Example1.2.12

Say a recipe calls for \(\frac{2}{3}\) cup of milk, but we’d like to double the recipe. One way to measure this out is to fill a measuring cup to \(\frac{2}{3}\text{,}\) two times:

a rectangle that is taller than it is wide; it is subdivided into three equally sized rectangles stacked vertically; the bottom two subrectangles are colored blue to indicate having liquid in them
a rectangle that is taller than it is wide; it is subdivided into three equally sized rectangles stacked vertically; the bottom two subrectangles are colored blue to indicate having liquid in them

Altogether there are four thirds of a whole here. So \(\frac{2}{3} \cdot 2 = \frac{4}{3}\text{.}\) The figure shows \(\frac{2}{3}\) of two wholes. Two wholes can be written as \(2\text{,}\) or as the fraction \(\frac{2}{1}\text{.}\) So mathematically, our figure says

\begin{equation*} \frac{2}{3} \cdot 2 = \frac{2}{3} \cdot \frac{2}{1} = \frac{4}{3}\text{.} \end{equation*}
in-context