Algebraic Fractions

An algebraic fraction is a fraction whose top, bottom, or both contain letters. Examples include $\dfrac{3x}{6}$, $\dfrac{2a}{5b}$ and $\dfrac{4xy}{8x}$.

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.

To simplify an algebraic fraction, look for the largest number that divides the numerator and denominator, then look for any letters that appear in both.

A good order is:

1. Cancel any common number factor.
2. Cancel letters that appear on the top and the bottom, one at a time.
3. Check that nothing is left to cancel.

For example, simplify $\dfrac{12a^{2}b}{18ab^{2}}$.

• 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 $x$.

To multiply algebraic fractions, multiply the numerators together and the denominators together. To keep numbers small, cancel before you multiply whenever you can.

For example, $\frac{3x}{4} \times \frac{8}{9x^{2}}.$ Cancel a $3$ from the top of the first and bottom of the second; cancel a $4$ from the bottom of the first and the top of the second; cancel an $x$ from each. You are left with $\frac{1}{1} \times \frac{2}{3x} = \frac{2}{3x}.$

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.

If two algebraic fractions already share the same denominator, just add or subtract the numerators and keep the denominator the same: $\frac{2x}{7} + \frac{3x}{7} = \frac{5x}{7}.$

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.

Many short problems lead naturally to an algebraic fraction. Reading the question carefully tells you whether you need to simplify, combine, or solve.

Some typical situations:

• Sharing equally: "Three friends share a bag of $x$ sweets equally" gives $\dfrac{x}{3}$ sweets each.
• Rate of change: "A car covers $d$ km in $t$ hours" gives an average speed of $\dfrac{d}{t}$ km/h.
• Combining quantities: "Bag A weighs $\dfrac{x}{2}$ kg and Bag B weighs $\dfrac{x}{3}$ kg" leads to $\dfrac{x}{2} + \dfrac{x}{3} = \dfrac{5x}{6}$ kg.
• Scaling: "If the cost per item is $\dfrac{C}{n}$, then $5$ items cost $\dfrac{5C}{n}$."

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.

Algebraic fractions are not just a topic on their own — they are the language of many later subjects.

• Solving equations often involves clearing an algebraic fraction by multiplying both sides by the denominator.
• Rearranging formulae for physics and finance uses the same multiply, divide and simplify steps.
• Ratios and proportion can be expressed as algebraic fractions, like $\dfrac{x}{y}$.
• Rates and speeds — distance, time, density and pressure — are written as fractions all the time.

Practise these now and the algebra in IGCSE Maths, Physics and Chemistry will feel much more familiar.