Constant Rate Examples in Math

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

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

Concept Recap

A constant rate of change means the output increases (or decreases) by the same fixed amount for every unit increase in the input — the hallmark of a linear function.

Constant rate means steady, uniform progress — like a car traveling at a fixed speed: every hour, the same number of miles is added to the total.

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: A constant rate adds the same fixed amount to the output every time the input goes up by one unit.

Common stuck point: The procedure for constant rate is the easy part; the trap is checking only one pair of points and declaring it constant. Asking "Does every equal step in the input add the exact same amount to the output?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does every equal step in the input add the exact same amount to the output?

Worked Examples

Example 1

easy

A car travels at a constant speed of

60

km/h. Write the distance function

d(t)

, find

d(2.5)

, and compute the rate of change from

t=1

t=3

Answer

d(t)=60t

;

d(2.5)=150

km; rate

=60

km/h

First step

Constant rate

60

km/h means distance

=

rate

\times

time:

d(t) = 60t

Full solution

2
Evaluate: $d(2.5) = 60 \times 2.5 = 150$ km.
3
Average rate of change from $t=1$ to $t=3$ : $\frac{d(3)-d(1)}{3-1} = \frac{180-60}{2} = \frac{120}{2} = 60$ km/h. (Constant rate means average rate equals instantaneous rate.)

A constant rate of change produces a linear function. For any linear function

f(x)=mx+b

, the average rate of change between any two points always equals the slope

m

Example 2

medium

Find the equation of the line passing through

(2, 5)

and

(6, 13)

, interpret the slope, and write it in slope-intercept form

y = mx + b

Example 3

medium

A gym charges a $30 sign-up plus $25/month. Write

C(m)

and find when

C(m) = 180

Example 4

medium

A taxi charges $3 base plus $2 per mile. Write

F(m)

and find

F(7)

Example 5

medium

A line of constant rate

3

passes through

(2, 11)

. Find

y

when

x = 9

Example 6

hard

A line through

(1, 4)

and

(5, k)

has constant rate

3

. Find

k

Example 7

challenge

A cyclist rides at constant

12

mph. A runner

5

miles ahead moves the same direction at constant

4

mph. When does the cyclist catch the runner?

Practice Problems

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

Example 1

easy

A plumber charges a

\$50

flat fee plus

\$75

per hour. Write

C(h)

and find the cost for

3

hours. At what hour does the cost reach

\$275

Example 2

medium

Determine whether the data is consistent with a constant rate of change:

x: 0, 2, 4, 6

and

y: 3, 7, 12, 15

. If not, find where it fails.

Example 3

easy

A car adds

60

miles every hour. What is its rate of change?

Example 4

easy

Table

(0,2),(1,5),(2,8),(3,11)

. What is the constant rate?

Example 5

easy

For

y=4x+1

, what is the rate of change?

Example 6

easy

Does a constant rate of

3

mean the output is always

3

Example 7

easy

A line has constant rate

5

. What shape is its graph?

Example 8

easy

If the rate is

-2

per unit, is the output increasing or decreasing?

Example 9

easy

Over

5

hours at a constant

10

mi/hr, how far does a car travel?

Example 10

easy

Is the rate of

y=7

(a horizontal line) constant?

Example 11

medium

A table shows

(0,4),(2,10),(4,16)

. Find the constant rate per unit input.

Example 12

medium

A phone plan costs

20

plus

5

per GB. Write the formula and identify the constant rate.

Example 13

medium

Verify whether

(1,3),(2,5),(3,8)

has a constant rate.

Example 14

medium

A tank drains at a constant rate, going from

100

L to

40

L in

3

minutes. Find the rate.

Example 15

medium

A line passes through

(2,9)

and

(6,21)

. Find its constant rate.

Example 16

medium

f

has constant rate

4

and

f(0)=2

, find

f(10)

Example 17

medium

Does data that looks straight over

[0,1]

guarantee a constant rate over

[0,10]

Example 18

medium

A worker packs

15

boxes per hour at a constant rate. How long to pack

90

boxes?

Example 19

challenge

A line has constant rate

m

, passes through

(1,7)

, and

f(4)=19

. Find

m

and

f(0)

Example 20

challenge

Two runners start together. A runs at constant

6

mph, B at constant

8

mph. After how long is B exactly

1

mile ahead?

Example 21

challenge

A candle is

24

cm tall and burns at a constant rate, reaching

18

cm after

2

hours. When does it fully burn out?

Example 22

medium

A line drops from

(0,20)

(4,8)

. Find its constant rate.

Example 23

easy

A printer prints

12

pages per minute at a constant rate. How many pages in

7

minutes?

Example 24

easy

A tap fills a bucket at

2

L per minute. How long to fill

15

Example 25

easy

A train travels

180

miles in

3

hours at constant speed. What is its rate?

Example 26

easy

f(x) = 2x - 1

, find the rate of change between

x=3

and

x=8

Example 27

medium

A line passes through

(-2, 5)

and

(4, -7)

. Find the constant rate of change.

Example 28

medium

A pool loses water at a constant

4

L/min. After how many minutes is

100

L lost?

Example 29

medium

Check whether

y = x^2

has a constant rate of change on

[0, 4]

Example 30

medium

A line has constant rate

-\tfrac{1}{2}

and passes through

(0, 6)

. Find

f(10)

Example 31

medium

A factory produces widgets at constant

50

/hour. If they need

1{,}200

widgets, how many hours of work are needed?

Example 32

medium

A 12-cup coffee maker brews at a constant rate, finishing in

6

minutes. Find the rate in cups/min.

Example 33

hard

A plant grows at a constant rate. After

4

days it is

10

cm and after

9

days it is

22.5

cm. Find the daily growth rate and the height at day

0

Example 34

hard

A car travels at

40

mi/h for

2

hours and then at

60

mi/h for

3

hours. Is the trip a constant-rate motion? What is the average rate?

Example 35

hard

A printer prints at

20

pages/min. After how many minutes will it have printed

130

pages, given

10

are already in the tray at start?

Example 36

hard

A car decelerates at a constant rate, slowing from

30

m/s to

0

m/s in

6

s. Find the rate of change of velocity.

Example 37

challenge

Two pipes fill a tank at constant rates: pipe A at

5

L/min, pipe B at

3

L/min. How long to fill a

200

L tank with both open?

Example 38

challenge

A linear function

f

satisfies

f(2) = 5

and

f(7) = 20

. Find a formula for

f(x)

and the value

x

for which

f(x) = 50

← Back to Constant Rate Formula Explained Practice Mode

Related Concepts

Rate of Change Linear Functions Linear Relationship Changing Rate

Background Knowledge

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

rate of changelinear functions