Multiplying and dividing fractions

Multiplying fractions

"Half of three fourths" is \(\frac{1}{2} \cdot \frac{3}{4}\). Cut three fourths in half and you get three eighths, which matches the rule.

No common denominator needed. But it pays to simplify before multiplying: any numerator can be divided by a common factor with any denominator, even across the two fractions. This is often called cross-canceling.

A whole number is a fraction over \(1\): \(12 \cdot \frac{3}{4} = \frac{12}{1} \cdot \frac{3}{4} = \frac{3}{1} \cdot \frac{3}{1} = 9\).

Reciprocals

The reciprocal of \(\frac{4}{7}\) is \(\frac{7}{4}\). The reciprocal of \(5\) is \(\frac{1}{5}\). The reciprocal of \(-\frac{2}{3}\) is \(-\frac{3}{2}\): the sign stays. Zero has no reciprocal.

Dividing fractions

It makes sense with a simple case: how many quarter-hours are in \(3\) hours? Twelve. And indeed \(3 \div \frac{1}{4} = 3 \cdot 4 = 12\).

Mixed numbers

To multiply or divide mixed numbers, convert them to improper fractions first. Multiplying the whole parts and the fraction parts separately doesn't work.

Powers of fractions

Negative exponents

The quotient rule \(\frac{b^m}{b^n} = b^{m-n}\) leads to negative exponents. For example \(\frac{2^3}{2^5} = 2^{-2}\), but the same fraction is \(\frac{8}{32} = \frac{1}{4}\). So \(2^{-2}\) must equal \(\frac{1}{4}\).

So \(10^{-3} = \frac{1}{1{,}000}\) and \(\left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^2 = \frac{25}{4}\). A negative exponent does not make the answer negative.

Practice