Mixed numbers

When you order two and a half pizzas, you don't say five halves. You say 2 and another half. That is exactly what a mixed number is: a whole part and a fraction, right next to each other. It is often clearer than a large fraction, and you can switch between the two forms at any time.

Proper and improper fractions

In a proper fraction the numerator is smaller than the denominator, for example $\frac{3}{4}$. If the numerator is greater than or equal to the denominator, the fraction is called improper, for example $\frac{9}{4}$. So an improper fraction contains at least one whole. That whole is exactly what we pull out next.

From improper fraction to mixed number

Divide the numerator by the denominator. The result is the whole part, the remainder stays as a fraction. In $\frac{9}{4}$ the 4 fits into 9 twice, that makes 8, and a remainder of 1 is left. So $\frac{9}{4} = 2\frac{1}{4}$. The denominator never changes here.

Back to an improper fraction

For the way back you compute: whole part times denominator, plus numerator. So $2\frac{3}{4}$ becomes $2 \cdot 4 + 3 = 11$, the denominator stays 4, giving $\frac{11}{4}$. You often need this form for calculating, because a single fraction is easier to multiply and divide.

Adding mixed numbers

The easiest way is to add the whole parts and the fractions separately. In $1\frac{1}{4} + 2\frac{1}{4}$ the wholes give 3 and the quarters together give $\frac{2}{4} = \frac{1}{2}$, so $3\frac{1}{2}$. If the fractions have different denominators, put them over the same denominator first. If the fractional part then comes to more than a whole, convert it and add that whole on top. In $1\frac{1}{2} + 2\frac{3}{4}$ the $\frac{1}{2}$ becomes $\frac{2}{4}$, and $\frac{2}{4} + \frac{3}{4} = \frac{5}{4}$. Together with the wholes that gives $3\frac{5}{4} = 4\frac{1}{4}$.