8. Multiplying and dividing fractions — Fractions

Multiplication

To multiply fractions, multiply straight across: numerator times numerator and denominator times denominator.

\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{a \cdot c}{b \cdot d}

Example. Multiplying directly and simplifying at the end:

\dfrac{3}{5} \cdot \dfrac{10}{4} = \dfrac{3 \cdot 10}{5 \cdot 4} = \dfrac{30}{20} = \dfrac{3}{2}

It is easier to factorise before multiplying and cross out the factors that appear both on top and underneath:

\dfrac{3}{5} \cdot \dfrac{10}{4} = \dfrac{3 \cdot \cancel{2} \cdot \cancel{5}}{\cancel{5} \cdot \cancel{2} \cdot 2} = \dfrac{3}{2}

This is where it really pays off, with big numbers:

\dfrac{15}{27} \cdot \dfrac{30}{48} \cdot \dfrac{24}{45} = \dfrac{(3 \cdot 5) \cdot (2 \cdot 3 \cdot 5) \cdot (2^3 \cdot 3)}{3^3 \cdot (2^4 \cdot 3) \cdot (3^2 \cdot 5)} = \dfrac{2^4 \cdot 3^3 \cdot 5^2}{2^4 \cdot 3^6 \cdot 5} = \dfrac{5}{3^3} = \dfrac{5}{27}

Multiplying first gives $\dfrac{10\,800}{58\,320}$, which then has to be simplified. Factorising first, the result comes out irreducible.

Division

To divide fractions, multiply crosswise: the numerator of the first by the denominator of the second, and the denominator of the first by the numerator of the second.

\dfrac{a}{b} : \dfrac{c}{d} = \dfrac{a \cdot d}{b \cdot c}

It is the same as multiplying the first by the reciprocal of the second. The reciprocal is obtained by swapping numerator and denominator.

\dfrac{a}{b} : \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c} = \dfrac{a \cdot d}{b \cdot c}

Example. With several divisions in a row, work from left to right:

\dfrac{5}{3} : \dfrac{2}{7} : \dfrac{4}{5} = \dfrac{35}{6} : \dfrac{4}{5} = \dfrac{175}{24}

Or all at once: the first fraction stays as it is and all the others are flipped.

\dfrac{5}{3} : \dfrac{2}{7} : \dfrac{4}{5} = \dfrac{5}{3} \cdot \dfrac{7}{2} \cdot \dfrac{5}{4} = \dfrac{175}{24}

A fraction of fractions. A fraction whose numerator and denominator are fractions is a division:

\dfrac{\;\dfrac{5}{6}\;}{\;\dfrac{4}{3}\;} = \dfrac{5}{6} : \dfrac{4}{3} = \dfrac{5 \cdot 3}{6 \cdot 4} = \dfrac{15}{24} = \dfrac{5}{8}

Tip: the outer numbers ($5$ and $3$) go to the numerator and the inner numbers ($6$ and $4$) to the denominator.