Algebraic Fractions
Algebraic Fractions in NSW Year 10 Core maths is about adding, subtracting, multiplying and dividing fractions that contain pronumerals (NESA outcome MA5‑ALG‑C‑01). Most questions have numerical denominators, so the same lowest common denominator method used for ordinary fractions still works.
You will combine fractions such as \(\frac{x+4}{3} + \frac{2x-1}{5}\), take care with signs when subtracting a binomial numerator, multiply fractions and divide by multiplying by the reciprocal, and cancel common factors. A few harder questions factorise before cancelling. You will also write the perimeter of a triangle with fractional side lengths as a single fraction, and spot the error in a student's working. It is part of Year 10 Core Algebra.
Theory
An algebraic fraction is a fraction with a pronumeral in it, such as \(\dfrac{x + 3}{4}\). This Year 10 maths page shows how to add and subtract them with a lowest common denominator, how to multiply and divide them by cancelling common factors, and how to leave the answer in simplest form.
An algebraic fraction has a pronumeral in the numerator, the denominator or both. The rules are the same as for number fractions.
To add or subtract, the fractions need the same denominator. Use the lowest common denominator (LCD): the smallest number that every denominator divides into. The LCD of \(4\) and \(6\) is \(12\), not \(24\).
When a numerator has two terms, such as \(x - 1\), treat it as one quantity: keep it in brackets while you multiply it.
To multiply, cancel common factors, then multiply the numerators and the denominators. To divide, multiply by the reciprocal of the second fraction.
A fraction is in simplest form when the numerator and the denominator have no common factor. You can only cancel a factor of the whole numerator and the whole denominator.
Adding or subtracting with the LCD, and multiplying and dividing:
With \(bd\) replaced by the LCD when \(b\) and \(d\) share a factor.
How to add or subtract algebraic fractions
- Find the LCD of the denominators.
- Rename each fraction: multiply its numerator, in brackets, by the factor that turns its denominator into the LCD. Write a whole number as a fraction over the LCD.
- Combine over one denominator. Expand the brackets, taking care with a minus sign in front of a numerator.
- Collect like terms in the numerator.
- Simplify: if every term of the numerator shares a factor with the denominator, factorise and cancel.
How to multiply or divide
- Flip the second fraction if it is a division.
- Factorise any numerator or denominator that has a common factor.
- Cancel factors common to a numerator and a denominator, then multiply across.
(i) \(\dfrac{2a}{3} + \dfrac{a}{4}\)
(ii) \(\dfrac{5m}{6} - \dfrac{2n}{9}\)
(i) The LCD of \(3\) and \(4\) is \(12\). Rename each fraction, then add the numerators.
| \(\dfrac{2a}{3} + \dfrac{a}{4}\) | \(=\) | \(\dfrac{8a}{12} + \dfrac{3a}{12}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{11a}{12}}\) |
(ii) The LCD of \(6\) and \(9\) is \(18\). \(m\) and \(n\) are not like terms, so the numerator stays as two terms.
| \(\dfrac{5m}{6} - \dfrac{2n}{9}\) | \(=\) | \(\dfrac{15m}{18} - \dfrac{4n}{18}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{15m - 4n}{18}}\) |
The LCD is \(12\). Multiply each numerator, in brackets, by the factor that makes its denominator \(12\).
| \(\) | \(\) | \(\dfrac{2x + 5}{3} - \dfrac{x - 2}{4}\) |
| \(\) | \(=\) | \(\dfrac{4(2x + 5) - 3(x - 2)}{12}\) |
| \(\) | \(=\) | \(\dfrac{8x + 20 - 3x + 6}{12}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{5x + 26}{12}}\) |
\(-3 \times (-2) = +6\). Writing \(-6\) here is the most common mistake.
(i) \(\dfrac{x - 2}{6} \times \dfrac{9}{4}\)
(ii) \(\dfrac{4x + 8}{5} \div \dfrac{3x + 6}{10}\)
(i) No common denominator is needed. Cancel the common factor \(3\) from \(9\) and \(6\), then multiply.
| \(\dfrac{x - 2}{6} \times \dfrac{9}{4}\) | \(=\) | \(\dfrac{x - 2}{2} \times \dfrac{3}{4}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{3(x - 2)}{8}}\) |
(ii) Multiply by the reciprocal of the second fraction. Factorise each numerator so the common bracket \((x + 2)\) can cancel.
| \(\) | \(\) | \(\dfrac{4x + 8}{5} \div \dfrac{3x + 6}{10}\) |
| \(\) | \(=\) | \(\dfrac{4(x + 2)}{5} \times \dfrac{10}{3(x + 2)}\) |
| \(\) | \(=\) | \(\dfrac{4 \times 2}{3}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{8}{3}}\) |
The LCD of \(2\) and \(5\) is \(10\). Write the whole number \(2\) as \(\dfrac{20}{10}\).
| \(\) | \(\) | \(\dfrac{x}{2} - \dfrac{x + 3}{5} + 2\) |
| \(\) | \(=\) | \(\dfrac{5x - 2(x + 3) + 20}{10}\) |
| \(\) | \(=\) | \(\dfrac{5x - 2x - 6 + 20}{10}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{3x + 14}{10}}\) |
Check with \(x = 2\): the question gives \(1 - 1 + 2 = 2\) and the answer gives \(\dfrac{20}{10} = 2\).
Common pitfalls
Frequently asked questions
How do you add algebraic fractions?
Find the lowest common denominator, rename each fraction so it has that denominator, then add the numerators and collect like terms. Keep each numerator in brackets while you rename it.
Why do you put brackets around the numerator?
Because the whole numerator is multiplied by the same number. For \(\dfrac{x - 2}{4}\) renamed over \(12\), the numerator becomes \(3(x - 2)\), and a minus sign in front changes both signs.
How do you divide algebraic fractions?
Multiply the first fraction by the reciprocal of the second. Then factorise, cancel any common factors and multiply across.
Can you cancel the x in (x + 3) over x?
No. The \(x\) on top is only one term of the numerator, not a factor of it. You can only cancel a factor of the whole numerator and the whole denominator.
What is the lowest common denominator?
The smallest number that every denominator divides into exactly. For \(6\) and \(9\) it is \(18\). Using a bigger common denominator still works, but you have more simplifying to do at the end.