Algebraic Fractions

In Grade 7 you met algebraic fractions — fractions with letters in the numerator or denominator. You simplified them by cancelling common factors, and combined simple ones with the four operations.

This year you go deeper. The fractions are a little harder: their tops and bottoms may need factorising before you can cancel, and their denominators may be brackets like $(x + 2)$.

Here is the reassuring part, just as before: every rule still works. Simplifying, multiplying, dividing, adding and subtracting all behave exactly as they do for number fractions. The letters simply come along for the ride.

Keep your Grade 7 toolkit close — the golden rule of cancelling and the common-denominator method are still at the heart of everything.

In Grade 7 you learned the golden rule: you may only cancel factors, things that are multiplied — never single terms in a sum.

So in $\frac{3x + 6}{3}$ you cannot cancel the 3s directly, because on top the 6 is added. But there is a clever move: factorise the top first.

The top $3x + 6$ factorises to $3(x + 2)$. Now the fraction is $\frac{3(x + 2)}{3}$, and the factor 3 is multiplied — so it cancels, leaving $x + 2$.

The Stage 9 method is:

1. Look for a common factor in the numerator (or denominator).
2. Factorise to turn the sum into a product.
3. Now cancel any factor shared by top and bottom.

For example, $\frac{4x + 8}{6}$ — factorise the top to $4(x + 2)$, then both $4$ and $6$ share a factor of $2$, giving $\frac{2(x + 2)}{3}$.

Factorising first is the key that unlocks cancelling in many Stage 9 fractions.

Multiplying algebraic fractions follows the rule you already know — multiply the tops together and the bottoms together, then simplify.

$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

This year the fractions may include brackets, so watch them carefully.

Take $\frac{x + 1}{4} \times \frac{2}{x + 1}$. Multiply tops: $2(x + 1)$. Multiply bottoms: $4(x + 1)$. That gives $\frac{2(x + 1)}{4(x + 1)}$.

Now cancel. The whole bracket $(x + 1)$ appears top and bottom — it is a factor, so it cancels. The numbers $2$ and $4$ also share a factor of $2$. The answer simplifies to $\frac{1}{2}$.

A neat shortcut: cancel before you multiply wherever a factor appears on a top and a bottom. It keeps the working small and tidy.

Dividing by a fraction uses the same clever move as always: flip the second fraction, then multiply. The flipped fraction is its reciprocal.

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Take $\frac{x}{6} \div \frac{x}{3}$. Flip the second fraction to $\frac{3}{x}$, then multiply: $\frac{x}{6} \times \frac{3}{x} = \frac{3x}{6x}$. Cancelling the common factor $3x$ leaves $\frac{1}{2}$.

The order matters: it is always the second fraction — the one you are dividing by — that gets flipped, never the first.

Dividing a fraction by a whole expression also works: treat the expression as a fraction over 1. So $\frac{x}{4} \div 3$ becomes $\frac{x}{4} \times \frac{1}{3} = \frac{x}{12}$.

Adding and subtracting algebraic fractions needs a common denominator — the bottoms must match before you combine the tops.

If the denominators differ, rewrite both fractions over a common denominator. With subtraction, take real care with the signs on the top.

For $\frac{2x}{3} - \frac{x}{4}$, a common denominator is 12. Rewrite each: $\frac{2x}{3} = \frac{8x}{12}$ and $\frac{x}{4} = \frac{3x}{12}$. Now subtract the tops: $\frac{8x - 3x}{12} = \frac{5x}{12}$.

When a numerator is a bracket, expand it carefully. For $\frac{x + 3}{2} - \frac{x}{5}$, a common denominator is 10: rewrite as $\frac{5(x + 3)}{10} - \frac{2x}{10}$. Expand the top: $\frac{5x + 15 - 2x}{10} = \frac{3x + 15}{10}$.

The key rules stay the same: only the numerators are combined, the common denominator stays, and subtraction affects every term on the top.

Getting comfortable with algebraic fractions now pays off across a great deal of maths ahead:

• Solving equations with fractions becomes routine once you can combine and clear them.
• Rearranging formulae — making a different letter the subject — often produces algebraic fractions to simplify.
• Rates and ratios such as speed, density and proportion are written as algebraic fractions.
• In science, formulae like $\text{speed} = \frac{\text{distance}}{\text{time}}$ are algebraic fractions you will rearrange often.

The reassuring message stays the same: nothing here is genuinely new — every rule is one you already use for number fractions. Factorising is simply an extra tool that lets cancelling work in more places.

If algebraic fractions feel hard later, it is almost always the cancelling rule — only factors, never terms in a sum — that needs a refresh. Come back to this guide whenever you need it; there is no prize for rushing.