $\displaystyle{\frac{34}{3}}$ .

Examples of mixed numbers include $5\displaystyle{\frac{2}{3}}$ and $3\displaystyle{\frac{2}{11}}$.


Let’s get back to the pizza notation used in the rest of the lessons about fractions. If you ate one half of a pizza and another pizza arrives, how many pizzas do you have? You have $ 1 + \displaystyle{\frac{1}{2} = \frac{3}{2}}$ pizza. But it’s kind of silly to say you have three halves of a pizza when you can just say you have one and a half, it’s easier and more understandable.


Example 1. Turn the fraction $\displaystyle{\frac{51}{4}}$ into a mixed number.

 $\displaystyle{\frac{51}{4}} = 12.75$

This means that our improper fraction contains $12$ wholes.

How do we get our proper fraction that goes with those wholes? Since you know that one whole contains $4$ parts (from the denominator in your fraction) and you have $12$ wholes, multiplying those two, you get $48$. You have $48$ parts in your wholes, but in your fraction you have $51$. You simply subtract those two and get your fraction.

$51 – 48 = 3$

This leads us to our final solution:

$\displaystyle{\frac{51}{4} = 12 \frac{3}{4}}$.

Example 2. Convert $12 \displaystyle{\frac{3}{4}}$ back to an improper fraction.

From the denominator of the proper fraction which is a part of the given number, we know that one whole contains $4$ parts, and that we have $12$ wholes and three parts. This means that we have to simply multiply $12$ with $4$ and add $3$.

$ 12 \displaystyle{\frac{3}{4}}$

$= \displaystyle{\frac{(12 \cdot 4 + 3)}{4}}$

$= \displaystyle{\frac{(48+3)}{4}}$

$= \displaystyle{\frac{51}{4}}$

Of course, now the question occurs how do you do all those mathematical operations with mixed numbers. The answer is you don’t. Always convert them into improper fraction, and then if there is a need, convert your solution back to a mixed number.