Substitution and formulas

Substitute values into formulas to find unknown quantities. Rearrange formulas and apply them to multi-step problems including area, perimeter, speed, and Pythagoras' theorem.

Worked examples

Substituting into a formula

Straightforward

Problem

The area of a rectangle is given by

A = lw

. Find

A

when

l = 9

and

w = 5

Rearranging a formula to find an unknown

Moderate

Problem

The formula

F = \dfrac{9}{5}C + 32

converts Celsius to Fahrenheit. Find the Celsius temperature

C

when

F = 68

Multi-step formula — Pythagoras' theorem

Challenging

Problem

A right-angled triangle has a hypotenuse of

13

cm and one leg of

5

cm. Find the length of the other leg using Pythagoras' theorem:

c^2 = a^2 + b^2

Practise

Q1·Straightforward

The area of a rectangle is given by

A = lw

.

Find

A

when

l = 7

and

w = 4

Q2·Straightforward

The perimeter of a rectangle is given by

P = 2(l + w)

.

Find

P

when

l = 6

and

w = 3

Q3·Straightforward

Given the formula

y = 3x + 2

, find the value of

y

when

x = 5

Q4·Straightforward

The formula

v = u + at

gives the final velocity of an object.

Find

v

when

u = 2

a = 3

, and

t = 4

Q5·Moderate

The circumference of a circle is given by

C = 2\pi r

.

Find the radius

r

when

C = 10\pi

Q6·Moderate

The formula to convert Celsius to Fahrenheit is

F = \dfrac{9}{5}C + 32

.

Find

F

when

C = 25

Q7·Moderate

The perimeter of a rectangle is given by

P = 2(l + w)

.

Find the width

w

when

P = 24

and

l = 8

Q8·Moderate

The formula

v = u + at

gives the final velocity of an object.

Find the acceleration

a

when

v = 20

u = 5

, and

t = 3

Q9·Challenging

Pythagoras' theorem states that

c^2 = a^2 + b^2

, where

c

is the hypotenuse.

Find the hypotenuse

c

of a right-angled triangle with legs

a = 6

and

b = 8

Q10·Challenging

The distance formula is

d = st

, where

d

is distance,

s

is speed, and

t

is time.

A car travels

150

km at a speed of

60

km/h. How many hours does the trip take?

Give your answer as a decimal.

Q11·Challenging

The area of a triangle is given by

A = \dfrac{1}{2}bh

.

Find the height

h

when

A = 30

and

b = 12

Q12·Challenging

The formula

v^2 = u^2 + 2as

relates velocity, acceleration, and distance.

Find the distance

s

when

v = 10

u = 0

, and

a = 2