Algebraic Fractions

The good news is that every rule you already use for number fractions still works. The only extra skill is spotting which letters and powers cancel.

• The numerator is the top of the fraction.
• The denominator is the bottom of the fraction.
• A factor is something that has been multiplied (not added). Only factors cancel.

For example, $\dfrac{6x}{9}$ has a common factor of $3$ on the top and bottom, so it simplifies to $\dfrac{2x}{3}$. You did not lose any value — you just wrote the same amount more neatly.

• Algebraic fractions follow the same rules as number fractions.
• Numerator is the top; denominator is the bottom.
• Only common factors (things multiplied) can be cancelled.
• Cancelling does not change the value of the fraction.

• The numbers $12$ and $18$ share a factor of $6$, giving $\dfrac{2a^{2}b}{3ab^{2}}$.
• $a^{2}$ on top and $a$ on the bottom share one $a$, giving $\dfrac{2ab}{3b^{2}}$.
• $b$ on top and $b^{2}$ on the bottom share one $b$, giving $\dfrac{2a}{3b}$.

So $\dfrac{12a^{2}b}{18ab^{2}} = \dfrac{2a}{3b}$.

If the numerator or denominator is a product like $2x(x + 1)$, treat the bracket as a single factor — it can only cancel with another copy of $(x + 1)$, not with a single $x$.

• Cancel any common number factor first.
• Then cancel letters that appear on the top and the bottom.
• Treat a bracket as a single factor — only cancel a matching bracket.
• Stop when there is no common factor left.

To divide by a fraction, you multiply by its reciprocal. The reciprocal of $\dfrac{a}{b}$ is $\dfrac{b}{a}$, so you simply flip the second fraction and change the divide to a multiply.

For instance, $\frac{x}{5} \div \frac{2x}{15} = \frac{x}{5} \times \frac{15}{2x} = \frac{15x}{10x} = \frac{3}{2}.$

Cancel as you go and the numbers stay friendly.

• Multiply: tops together, bottoms together.
• Cancel before you multiply to keep numbers small.
• Divide: flip the second fraction, then multiply.
• Always simplify your final answer.

If the denominators are different, you need a common denominator first. A safe choice is the product of the two denominators; an even better choice is the lowest common multiple (LCM).

A reliable method:

1. Find a common denominator.
2. Rewrite each fraction with that denominator by multiplying its top and bottom by whatever is needed.
3. Combine the numerators.
4. Simplify the result if possible.

For example, simplify $\dfrac{x}{3} + \dfrac{x}{4}$.

• The lowest common multiple of $3$ and $4$ is $12$.
• $\dfrac{x}{3} = \dfrac{4x}{12}$ and $\dfrac{x}{4} = \dfrac{3x}{12}$.
• Add the tops: $\dfrac{4x + 3x}{12} = \dfrac{7x}{12}$.

The same idea works for subtraction — just keep careful track of negative signs in the numerator.

• Same denominator: combine the tops, keep the bottom.
• Different denominators: rewrite each with a common denominator first.
• The lowest common multiple keeps the numbers small.
• Always simplify the final answer if you can.

The same toolkit handles all of them: write the expression, then simplify or combine using the rules from this note. If a value is missing, set the simplified expression equal to a number and solve.

A friendly approach is to do the algebra in steps and check with easy numbers at the end. Replace each letter with a small whole number (like $x = 6$) and see whether your simplified expression gives the same value as the original — if it does, your simplification is almost certainly right.

• Translate the words into an algebraic fraction first.
• Then simplify, combine or solve using the rules.
• Check your work by substituting a small number for each letter.
• Look back at the question to give a sensible final answer.