Learn4024Paper 2 · CalculatorAlgebraAlgebraic fractions and formulae

Algebra · Subtopic 2 of 8

Algebraic Fractions and Formulae

Algebraic fractions follow exactly the same rules as number fractions: factorise, cancel whole brackets, and use a common denominator to add. Changing the subject of a formula means undoing operations in reverse order, doing the same to both sides.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 35 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Substitute values into formulae.
Change the subject of a formula, including when the subject appears twice.
Simplify algebraic fractions by factorising.
Add and subtract algebraic fractions.

Pause and think

$\dfrac{6}{9}=\dfrac23$ because $3$ cancels. Does $\dfrac{x+6}{x+9}$ simplify to $\dfrac{x+2}{x+3}$ ?

Section 2 of 14

Core idea

Start hereWorking backwards

A circle's area is $A=\pi r^2$ . But what if you know the area of a pond and need its radius? Rearranging the formula, making $r$ the subject, turns one formula into a tool for a different question.

The big idea

Do the same operation to both sides. Undo in reverse order. If the subject appears twice: collect, factorise, divide.

1. The balance

Rearranging is like a balance: do the same to both sides

Whatever you do to one side, do to the other, and it stays balanced.

Worked example $r$ from $A=\pi r^2$

Make $r$ the subject.

Goal: $r=\ldots$
Strategy: Undo "square then ×π": ÷π then √.

$r^2=\frac A\pi$
Divide by $\pi$ .
$r=\sqrt{\frac A\pi}$
Square root (radius positive).

Answer

r=\sqrt{A/\pi}

✓ Check

r=2

A=4\pi

;

\sqrt{4\pi/\pi}=2

✓.

Worked example $a$ from $v^2=u^2+2as$

Make $a$ the subject.

Goal: $a=\ldots$
Strategy: Remove $u^2$ , then divide by $2s$ .

$v^2-u^2=2as$
Subtract $u^2$ .
$a=\frac{v^2-u^2}{2s}$
Divide.

Answer

a=\frac{v^2-u^2}{2s}

✓ Check

u=0

v=10

s=25

a=2

;

0+2(2)(25)=100

✓.

2. Subject appearing twice

Worked example $x$ from $y=\frac{x+2}{x-3}$

Make $x$ the subject.

Goal: $x=\ldots$
Strategy: Clear the fraction, collect $x$ terms, factorise.

$y(x-3)=x+2$
Multiply by $x-3$ .
$xy-3y=x+2$
Expand.
$xy-x=3y+2$
$x$ terms left, others right.
$x(y-1)=3y+2\Rightarrow x=\frac{3y+2}{y-1}$
Factorise, divide.

Answer

x=\frac{3y+2}{y-1}

✓ Check

x=4

y=6

;

\frac{20}5=4

✓.

Common misconceptionTaking roots term by term

\sqrt{v^2-u^2}\neq v-u

Pause and think

Make $h$ the subject of $V=\frac13\pi r^2h$ .

Summary

Undo in reverse order.
Twice: collect, factorise, divide.

Subject: The letter on its own.

HistoryViète's letters. Until about

1590

formulas were written in words. François Viète, a French lawyer and code-breaker for King Henri IV, was the first to use letters for known quantities as well as unknowns, so a single formula could describe every case at once.

Fun factYour phone rearranges formulas. A sat-nav knows

\text{time}=\dfrac{\text{distance}}{\text{speed}}

. When you ask "how fast must I drive to arrive by

9

?", it changes the subject to speed. Every "what do I need?" question is a rearranged formula.

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