Example 1.2.15

We could also use multiplication to decrease amounts. Suppose we needed to cut the recipe down to just one fifth. Instead of four of the \(\frac{2}{3}\) cup milk, we need one fifth of the \(\frac{2}{3}\) cup milk. So instead of multiplying by \(4\text{,}\) we multiply by \(\frac{1}{5}\text{.}\) But how much is \(\frac{1}{5}\) of \(\frac{2}{3}\) cup?

If we cut the measuring cup into five equal vertical strips along with the three equal horizontal strips, then in total there are \(3\cdot5=15\) subdivisions of the cup. Two of those sections represent \(\frac{1}{5}\) of the \(\frac{2}{3}\) cup.

In the end, we have \(\frac{2}{15}\) of a cup. The denominator \(15\) came from multiplying \(5\) and \(3\text{,}\) the denominators of the fractions we had to multiply. The numerator \(2\) came from multiplying \(1\) and \(2\text{,}\) the numerators of the fractions we had to multiply.

\begin{align*} \frac{1}{5}\cdot\frac{2}{3}\amp=\frac{1\cdot2}{5\cdot3}\\ \amp=\frac{2}{15} \end{align*}
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