The **product of two fractions** is the fraction of which the numerator is the product of the two numerators and the denominator is the product of the two denominators.

**Example**

\[\begin{aligned}\tfrac{3}{10}\times \tfrac{5}{7} &=\tfrac{3\times 5}{10\times 7}\\[0.1cm]&=\tfrac{15}{70}=\tfrac{3}{14}\\[0.2cm]\tfrac{5}{4}\times \tfrac{-2}{3} &=\tfrac{5\times (-2)}{4\times 3}\\[0.1cm]&=\tfrac{-10}{12}=-\tfrac{5}{6}\end{aligned}\]

Sometimes it helps to postpone the calculation of these products of numerators and denominators, and is it more convenient to first explore whether there are any factors common to the numerators and denominators that can be cancelled and then actually this operation. This prevents unnecessary arithmetical work with large numbers. However, in our solutions to the exercises we will not do this.

In order to get an idea how this works we look again at the first example \(\tfrac{3}{10}\times \tfrac{5}{7}=\tfrac{3}{14}\). In this case, the numerator of the second fraction is a divisor of the denominator of the first fraction. By carrying out this division we get: \(\tfrac{3}{10}\times \tfrac{5}{7}=\tfrac{3}{2}\times \tfrac{1}{7}\). The product of the fractions can now be computed easily by calculating the product of the numerators and the product of the denominators: \(\tfrac{3}{2}\times \tfrac{1}{7}=\tfrac{3\times 1}{2\times 7}=\tfrac{3}{14}\).

Calculate \(\frac{49}{25}\times\frac{30}{21}\) and simplify the answer as much as possible.

\[\begin{aligned}\frac{49}{25}\times\frac{30}{21}&=\frac{49\times 30}{25\times 21} &\blue{\text{collection of numerators and denominators}}\\ \\ &=\frac{1470}{525}&\blue{\text{multiplication in numerator and denominator}}\\ \\&=\frac{14}{5} &\blue{\text{simplification of the fraction}}\end{aligned}\]

Division by a fraction is the same as multiplication by the inverted fraction.

The **inverted fraction **is obtained by swapping the numerator and denominator of the fraction.

**Example**

\[\begin{aligned}\tfrac{2}{3}\div\tfrac{5}{7} &=\tfrac{2}{3}\times \tfrac{7}{5}=\tfrac{14}{15}\\[0.2cm]\tfrac{2}{3}\div\tfrac{-4}{9} &=\tfrac{2}{3}\times \tfrac{9}{-4}=-\tfrac{3}{2}\end{aligned}\]

In general, two numbers are reciprocal if their product is equal to \(1\). The number \(3\) is the reciprocal of \(\tfrac{1}{3}\) and the fraction \(\tfrac{2}{3}\) is the reciprocal, or inverted fraction, of \(\tfrac{3}{2}\).

Calculate \(\frac{1}{3}\div\frac{1}{2}\) and simplify the answer as much as possible.

\[\begin{aligned} \frac{1}{3}\div\frac{1}{2}&=\frac{1}{3}\times\frac{2}{1}&\blue{\text{conversion to a product of fractions}}\\ \\ &=\frac{1\times 2}{3\times 1}&\blue{\text{collection of numerators and denominators}}\\ \\ &=\frac{2}{3}&\blue{\text{multiplication in numerator and denominator}}\end{aligned}\]

Instead of the operator \(\div\) for division, one also uses the operators \(\mathrm{:}\) and \(/\). Sometimes one uses a horizontal line, too; the following example contains a rational number with a fraction in the numerator and a fraction in the denominator.

Calculate \(\displaystyle\frac{\displaystyle\;\frac{6}{5}\;}{\displaystyle\frac{9}{10}}\) and simplify the answer as much as possible.

\[\begin{aligned}\displaystyle\frac{\displaystyle\;\frac{6}{5}\;}{\displaystyle\frac{9}{10}}&=\frac{6}{5}\div\frac{9}{10}&\blue{\text{change of notation}}\\ &=\frac{6}{5}\times\frac{10}{9}&\blue{\text{conversion to a product of fractions}}\\ \\ &=\frac{6\times 10}{5\times 9}&\blue{\text{collection of numerators and denominators}}\\ \\ &=\frac{60}{45}&\blue{\text{multiplication in numerator and denominator}}\\ \\&=\frac{4}{3} &\blue{\text{simplification of the fraction}}\end{aligned}\]

Multiplication and Division (32:50)