Constant of Proportionality Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Constant of Proportionality.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The constant ratio $k$ between two proportional quantities: if $y = kx$ , then $k$ is the constant of proportionality.

If $y$ is always 3 times $x$ , the constant of proportionality is 3.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: It is the one number $k$ in $y=kx$ that turns any $x$ into its matching $y$ .

Common stuck point: The procedure for constant of proportionality is the easy part; the trap is computing $k$ as a difference $y-x$ . Asking "Does $\frac{y}{x}$ give the same number for every pair in the data?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does $\frac{y}{x}$ give the same number for every pair in the data?

Worked Examples

Example 1

easy

A car travels 60 miles per hour. Write the equation relating distance

d

and time

t

. What is the constant of proportionality?

Answer

d = 60t

;

k = 60

First step

The relationship is

d = k \cdot t

where

k

is the constant of proportionality.

Full solution

2
Here, speed = 60 mph, so $k = 60$ .
3
Equation: $d = 60t$ .
4
In 3 hours: $d = 60 \times 3 = 180$ miles.

y = kx

k

is the constant of proportionality — the unit rate. Here

k = 60

miles per hour.

Example 2

medium

The table shows

x

and

y

: (2, 10), (4, 20), (6, 30). Is this proportional? Find

k

Example 3

medium

Spring stretches proportionally to force.

4

N stretches it

12

cm. Find

k

(cm/N) and stretch from

9

Example 4

hard

A map uses a scale:

2

cm represents

5

km. Find

k

(km per cm) and the real distance for

7.5

cm on the map.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Apples cost $0.75 each. Write the equation for cost

C

given quantity

q

. Find the cost of 8 apples.

Example 2

medium

y = kx

and

y = 35

when

x = 7

, find

k

and predict

y

when

x = 12

Example 3

easy

y = 5x

, what is the constant of proportionality?

Example 4

easy

A table shows

x=2, y=6

for a proportional relation

y=kx

. Find

k

Example 5

easy

y = kx

and

k = \frac{3}{4}

, find

y

when

x = 8

Example 6

easy

Apples cost $2 each. Write

k

for total cost

c = kn

where

n

is the number of apples.

Example 7

easy

For

y = kx

with

x=4, y=10

, find

k

Example 8

easy

A car travels

120

miles in

2

hours at constant speed. What is

k

(speed) in

d = kt

Example 9

easy

y = kx

, if doubling

x

doubles

y

, is

k

constant?

Example 10

easy

Find

k

for

y = kx

given

x = 7, y = 21

Example 11

medium

A table has

(x,y)

(2,8),(3,12),(5,20)

. Is it proportional, and if so find

k

Example 12

medium

A table has

(x,y)

(1,3),(2,6),(3,10)

. Find

k

from the first row and test if proportional.

Example 13

medium

y = kx

and

y = 18

when

x = 6

, find

y

when

x = 10

Example 14

medium

A recipe uses

k

cups of flour per cookie.

24

cookies use

6

cups. Find

k

and the flour for

40

cookies.

Example 15

medium

Graph of

y=kx

passes through

(4, 10)

. Find

k

and the

y

x=6

Example 16

medium

Two quantities satisfy

y = kx

. When

x=3

y=7.5

. Is

k

a whole number? Find it.

Example 17

medium

A spring stretches proportionally:

2

N stretches it

5

cm. Find

k

(cm per N) and the stretch for

7

Example 18

challenge

A table is proportional with constant

k

. Rows:

(x,y) = (a, 12)

and

(6, 18)

. Find

k

and

a

Example 19

challenge

Quantities satisfy

y = kx

. If increasing

x

4

increases

y

10

, find

k

Example 20

challenge

Prove that if

y = kx

then the ratio

y/x

is the same for every

x \neq 0

Example 21

medium

A printer prints proportionally:

90

pages in

3

minutes. Find

k

(pages/min) and pages in

7

minutes.

Example 22

medium

Currency converts proportionally:

\$50

buys

40

euros. Find

k

(euros per dollar) and euros for

\$80

Example 23

easy

y = kx

with

x=3

and

y=15

. Find

k

Example 24

easy

A bag of

5

oranges costs $3. Find

k

in cost

= k \cdot

(number of oranges).

Example 25

easy

A bike travels

36

miles in

3

hours at constant speed. Find

k

d=kt

Example 26

easy

y = kx

with

x=10

and

y=4

. Find

k

Example 27

medium

A table:

(x,y) = (2,9),(4,18),(6,27)

. Is it proportional? Find

k

Example 28

medium

A table:

(x,y) = (1,5),(2,9),(3,15)

. Proportional? If so

k

, else explain.

Example 29

medium

y

is proportional to

x

y=21

when

x=3

. Find

y

when

x=8

Example 30

medium

A recipe scales proportionally:

3

cups flour make

24

cookies. How many cookies from

5

cups?

Example 31

medium

A car uses

6

gallons to drive

180

miles. Find

k

(miles per gallon) and miles from

10

gallons.

Example 32

medium

Graph of

y=kx

passes through

(5, 12)

. Find

y

when

x = 15

Example 33

medium

Currency: $80 buys

70

euros. Find

k

(euros per dollar) and euros from $200.

Example 34

medium

k

pages/minute, a printer made

84

pages in

4

minutes. Find

k

Example 35

medium

y = kx

has

k = 0.25

. Find

y

when

x=20

and

x

when

y=11

Example 36

hard

A line through origin passes through

(a, 12)

with

k = 4

. Find

a

Example 37

hard

Two quantities are proportional. Increasing

x

6

increases

y

15

. Find

k

Example 38

hard

y=kx

and another point

(8, 20)

is on the graph. A new point

(x, y)

satisfies

y = 17.5

. Find

x

Example 39

hard

y = kx

. The graph passes through

(2, 5)

and another point on the graph is

(p, p+3)

. Find

p

Example 40

hard

Three quantities:

y \propto x

with

k_1

, and

z \propto y

with

k_2

. If

k_1 = 3

and

k_2 = 4

, what is the constant relating

z

and

x

Example 41

challenge

For

y=kx

, prove that doubling

x

doubles

y

Example 42

challenge

A relation has

(1, 4), (2, 8), (3, 13)

. Decide proportionality; if not, what's the smallest change to the last

y

to make it proportional?

Example 43

challenge

y = kx + b

describes a line. For what

b

is the relation a proportional relationship?

← Back to Constant of Proportionality Formula Explained Practice Mode

Related Concepts

Proportional Relationships Ratios Linear Functions Slope

Background Knowledge

These ideas may be useful before you work through the harder examples.

proportionalityratios