Constant vs Variable Examples: 42 Problems with Answers

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

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

Concept Recap

Constants are symbols with fixed, unchanging values; variables are symbols whose values can change or are yet to be determined.

$\pi \approx 3.14159$ is always the same (constant). $x$ can be anything (variable).

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 holds one fixed value; a variable's value can change or is yet unknown.

Common stuck point: The procedure for constant vs variable is the easy part; the trap is trying to solve for a named constant. Asking "Could this symbol's value be different in another situation, or is it locked forever?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Could this symbol's value be different in another situation, or is it locked forever?

Worked Examples

Example 1

easy

C = 2\pi r

, identify the constants and variables.

Answer

Constants:

2, \pi

. Variables:

C, r

First step

\pi \approx 3.14159

is a constant—it never changes.

Full solution

2
$2$ is a constant coefficient.
3
$r$ (radius) is a variable—it can be any positive number.
4
$C$ (circumference) is a variable—it depends on $r$ .

Constants have fixed values that don't change. Variables can take different values. In formulas, Greek letters like

\pi

and

e

are usually constants.

Example 2

medium

In the equation

y = 3x + 5

, someone says '3 is a variable because it could be any slope.' Is this correct?

Example 3

medium

A = \pi r^2

, which symbols change, which are fixed, and what is special about

\pi

Example 4

medium

g(t) = at + b

where

a = 4

and

b = 7

are fixed, which symbols are constants and which is the variable?

Example 5

medium

In the formula

d = vt

, when does

v

play the role of a constant and when a variable?

Example 6

hard

Explain why in

ax^2 + bx + c

a, b, c

are constants but

x

is a variable, even though we sometimes change

a, b, c

to study families of parabolas.

Example 7

challenge

f(x) = c

for an unknown constant

c

, what is the rate of change of

f

, and what is

f(100) - f(7)

Practice Problems

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

Example 1

easy

F = ma

(Newton's second law), which symbols are variables?

Example 2

medium

Is 0 a constant or a variable?

Example 3

easy

3x+5

, which symbol is the variable?

Example 4

easy

3x+5

, name the constants.

Example 5

easy

\pi

a constant or a variable?

Example 6

easy

e

a constant or a variable?

Example 7

easy

y=7

, can

y

vary?

Example 8

easy

2r

, if

r

is a length that can change, what is

r

Example 9

easy

5x^2

, which is the constant coefficient?

Example 10

easy

Is the number 12 a constant or a variable?

Example 11

medium

E=mc^2

, classify

m

and

c

in physics.

Example 12

medium

A letter

c

is a constant in

E=mc^2

but a variable in

y=cx

. Explain in one line.

Example 13

medium

In the term

kx

where

k

is a fixed proportionality constant, can you solve for

k

from

k=k

Example 14

medium

A=\pi r^2

, classify each symbol.

Example 15

medium

f(x)=3x+k

for a fixed but unspecified

k

, what is

k

called?

Example 16

medium

\sqrt{2}\,x

, is

\sqrt{2}

a constant?

Example 17

medium

In the sequence rule

a_n=2n+1

, which symbol varies and which is fixed structure?

Example 18

challenge

y=ax+b

, explain why

a

and

b

are constants for one line yet behave like variables across the family of all lines.

Example 19

challenge

Given

kx=12

where

k=4

is a known constant, find the variable

x

; then say what would change if instead

x

were known and

k

unknown.

Example 20

challenge

Explain why in

y=mx

the ratio

\frac{y}{x}

is constant even though

x

and

y

are variables.

Example 21

medium

P=4s

for a square, classify

P

s

, and 4.

Example 22

medium

y=\frac{1}{2}x+3

, identify the constants.

Example 23

easy

9 - 2x

, list the constants.

Example 24

easy

V = lwh

, classify each symbol if a box's dimensions can change.

Example 25

easy

1

a constant?

Example 26

medium

In a chemistry formula

PV = nRT

, identify the constant.

Example 27

medium

f(x) = 5

, does the value of

f

change as

x

changes?

Example 28

medium

Solve for the variable in

3x + 5 = 14

Example 29

medium

h(x) = 3

, what is the rate of change of the output?

Example 30

medium

5x + 3y = 30

, which letters are variables?

Example 31

hard

Given

y = kx

with

k

a fixed proportionality constant. If

(x, y) = (4, 12)

lies on this graph, find

k

and then

y

when

x = 9

Example 32

hard

f(x) = \sqrt{2} \cdot x + \pi

, classify every symbol.

Example 33

hard

T = T_0 + kt

where

T_0

and

k

are fixed for one experiment, find

T

t = 8

T_0 = 20

and

k = 1.5

Example 34

challenge

In the equation

ax + b = 0

with

a \ne 0

constant and

b

constant, give a formula for the variable

x

Example 35

challenge

y = a\sin(bx + c) + d

, classify each symbol if you study one specific sine curve.

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Related Concepts

Variables Parameter Function Families

Background Knowledge

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

variables