Direct proportion

Identify the constant of proportionality from tables and equations. Write and apply equations of the form

y = kx

to solve direct proportion problems.

Worked examples

Identify the constant of proportionality from a table

Straightforward

Problem

The table shows values of

x

and

y

\begin{array}{c|ccc} x & 2 & 3 & 5 \\ \hline y & 14 & 21 & 35 \end{array}

Find the constant of proportionality

k

and write the equation relating

y

and

x

Find missing values using $y = kx$

Moderate

Problem

y

is directly proportional to

x

. When

x = 5

y = 30

. (a) Find

k

. (b) Find

y

when

x = 9

Solve a direct proportion word problem

Challenging

Problem

The amount of paint

L

litres needed to cover a wall is directly proportional to the area

A

^2

being painted. 12 litres covers 48 m

^2

. How many litres are needed to cover 80 m

^2

Practise

Q1·Straightforward

The equation

y = 6x

shows that

y

is directly proportional to

x

. What is the constant of proportionality

k

Q2·Straightforward

The table shows values of

x

and

y

\begin{array}{c|ccc} x & 1 & 2 & 3 \\ \hline y & 3 & 6 & 9 \end{array}

What is the constant of proportionality

k

Q3·Straightforward

The equation

y = 0.5x

shows direct proportion. What is the constant of proportionality

k

Q4·Straightforward

A table shows

x = 4, y = 28

and

x = 6, y = 42

. What is the constant of proportionality

k

Q5·Moderate

y

is directly proportional to

x

. When

x = 4

y = 20

. Find the value of

y

when

x = 7

Q6·Moderate

y

is directly proportional to

x

. When

x = 6

y = 18

. Find the value of

x

when

y = 30

Q7·Moderate

A recipe requires flour directly proportional to the number of people served. 300 g of flour serves 4 people. How many grams of flour are needed to serve 10 people?

Q8·Moderate

y = kx

. When

x = 3

y = 24

. Find

y

when

x = 8

Q9·Challenging

The cost

C

dollars to hire a piece of equipment is directly proportional to the number of days

n

. Hiring for 5 days costs

\$35

. What is the cost of hiring for 12 days?

Q10·Challenging

The distance

d

km travelled at a constant speed is directly proportional to the time

t

hours. After 3 hours,

d = 195

km. How far is travelled in 5 hours?

Q11·Challenging

The mass

m

grams of a wire is directly proportional to its length

l

cm. A wire 8 cm long has a mass of 52 g. What length of wire, in cm, has a mass of 130 g?

Q12·Challenging

A worker's wages

W

dollars are directly proportional to the number of hours

h

worked. Working 6 hours earns

\$54

. How many hours must be worked to earn

\$126