Adding and Subtracting Algebraic Fractions
Adding and subtracting algebraic fractions uses a common denominator, just as number fractions do, so \(\frac{x}{3} + \frac{x}{5} = \frac{5x}{15} + \frac{3x}{15} = \frac{8x}{15}\). This NSW Year 9 Core maths subtopic is Stage 5 Core content under NESA outcome MA5‑ALG‑C‑01; a pronumeral underneath reaches into MA5‑ALG‑P‑01.
You will add and subtract fractions that already share a denominator, including ones like \(\frac{4}{d} + \frac{3}{d}\), spot when one denominator divides into the other, and find a common denominator for unrelated ones such as \(\frac{7a}{8} + \frac{5a}{6}\). You will also add a whole number by writing it over the denominator, and add fractions in different pronumerals, such as \(\frac{2x}{3} + \frac{y}{2}\), where the top cannot be collected. It is part of Year 9 Core Algebra: Expanding and Factorising.
Theory
Adding or subtracting algebraic fractions needs a common denominator, exactly as with number fractions. This Year 9 maths page shows the three cases: denominators that already match, one that divides the other, and unrelated ones that need the lowest common denominator, plus whole numbers and different pronumerals.
Fractions can only be added or subtracted when they have the same denominator. Then the numerators are added or subtracted and the denominator stays the same.
The lowest common denominator (LCD) is the smallest number that every denominator divides into. For \(4\) and \(6\) it is \(12\).
To rewrite a fraction over the LCD, multiply its top and bottom by the same number. This makes an equivalent fraction: \(\dfrac{x}{3} = \dfrac{4x}{12}\).
Like terms such as \(4x\) and \(3x\) can be collected into \(7x\). Unlike terms such as \(4x\) and \(3y\) cannot, so they stay as two terms on top.
Same denominator:
Different denominators (multiplying them always gives a common one):
| Denominators | LCD | Multiply |
|---|---|---|
| \(3\) and \(3\) | \(3\) | nothing |
| \(5\) and \(10\) | \(10\) | the fifths by \(2\) |
| \(6\) and \(9\) | \(18\) | sixths by \(3\), ninths by \(2\) |
| \(4\) and \(6\) | \(12\) | quarters by \(3\), sixths by \(2\) |
How to add or subtract algebraic fractions
- Find the lowest common denominator of all the fractions (a whole number has denominator \(1\)).
- Rewrite each fraction over the LCD: multiply its top and bottom by the same number.
- Add or subtract the numerators and keep the denominator.
- Collect like terms on top, then cancel any factor common to the whole top and the bottom.
(i) \(\dfrac{5k}{8} + \dfrac{k}{8}\) and
(ii) \(\dfrac{9}{y} - \dfrac{4}{y}\).
(i) Add the numerators, keep the denominator, then simplify:
| \(\dfrac{5k}{8} + \dfrac{k}{8}\) | \(=\) | \(\dfrac{6k}{8}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{3k}{4}}\) |
(ii) A pronumeral underneath works the same way when it is the same in both:
| \(\dfrac{9}{y} - \dfrac{4}{y}\) | \(=\) | \(\dfrac{9 - 4}{y}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{5}{y}}\) |
The denominator stays. It is never added.
\(5\) goes into \(10\), so only the fifths change. Multiply top and bottom by \(2\):
| \(\dfrac{2n}{5}\) | \(=\) | \(\dfrac{4n}{10}\) |
| \(\dfrac{4n}{10} - \dfrac{n}{10}\) | \(=\) | \(\boldsymbol{\dfrac{3n}{10}}\) |
\(3\) and \(10\) share no factor, so the answer is finished.
The lowest number that \(6\) and \(9\) both go into is \(18\). Whatever multiplies the bottom multiplies the top:
| \(\dfrac{5c}{6}\) | \(=\) | \(\dfrac{15c}{18}\) |
| \(\dfrac{c}{9}\) | \(=\) | \(\dfrac{2c}{18}\) |
| \(\dfrac{15c}{18} - \dfrac{2c}{18}\) | \(=\) | \(\boldsymbol{\dfrac{13c}{18}}\) |
Multiplying the denominators, \(6 \times 9 = 54\), also works, but then the answer has to be simplified.
(i) \(\dfrac{2x}{5} + \dfrac{y}{4}\) and
(ii) \(\dfrac{x}{4} + \dfrac{x}{6} - 2\).
(i) The LCD of \(5\) and \(4\) is \(20\):
| \(\dfrac{8x}{20} + \dfrac{5y}{20}\) | \(=\) | \(\boldsymbol{\dfrac{8x + 5y}{20}}\) |
\(8x\) and \(5y\) are not like terms, so the top stays as two terms.
(ii) The LCD of \(4\), \(6\) and \(1\) is \(12\). The \(2\) is \(\dfrac{2}{1} = \dfrac{24}{12}\):
| \(\) | \(=\) | \(\dfrac{3x}{12} + \dfrac{2x}{12} - \dfrac{24}{12}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{5x - 24}{12}}\) |
Common pitfalls
Frequently asked questions
How do you add algebraic fractions?
Rewrite the fractions over a common denominator, add the numerators, keep the denominator, then simplify.
How do you find the lowest common denominator?
List multiples of the larger denominator until you reach one that the other denominator also divides into. For \(4\) and \(6\), the multiples of \(6\) are \(6\) and \(12\), and \(4\) divides into \(12\).
Why can't you add the denominators?
The denominator tells you the size of the pieces. A third and a quarter are different sizes, so they must be changed into the same size pieces, such as twelfths, before they can be counted together.
How do you add a whole number to an algebraic fraction?
Write the whole number over the same denominator. For example \(2 = \dfrac{24}{12}\), so \(\dfrac{x}{12} + 2 = \dfrac{x + 24}{12}\).