Equivalence Examples in Math

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

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

Concept Recap

When two expressions, numbers, or objects represent the same value or are interchangeable in every relevant context.

$\frac{1}{2}$ , $0.5$ , and $50\%$ are equivalent—different forms, same value.

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: Two things are equivalent when they always stand for the same value and can replace each other anywhere.

Common stuck point: The procedure for equivalence is the easy part; the trap is judging value by appearance. Asking "Do the two expressions name the exact same value in every context?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do the two expressions name the exact same value in every context?

Worked Examples

Example 1

easy

Are

3 \times 4

and

6 \times 2

equivalent? Show your work.

Answer

Yes, both equal 12

First step

Calculate

3 \times 4 = 12

Full solution

2
Calculate $6 \times 2 = 12$ .
3
Both equal 12, so $3 \times 4 = 6 \times 2$ . They are equivalent.

Two expressions are equivalent when they name the same value. Here both equal 12 even though they look different.

Example 2

medium

Find a value of

n

that makes

4 \times n = 2 \times 10

true. Explain what equivalence means here.

Example 3

medium

Show

\frac{1}{2} + \frac{1}{4}

is equivalent to

\frac{3}{4}

Example 4

hard

Show

\frac{x^2 - 9}{x + 3}

is equivalent to

x - 3

for

x \ne -3

Example 5

hard

A student claims

\sqrt{x^2}

is equivalent to

x

. Find a counterexample and give the correct equivalent form.

Example 6

challenge

Use equivalence to convert

0.\overline{3}

to a fraction.

Practice Problems

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

Example 1

easy

15 + 7

equivalent to

11 + 11

? Show your work.

Example 2

medium

Find

n

such that

5 \times 8 = n + 20

Example 3

easy

Are

\frac{1}{2}

and

0.5

equivalent?

Example 4

easy

\frac{2}{4}

equivalent to

\frac{1}{2}

Example 5

easy

50\%

equivalent to

\frac{1}{2}

Example 6

easy

3+4

equivalent to

4+3

Example 7

easy

Write

\frac{3}{4}

as an equivalent fraction with denominator 8.

Example 8

easy

0.25

equivalent to

\frac{1}{4}

Example 9

easy

Are

\frac{5}{10}

and

\frac{1}{2}

equivalent?

Example 10

easy

\frac{2}{1}

equivalent to

2

Example 11

medium

Are

2(x+3)

and

2x+6

equivalent? Justify.

Example 12

medium

Are

x^2

and

2x

equivalent? Test more than one value.

Example 13

medium

Simplify and decide: is

\frac{6}{8}

equivalent to

\frac{9}{12}

Example 14

medium

Is the equation

2x+2 = 2(x+1)

an identity or a solvable equation?

Example 15

medium

Convert

\frac{7}{8}

to a decimal and confirm equivalence.

Example 16

medium

Are

\frac{a}{b}

and

\frac{2a}{2b}

equivalent for nonzero

b

? Why?

Example 17

medium

Order from least to greatest by finding equivalents:

\frac{1}{2}, 0.4, \frac{3}{5}

Example 18

medium

3(2x-1)+3

equivalent to

6x

? Justify.

Example 19

challenge

Show

\frac{x^2-1}{x-1}

is equivalent to

x+1

for

x\ne 1

, and explain the restriction.

Example 20

challenge

Prove

\frac{1}{2}+\frac{1}{3}

is equivalent to

\frac{5}{6}

Example 21

challenge

Determine all

x

for which

\frac{2x}{x}

is equivalent to

2

Example 22

medium

Are

\frac{3}{4}

and

\frac{75}{100}

equivalent? Justify.

Example 23

easy

6+5

equivalent to

7+4

Example 24

easy

\frac{4}{6}

equivalent to

\frac{2}{3}

Example 25

easy

Find

n

3 \times n = 12

Example 26

easy

25\%

equivalent to

\frac{1}{4}

Example 27

easy

Find

n

4 \times 9 = 6 \times n

Example 28

easy

0.75

equivalent to

\frac{3}{4}

Example 29

medium

Are

3(x+4)

and

3x + 12

equivalent? Justify.

Example 30

medium

Are

x + x + x

and

3x

equivalent?

Example 31

medium

Are

\frac{3}{5}

and

\frac{12}{20}

equivalent?

Example 32

medium

Find

n

so that

\frac{4}{6} = \frac{n}{18}

Example 33

medium

Are

2 \times (3 + 5)

and

2 \times 3 + 2 \times 5

equivalent?

Example 34

medium

Are

\frac{5}{8}

and

0.625

equivalent?

Example 35

medium

Are

\frac{2}{5}

and

\frac{6}{15}

equivalent?

Example 36

hard

Are

(x+2)^2

and

x^2 + 4

equivalent? Show with one test value.

Example 37

hard

Order from smallest to largest by finding equivalents:

\frac{2}{3}, \frac{5}{8}, 0.7

Example 38

hard

Are

\frac{a+b}{2}

and

\frac{a}{2} + \frac{b}{2}

equivalent for all

a

and

b

Example 39

hard

Are

\frac{1}{x} + \frac{1}{y}

and

\frac{1}{x+y}

equivalent for all nonzero

x, y

Example 40

challenge

For what values of

x

are

\frac{x^2 - 1}{x - 1}

and

x + 1

equivalent?

← Back to Equivalence Formula Explained Practice Mode

Related Concepts

Equal Equivalent Fractions

Background Knowledge

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

equal