Rearranging Formulas

Make a different variable the subject of a formula, then use the rearranged formula to solve practical problems.

45 min Algebra Formulas and equations Lesson 4 of 13

f

Printable worksheet

Open this lesson's worksheet

Use the printable version for rearranging formulas, substituting after rearranging and checking with numbers.

Open printable version

Think First

The distance formula is $d = st$. If you know distance and time, how could you find speed?

Type your first rearrangement idea.

Write your first rearrangement idea in your book.

Write your idea in your book

Saved

Know

The subject of a formula is the variable written by itself.
Rearranging uses inverse operations.
A rearranged formula can be checked with a numerical example.

Understand

Changing the subject does not change the relationship.
Rearrange before substituting when the required variable is not the subject.
Each step must keep both sides equal.

Can Do

Rearrange $d = st$ for $s$ and $t$.
Rearrange $C = 2\pi r$ for $r$.
Use a rearranged formula in context.

r

Rearranged Forms

$d = st$

$s = \frac{d}{t}$ and $t = \frac{d}{s}$

$C = 2\pi r$

$r = \frac{C}{2\pi}$

$A = bh$

$b = \frac{A}{h}$ and $h = \frac{A}{b}$

Core Content

1. The Subject Is the Variable by Itself

In $d = st$, distance is the subject because $d$ is by itself.

If a question asks for speed, it is often clearer to rearrange the formula to make $s$ the subject before substituting values.

Key idea: Rearranging is solving a formula for a variable rather than solving for a number.

Worked Example 1

Rearrange distance equals speed times time

Rearrange $d = st$ to make $s$ the subject.

$d = st$

Divide both sides by $t$:

$\frac{d}{t} = \frac{st}{t}$

$s = \frac{d}{t}$

Use it: If $d = 180$ km and $t = 3$ h, then $s = \frac{180}{3} = 60$ km/h.

Worked Example 2

Rearrange a circumference formula

The circumference of a circle is $C = 2\pi r$. Rearrange the formula to make $r$ the subject.

$C = 2\pi r$

Divide both sides by $2\pi$:

$\frac{C}{2\pi} = r$

$r = \frac{C}{2\pi}$

Common error: Divide by the whole multiplier $2\pi$, not just by 2.

Worked Example 3

Rearrange and substitute

A rectangle has area $A = bh$. Its area is $84$ cm2 and its height is $7$ cm. Find the base length.

Make $b$ the subject: $b = \frac{A}{h}$

Substitute $A = 84$ and $h = 7$:

$b = \frac{84}{7} = 12$

Answer: The base length is 12 cm.

2. Check a Rearranged Formula With Numbers

To check $s = \frac{d}{t}$ from $d = st$, choose simple values. If $s = 5$ and $t = 4$, then $d = 20$. The rearranged formula gives $s = \frac{20}{4} = 5$, so it matches.

Communication habit: A rearranged formula should still give the same relationship when tested with the same values.

Activity

Rearrange, Substitute, Check

Rearrange $d = st$ to make $t$ the subject.
Use your formula to find $t$ when $d = 240$ km and $s = 80$ km/h.
Rearrange $A = bh$ to make $h$ the subject.
Use your formula to find $h$ when $A = 96$ m2 and $b = 12$ m.

Complete the rearranging practice in your book.

Revisit Distance, Speed and Time

From $d = st$, speed is $s = \frac{d}{t}$ and time is $t = \frac{d}{s}$. The variable you need determines which version is most useful.

Explain when to use each formula in your book.

Assessment

MC

Multiple Choice

Random questions from the lesson bank - feedback appears immediately.

SA

Short Answer

Rearrange first, then substitute and interpret.

1. Rearrange $d = st$ to make $t$ the subject, then find $t$ when $d = 150$ km and $s = 50$ km/h. 3 MARKS

Answer in your book.

2. Rearrange $C = 2\pi r$ to make $r$ the subject. Use it to find $r$ when $C = 31.4$ cm and $\pi \approx 3.14$. 4 MARKS

Answer in your book.

3. Explain why substituting before identifying the required subject can make a formula question harder. 2 MARKS

Answer in your book.

Subject Switch

Name the subject, choose the inverse operation, then check the rearranged formula with easy numbers.

Back to module