What is Division of Algebraic Expressions?

When we divide numbers, we split a quantity into equal parts. In algebra, we do exactly the same thing — but with expressions that contain variables.

You already know that a fraction is just another way of writing a division:

$\frac{a}{b} = a \div b$

This means everything you know about fractions applies directly to algebraic division. The expression on top is the numerator, and the expression on the bottom is the denominator.

Division of Algebraic Expressions

The process of splitting an algebraic expression (the numerator) by another expression or number (the denominator). A fraction bar is the most common way to write this: $\frac{numerator}{denominator}$ .

Here are some ways to read the expression $\frac{x}{3}$ :

" $x$ divided by 3"
" $x$ over 3"
"the quotient of $x$ and 3"
" $x$ on 3"

Note

The division symbol $\div$ actually looks like a fraction — a dot above a bar and a dot below. This is not a coincidence! It is a reminder that division and fractions are the same operation.

Writing Expressions as Fractions

Any division can be rewritten as a fraction, and any fraction can be rewritten as a division. This is a powerful tool in algebra because fractions are often easier to simplify.

Example

Write each division as a fraction:

$5 \div 6 = \frac{5}{6}$
$- 4 \div 7 = \frac{- 4}{7}$
$x \div 3 = \frac{x}{3}$

Write each fraction as a division:

$\frac{27}{9} = 27 \div 9 = 3$
$\frac{- 25}{5} = - 25 \div 5 = - 5$

Exam Tip

Practise rewriting fractions as divisions and vice versa until it feels automatic. This skill underpins almost every simplification step you will do in algebra.

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