Multiply and Divide Algebraic Fractions
Algebraic fractions are fractions with pronumerals in them, such as \(\frac{3x}{4}\). Multiplying and dividing them works just like it does with number fractions. This NSW Year 9 Core maths subtopic is Stage 5 Core content under NESA outcome MA5‑ALG‑C‑01; fractions with a pronumeral underneath reach into the Path outcome MA5‑ALG‑P‑01.
You will multiply straight across, cancel common numbers and pronumerals before you multiply, cancel powers such as \(\frac{y}{y^2} = \frac{1}{y}\), and divide by multiplying by the reciprocal, so \(\frac{x}{3} \div \frac{2}{x} = \frac{x^2}{6}\). You will also work chains of multiplication and division from left to right, as in \(5 \div \frac{3}{10p} \times 6p^2\), and treat a whole number as a fraction over \(1\). It is part of Year 9 Core Algebra: Expanding and Factorising.
Theory
Algebraic fractions multiply and divide exactly like number fractions, with no common denominator needed. This Year 9 maths page shows how to multiply tops and bottoms, cancel common numbers and pronumerals, divide by flipping the second fraction, and work through a chain of both.
An algebraic fraction is a fraction with a pronumeral in it, such as \(\dfrac{3x}{4}\) or \(\dfrac{5}{2y}\). The top is the numerator and the bottom is the denominator.
To multiply two fractions, multiply the numerators together and the denominators together. No common denominator is needed.
To cancel, divide the top and the bottom by the same factor. A number cancels with a number, and a pronumeral cancels with the same pronumeral. Cancelling never changes the value of the fraction.
The reciprocal of \(\dfrac{c}{d}\) is \(\dfrac{d}{c}\), the fraction turned upside down. Dividing by a fraction is the same as multiplying by its reciprocal.
Multiplying and dividing:
Cancelling powers of the same pronumeral, one for one:
How to multiply or divide algebraic fractions
- Change every \(\div\) into \(\times\) by flipping the fraction straight after it.
- Write any whole term as a fraction over \(1\).
- Cancel numbers with numbers and pronumerals with pronumerals, top against bottom.
- Multiply the tops together and the bottoms together.
- Check nothing else cancels. If every pronumeral has gone, the answer is a number.
Cancel before multiplying: \(5\) goes into \(5\) and \(10\), and \(3\) goes into \(3\) and \(6\).
| \(\dfrac{5a}{6} \times \dfrac{3}{10}\) | \(=\) | \(\dfrac{a}{2} \times \dfrac{1}{2}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{a}{4}}\) |
Cancelling first keeps the numbers small. Multiplying first gives \(\dfrac{15a}{60}\), which simplifies to the same answer.
Multiply across, then cancel the numbers and the pronumerals separately:
| \(\dfrac{2p}{q} \times \dfrac{q^2}{8}\) | \(=\) | \(\dfrac{2pq^2}{8q}\) |
| \(\) | \(=\) | \(\dfrac{pq^2}{4q}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{pq}{4}}\) |
\(2\) cancels into \(8\) to leave \(4\). One \(q\) on top cancels the \(q\) underneath, because \(q^2 = q \times q\).
Keep, change, flip, then cancel:
| \(\dfrac{9x}{4} \div \dfrac{3x}{8}\) | \(=\) | \(\dfrac{9x}{4} \times \dfrac{8}{3x}\) |
| \(\) | \(=\) | \(\dfrac{3}{1} \times \dfrac{2}{1}\) |
| \(\) | \(=\) | \(\mathbf{6}\) |
\(3\) cancels into \(9\) and \(3\), \(4\) into \(4\) and \(8\), and the \(x\) cancels too. Every pronumeral has gone, so the answer is just a number.
Change the division into a multiplication by flipping the fraction straight after it. The last fraction stays as it is.
| \(\) | \(=\) | \(\dfrac{2m}{5} \times \dfrac{10}{m} \times \dfrac{3}{4}\) |
| \(\) | \(=\) | \(\dfrac{2}{1} \times \dfrac{2}{1} \times \dfrac{3}{4}\) |
| \(\) | \(=\) | \(\dfrac{12}{4}\) |
| \(\) | \(=\) | \(\mathbf{3}\) |
A whole number or a single term joins a chain as a fraction over \(1\): \(6 = \dfrac{6}{1}\) and \(4ab = \dfrac{4ab}{1}\).
Common pitfalls
Frequently asked questions
How do you multiply algebraic fractions?
Multiply the numerators together and the denominators together, then cancel any common factors. It is quicker to cancel before you multiply.
How do you divide algebraic fractions?
Keep the first fraction, change \(\div\) to \(\times\), and flip the second fraction. Then multiply and cancel as usual.
Can you cancel pronumerals in a fraction?
Yes, if the same pronumeral is a factor of the top and the bottom. For example \(\dfrac{y}{y^2} = \dfrac{1}{y}\).
Do you need a common denominator to multiply fractions?
No. A common denominator is only needed for adding and subtracting fractions.