Lesson 2.2 · 25 min

Algebraic Fractions

Adding and subtracting fractions with letters in them, and simplifying by factorising and cancelling.

In this lesson you will learn to…

Add and subtract algebraic fractions using a common denominator
Write a whole-number or letter term as a fraction to combine it with another
Simplify an algebraic fraction by factorising the numerator and cancelling
Avoid the most common cancelling mistake

Why this matters

Formulae in science are full of fractions with letters in them — speed is $\frac{d}{t}$ , density is $\frac{m}{V}$ . As soon as you combine two of them, for example the total time for a journey done in two parts, you need to add algebraic fractions.

It is the same as number fractions

Everything you already know about fractions still applies. Compare:

Numbers	Letters
$\frac{2}{7} + \frac{3}{7} = \frac{5}{7}$	$\frac{w}{6} + \frac{p}{6} = \frac{w + p}{6}$
$\frac{1}{3} + \frac{2}{7} = \frac{7}{21} + \frac{6}{21} = \frac{13}{21}$	$\frac{a}{5} + \frac{a}{3} = \frac{3a}{15} + \frac{5a}{15} = \frac{8a}{15}$

If you are ever unsure about the algebra, try the same question with numbers first.

A real problem

Avril cycles 10 km at $x$ km/h, then 14 km at $y$ km/h. Time is distance ÷ speed, so the total time is

$\frac{10}{x} + \frac{14}{y} = \frac{10y}{xy} + \frac{14x}{xy} = \frac{10y + 14x}{xy} \text{ hours.}$

The cancelling trap

When you simplify $\frac{10x + 5}{5}$ , think of it as $\frac{1}{5}(10x + 5)$ . Every term in the bracket is divided by 5:

$\frac{10x + 5}{5} = 2x + 1$

A very common wrong answer is to cross out the two 5s and write $10x$ . That is only allowed when the 5 multiplies the whole numerator — which is exactly why you factorise first.

Check it with a number. Put $x = 1$ : $\frac{10 + 5}{5} = 3$ . Then $2x + 1 = 3$ ✓ but $10x = 10$ ✗.

Negative signs and subtraction

When you subtract a fraction with a bracket on top, the minus sign applies to everything in that bracket:

$\frac{x - 2}{5} - \frac{2x - 1}{7} = \frac{7(x - 2) - 5(2x - 1)}{35} = \frac{7x - 14 - 10x + 5}{35} = \frac{-3x - 9}{35}$

Notice that $-5 \times -1 = +5$ .

Key rules and formulas

Same denominator — add the tops
$\frac{w}{6} + \frac{p}{6} = \frac{w + p}{6}$
Exactly as with number fractions. Subtracting works the same way: $\frac{3}{t} - \frac{m}{t} = \frac{3 - m}{t}$ .
Different denominators — make them the same first
$\frac{2}{b} + \frac{b}{7} = \frac{14}{7 b} + \frac{b ^{2}}{7 b} = \frac{14 + b ^{2}}{7 b}$
Find a common denominator (often the two denominators multiplied), then use equivalent fractions.
A whole term is a fraction too
$x = \frac{5 x}{5}$ and $3 = \frac{3 x}{x}$ . Write it with the denominator you need, then add.
Cancel factors, never terms
$\frac{35 x + 10}{15} = \frac{5 ( 7 x + 2 )}{15} = \frac{7 x + 2}{3}$
Factorise the top first. You may only cancel something that multiplies the whole numerator. $\frac{10 x + 5}{5}$ is not $10 x$ .

Worked examples

Worked example 1

Adding with letter denominators

Simplify a

\frac{2}{b} + \frac{b}{7}

\frac{x}{5} + \frac{4}{y}

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Adding two fractions with brackets

Simplify

\frac{x + 3}{4} + \frac{x - 1}{5}

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Simplifying by factorising

Simplify

\frac{24 x ^{2} + 20 x}{64 x}

Try it yourself first, then reveal the solution one step at a time.

Practice: algebraic fractions

11 questions

Type fractions with brackets round the top and bottom, e.g. (14 + b^2)/(7b). Any equivalent answer is marked correct.

Question 1 of 11Type the answerFoundation
Warm-up with numbers: work out $\frac{5}{7} - \frac{2}{3} + \frac{1}{4}$ .
Your answer
Question 2 of 11Algebraic answer
Simplify $\frac{a}{5} + \frac{a}{3}$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.
Question 3 of 11Algebraic answer
Simplify $\frac{2 x}{3} - \frac{y}{6}$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.
Question 4 of 11Algebraic answer
Simplify $x + \frac{2 x}{5}$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.
Question 5 of 11Step by step
Add these fractions one step at a time.
Simplify $\frac{x + 3}{4} + \frac{x - 1}{5}$ .
1. What is the lowest common denominator?
  Step 1 answer
Question 6 of 11Algebraic answerChallenge
Avril cycles 10 km at $x$ km/h and then 14 km at $y$ km/h. Write the total time for the journey, in hours, as a single fraction.
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.
Question 7 of 11Multiple choice
Which is the correct simplification of $\frac{10 x + 5}{5}$ ?
Question 8 of 11Choose all that apply
Select every expression that is equivalent to $2 x + 3$ .
Select every correct answer — there may be more than one.
$\frac{2 x + 9}{3}$ $\frac{1}{4} (8 x + 12)$ $\frac{1}{2 x + 3}$ $\frac{4 x + 12}{2}$ $\frac{x + 1.5}{0.5}$ $\frac{2 x + 15 + 8 x}{5}$
Question 9 of 11Fill the gaps
Complete the working to simplify $\frac{30 x + 15}{10}$ .
$\frac{30 x + 15}{10} = \frac{□ ( 6 x + □ )}{10}$ — the boxes are Gap a and Gap b — $= \frac{6 x + 3}{□}$ where the box is Gap c.
Question 10 of 11Algebraic answerChallenge
Simplify $\frac{24 x ^{2} + 20 x}{64 x}$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.
Question 11 of 11True or false
Franco says $\frac{4 ( 25 p + 10 )}{80}$ simplifies to $\frac{25 p + 10}{20}$ and he is finished.

In this lesson you will learn to…

Why this matters

It is the same as number fractions

A real problem

The cancelling trap

Negative signs and subtraction

Key rules and formulas

Same denominator — add the tops

Different denominators — make them the same first

A whole term is a fraction too

Cancel factors, never terms

Worked examples

Practice: algebraic fractions