Direct variation

Identify direct variation from tables of values, find the constant of variation k, write equations of the form y = kx, and apply direct variation to multi-step problems including unit conversions.

Worked examples

Identifying direct variation and stating k

Straightforward

Problem

A table of values shows: when

x = 2

y = 8

; when

x = 4

y = 16

; when

x = 6

y = 24

; when

x = 8

y = 32

. Determine whether

y

varies directly with

x

and, if so, state the constant of variation

k

Finding k and using y = kx to find a missing value

Moderate

Problem

y

varies directly with

x

. When

x = 5

y = 35

. Find the value of

y

when

x = 12

Multi-step direct variation problem with unit conversion

Challenging

Problem

The volume of water

V

(in litres) pumped varies directly with time

t

(in minutes). A pump moves 840 litres in 24 minutes. How many kilolitres does the pump move in 1.5 hours?

Practise

Q1·Straightforward

A table of values shows: when

x = 1

y = 5

; when

x = 2

y = 10

; when

x = 3

y = 15

; when

x = 4

y = 20

. Calculate the constant of variation

k

Q2·Straightforward

A table shows the data pairs

(x, y)

(3, 9)

(6, 18)

(9, 27)

. Does

y

vary directly with

x

? If so, state

k

No, the values do not show direct variationYes, $k = 3$Yes, $k = 6$Yes, $k = 9$

Q3·Straightforward

A quantity

y

varies directly with

x

. When

x = 7

y = 42

. State the constant of variation

k

Q4·Straightforward

Which statement correctly describes a direct variation relationship between

y

and

x

The ratio $\frac{y}{x}$ is constant for all pairs of valuesThe graph of $y$ against $x$ is a straight line that does not pass through the originAs $x$ increases, $y$ always decreasesThe value of $y$ equals the value of $x$ for every data point

Q5·Moderate

y

varies directly with

x

. When

x = 4

y = 28

. Find the value of

y

when

x = 11

Q6·Moderate

y

varies directly with

x

. When

x = 3

y = 21

. Find the value of

x

when

y = 63

Q7·Moderate

The cost

C

of printing leaflets varies directly with the number of leaflets

n

printed. It costs

\$36

to print 60 leaflets. Find the cost (in dollars) of printing 150 leaflets.

Q8·Moderate

A spring stretches by a length

L

(in cm) that varies directly with the mass

m

(in kg) attached to it. A mass of 5 kg stretches the spring by 8 cm. How far (in cm) will the spring stretch when a mass of 15 kg is attached?

Q9·Challenging

A car's fuel consumption varies directly with the distance driven. The car uses 9 litres of fuel for every 120 km. How many litres of fuel are needed for a 400 km journey?

Q10·Challenging

The mass of a steel rod varies directly with its length. A rod of length 250 cm has a mass of 15 kg. Find the mass, in grams, of a rod of length 80 cm.

Q11·Challenging

The volume of waste water

V

(in litres) produced by a factory varies directly with the time

t

(in hours) the machines run. In one 8-hour shift, the factory produces 2400 litres of waste water. The machines run for two 8-hour shifts per day, five days per week. Find the total volume of waste water produced in one week, in kilolitres (kL).

Q12·Challenging

A car's fuel consumption varies directly with the distance driven. On one trip, the car travels 350 km and uses 28 litres of fuel. On another trip, the car uses 42 litres. How far (in km) did the car travel on the second trip?