Algebraic fractions

What an algebraic fraction is Extended

An algebraic fraction is a fraction with letters in the numerator, the denominator or both, such as $\frac{x}{3}$, $\frac{2}{x+1}$ or $\frac{x^2-2x}{x^2-5x+6}$.

Every letter stands for a number, so an algebraic fraction obeys exactly the same rules as a number fraction. Whenever you are unsure what to do, ask what you would do with $\frac{3}{4}$ and $\frac{5}{6}$, and do the same thing.

One extra rule: the denominator can never be zero. $\frac{2}{x+1}$ has no value when $x=-1$. You do not usually have to state this in an answer, but it explains why cancelling a factor such as $(x-2)$ is only valid when $x\ne 2$.

Simplifying by cancelling common factors Extended

You simplify $\frac{12}{18}$ by dividing the top and the bottom by a common factor, 6, to get $\frac{2}{3}$. With algebra it is the same: divide the numerator and the denominator by a common factor.

For a single term on top and bottom, cancel numbers and letters separately:

$$\frac{12x^{3}y}{18x{y}^2}=\frac{2x^2}{3y}$$

(12 and 18 share 6; $x^3÷x=x^2$; one $y$ cancels, leaving $y$ on the bottom.)

When the top or the bottom has more than one term, factorise first. You may cancel only a factor that multiplies the whole of the numerator and the whole of the denominator, never a single term inside a sum.

This is the most common mistake in the topic. $\frac{x+6}{x+3}$ does not simplify to $2$, and it does not simplify to $\frac{6}{3}$. Test it with $x=2$: $\frac{8}{5}$ is not 2. The $x$ in $x+6$ is a term, not a factor, so it cannot be cancelled.

The method:

1. Factorise the numerator fully.
2. Factorise the denominator fully.
3. Cancel any bracket (or number or letter) that is a factor of both.
4. Leave the answer in factorised form unless asked otherwise.

Multiplying and dividing Extended

Multiplying: multiply the numerators together and the denominators together, then simplify. It is easier to cancel common factors before multiplying.

$$\frac{3a}{4}\times \frac{9a}{10}=\frac{27a^2}{40}$$

Nothing cancels here (3 and 9 are both on top; 4 and 10 are both on the bottom), so $\frac{27a^2}{40}$ is the answer.

Dividing: dividing by a fraction is the same as multiplying by its reciprocal, the fraction turned upside down. Keep the first fraction, change $÷$ to $\times$, flip the second.

$$\frac{3a}{4}÷\frac{9a}{10}=\frac{3a}{4}\times \frac{10}{9a}=\frac{30a}{36a}=\frac{5}{6}$$

The $a$ cancels completely, so the answer is a number. That is fine: it means the result is $\frac{5}{6}$ whatever the value of $a$ (as long as $a\ne 0$).

With brackets, factorise every numerator and denominator first, then cancel across the whole product: any factor on top can cancel with the same factor anywhere on the bottom.

Adding and subtracting Extended

You can only add fractions that have the same denominator. For $\frac{1}{3}+\frac{1}{2}$ you write both as sixths: $\frac{2}{6}+\frac{3}{6}=\frac{5}{6}$. Algebraic fractions work the same way:

1. Find a common denominator. With number denominators use their lowest common multiple; with algebraic denominators that share no factor, multiply them.
2. Rewrite each fraction over the common denominator, multiplying its numerator by whatever its denominator was multiplied by.
3. Add or subtract the numerators. Put every numerator in a bracket first, so that a minus sign in front of a fraction applies to all of its numerator.
4. Expand, collect like terms, and see whether the result factorises and cancels.

$$\frac{x}{3}+\frac{x-4}{2}=\frac{2x}{6}+\frac{3(x-4)}{6}=\frac{2x+3x-12}{6}=\frac{5x-12}{6}$$

Subtraction needs the bracket:

$$\frac{2x}{3}-\frac{3(x-5)}{2}=\frac{4x-9(x-5)}{6}=\frac{4x-9x+45}{6}=\frac{45-5x}{6}$$

The $-9\times -5=+45$ is exactly where marks are lost. Finally, never add the denominators: $\frac{x}{3}+\frac{x}{2}$ is not $\frac{2x}{5}$.