# Algebraic Fractions

> Simplifying, multiplying, dividing, adding and subtracting algebraic fractions, and solving equations that contain fractions.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/algebraic-fractions/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The shared pizza bill
Three friends split a pizza bill. Later, two more friends arrive and the bill is split five ways instead. Josh writes the cost per person as $\dfrac{c}{3}$ before and $\dfrac{c}{5}$ after, and wants to know how much each of the first three saved.

That is $\dfrac{c}{3} - \dfrac{c}{5}$, a subtraction of **algebraic fractions**. The rules are exactly the ones you already use for number fractions like $\dfrac{1}{3} - \dfrac{1}{5}$. The letters just come along for the ride.

**In short:** An algebraic fraction is a fraction with letters in it, like $\dfrac{x + 2}{x^2 - 4}$. Factorise first and cancel common factors (never single terms), add and subtract over a common denominator, and remember that a value making a denominator zero is never allowed.

## Starting point: Years 9 and 10

### Simplifying

Cancel **common factors**, never single terms. Factorise first, then cancel.

$$\frac{6x^2}{9x} = \frac{2x}{3} \qquad \frac{4x + 8}{4} = \frac{4(x + 2)}{4} = x + 2$$

### Worked example: Factorise, then cancel
$$\frac{x^2 - 9}{x^2 + 5x + 6} = \frac{(x - 3)(x + 3)}{(x + 2)(x + 3)} = \frac{x - 3}{x + 2}$$

### Watch out: Cancel factors, not terms
$\dfrac{x + 6}{3}$ is **not** $x + 2$. The 3 is not a factor of the whole top. Only $\dfrac{3x + 6}{3} = x + 2$.

### Multiplying and dividing

- **Multiply:** factorise, cancel across, then multiply tops and bottoms.
- **Divide:** flip the second fraction and multiply.

$$\frac{2x}{5} \times \frac{10}{x^2} = \frac{20x}{5x^2} = \frac{4}{x}$$

$$\frac{x + 1}{6} \div \frac{x + 1}{3} = \frac{x + 1}{6} \times \frac{3}{x + 1} = \frac{1}{2}$$

### Adding and subtracting

Find a **common denominator**, rewrite each fraction, then add the tops.

$$\frac{c}{3} - \frac{c}{5} = \frac{5c}{15} - \frac{3c}{15} = \frac{2c}{15}$$

So each of Josh's first three friends saved $\dfrac{2c}{15}$.

### Worked example: Brackets on top
$$\frac{x + 2}{4} + \frac{x - 1}{3} = \frac{3(x + 2) + 4(x - 1)}{12} = \frac{7x + 2}{12}$$

### Watch out: Subtracting a whole top
In $\dfrac{x}{2} - \dfrac{x - 3}{5}$, the minus applies to all of $x - 3$. Use brackets: $\dfrac{5x - 2(x - 3)}{10} = \dfrac{3x + 6}{10}$.

### Solving equations with fractions

Multiply **every term** by the lowest common denominator to clear the fractions.

### Worked example: Clear the fractions
Solve $\dfrac{x}{2} + \dfrac{x}{3} = 10$. Multiply every term by 6:

$$3x + 2x = 60 \;\Rightarrow\; x = 12$$

### Worked example: Brackets matter
Solve $\dfrac{2x - 1}{3} - \dfrac{x + 2}{4} = 1$. Multiply by 12:

$$4(2x - 1) - 3(x + 2) = 12 \;\Rightarrow\; 5x - 10 = 12 \;\Rightarrow\; x = 4.4$$

### Key idea: Same rules as number fractions
Factorise first. Cancel only factors. Common denominators to add. Flip and multiply to divide. Clear fractions in equations by multiplying every term.

### Calculator tip: Use the fraction key, and test a number
The fraction key (`□/□`) keeps answers exact, such as $\frac{7}{12}$ instead of 0.5833. Press `S⇔D` to see the decimal.

To check a simplified algebraic fraction, store a test value such as $x = 3$ and work out both the original expression and your answer. If they agree, and also agree for a second value like $x = -2$, your simplification is almost certainly right.

