Improper Fractions

A fraction like $\frac{7}{3}$, with a numerator at least as big as its denominator, is called improper. The same amount can be written as a mixed number: 2 wholes and 1 third more, said 'two and one third'. It means 2 plus one third, $2+\frac{1}{3}$, and it is written $2\tfrac{1}{3}$, the whole number right beside the fraction.

To turn a mixed number into an improper fraction, count everything in thirds. Each whole is 3 thirds, so 2 wholes are $\frac{6}{3}$. Add the 1 third: $\frac{6}{3}+\frac{1}{3}$ is 7 thirds, $\frac{7}{3}$. The bottom stays 3 because the pieces are still thirds; only their number grows.

In one step: multiply the whole number by the denominator, add the numerator, and keep the denominator: $2\cdot 3+1$ is 7, so it is 7 thirds. An improper fraction is a single fraction to work with, which is what multiplying and dividing fractions will need later in this section.