Mixed to Improper

A cook asks for "5 and 1/3 cups". That amount is called a mixed number: a whole number and a fraction joined by the word and. It is written $5\tfrac{1}{3}$, the whole number right beside the fraction, and it means plus: $5\tfrac{1}{3}$ is $5+\frac{1}{3}$.

To work with it as one fraction, count everything in thirds. Each whole holds 3 thirds, so 5 wholes hold $5\cdot 3$, which is 15 thirds. The line $5+\frac{1}{3}$ becomes $\frac{15}{3}+\frac{1}{3}$, and adding the tops gives $\frac{16}{3}$. As in Chapter 4, a fraction like $\frac{16}{3}$, with its top bigger than its bottom, is improper.

Why bother? A whole number and a fraction cannot be added top to top until they are counted in the same unit. Once the whole is written in thirds, it is just another fraction with the same denominator, and the rules for fractions take over. The shortcut: whole times denominator, plus the top, all over the denominator.