**Mind map**

```text
Algebraic fractions
  Simplify
    Factorise first
    Cancel factors only
  Multiply
    Cancel across
  Divide
    Flip and multiply
  Add and subtract
    Common denominator
    Brackets on tops
  Equations
    Multiply every term by LCD
    Check the answer
```

### Year 11: Letters in the denominator
With algebraic denominators, the common denominator is usually their product.

$$\frac{2}{x} + \frac{3}{x + 1} = \frac{2(x + 1) + 3x}{x(x + 1)} = \frac{5x + 2}{x(x + 1)}$$

**Equations with $x$ on the bottom.** Solve $\dfrac{4}{x} = \dfrac{3}{x - 1}$. Cross-multiply: $4(x - 1) = 3x$, so $x = 4$.

Solve $x + \dfrac{6}{x} = 5$. Multiply by $x$: $x^2 - 5x + 6 = 0$, so $x = 2$ or $x = 3$.

Always check the answers do not make a denominator zero. Those values are not in the domain.

**Finding unknown numerators.** To find $a$ and $b$ in $\dfrac{a}{x - 1} + \dfrac{b}{x + 3} = \dfrac{4x}{(x - 1)(x + 3)}$, join the left side: $a(x + 3) + b(x - 1) = 4x$. Put in $x = 1$: $4a = 4$, so $a = 1$. Put in $x = -3$: $-4b = -12$, so $b = 3$.

**Splitting off a whole number.** Rewrite the top so the bottom appears in it:
$$\frac{x + 5}{x + 1} = \frac{(x + 1) + 4}{x + 1} = 1 + \frac{4}{x + 1}$$

**Inequalities with $x$ on the bottom.** Do not multiply by $x - 4$, because it might be negative. Multiply by its square, which is always positive. For $\dfrac{x - 2}{x - 4} \leq 3$ with $x \neq 4$: $(x - 2)(x - 4) \leq 3(x - 4)^2$, so $(x - 4)\big[(x - 2) - 3(x - 4)\big] \leq 0$, which is $(x - 4)(10 - 2x) \leq 0$. Sketch this parabola: it is at or below zero for $x < 4$ or $x \geq 5$.

### Year 12: Splitting fractions for calculus
Calculus often needs the reverse: split one fraction into simpler pieces.

$$\frac{x^2 + 3x}{x} = x + 3 \qquad \frac{2x + 5}{\sqrt{x}} = 2x^{\frac{1}{2}} + 5x^{-\frac{1}{2}}$$

Then each piece can be differentiated or integrated with the power rule. And a fraction whose top is the derivative of its bottom integrates to a log:

$$\int \frac{2x + 3}{x^2 + 3x}\,dx = \ln\lvert x^2 + 3x \rvert + C$$

**Limits.** If putting in the value gives $\frac{0}{0}$, factorise and cancel first:
$$\lim_{x \to 3} \frac{x^2 + x - 12}{x - 3} = \lim_{x \to 3} \frac{(x + 4)(x - 3)}{x - 3} = \lim_{x \to 3} (x + 4) = 7$$

## Quick summary

- Factorise first, then cancel common factors, never single terms.
- Multiply: cancel across, then multiply. Divide: flip the second fraction and multiply.
- Add and subtract with a common denominator. Put brackets around each top.
- Clear fractions in an equation by multiplying every term by the lowest common denominator.
- Values that make a denominator zero are never allowed.

## Quiz

Try these questions from the chapter quiz.

1. Simplify $\dfrac{12x^3}{4x}$.
   - A. $3x^3$
   - B. $3x^2$
   - C. $8x^2$
   - D. $\dfrac{3}{x^2}$
   - Answer: B. $3x^2$. $12 \div 4 = 3$ and $x^3 \div x = x^2$.

2. Simplify $\dfrac{5x - 10}{5}$.
   - A. $x - 10$
   - B. $x - 2$
   - C. $5x - 2$
   - D. $x$
   - Answer: B. $x - 2$. $\dfrac{5(x - 2)}{5}$.

3. Simplify $\dfrac{x^2 - 4}{x - 2}$.
   - A. $x - 2$
   - B. $x + 2$
   - C. $x^2 - 2$
   - D. $x$
   - Answer: B. $x + 2$. $\dfrac{(x - 2)(x + 2)}{x - 2}$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/algebraic-fractions/#quiz