Equivalent Fractions Examples in Math

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

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

Concept Recap

Two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent if they represent the same value, which happens exactly when $a \times d = b \times c$ (cross-multiplication gives equal products).

Half a pizza is the same whether cut into 2 or 4 pieces: $\frac{1}{2} = \frac{2}{4}$ .

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: Equivalent fractions change the number of pieces named, not the amount covered.

Common stuck point: The procedure for equivalent fractions is the easy part; the trap is multiplying only the numerator. Asking "Did numerator and denominator change by the same factor?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Did numerator and denominator change by the same factor?

Worked Examples

Example 1

easy

Find three fractions equivalent to

\frac{3}{4}

Two of the three equivalent fractions for 3/4 are shown here.

Answer

\frac{6}{8},\; \frac{9}{12},\; \frac{15}{20}

First step

Multiply numerator and denominator by 2:

\frac{3 \times 2}{4 \times 2} = \frac{6}{8}

Full solution

2
Multiply by 3: $\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$ .
3
Multiply by 5: $\frac{3 \times 5}{4 \times 5} = \frac{15}{20}$ .

Multiplying both the numerator and denominator by the same nonzero number produces an equivalent fraction. The value of the fraction does not change because you are effectively multiplying by 1.

Example 2

medium

Simplify

\frac{36}{48}

to its lowest terms.

Example 3

medium

Find a fraction equivalent to

\frac{4}{9}

with denominator

27

Example 4

hard

Reduce

\frac{84}{126}

to simplest form.

Example 5

challenge

Prove that if

\frac{a}{b} = \frac{c}{d}

(with

b,d>0

), then

\frac{a+c}{b+d} = \frac{a}{b}

Practice Problems

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

Example 1

easy

Fill in the blank:

\frac{5}{6} = \frac{?}{30}

Example 2

hard

Are

\frac{14}{21}

and

\frac{10}{15}

equivalent? Show your reasoning.

Example 3

easy

Are

\frac{1}{2}

and

\frac{2}{4}

equivalent?

Both bars show the same shaded area — are these fractions equivalent?

Example 4

easy

Find an equivalent fraction to

\frac{2}{3}

with denominator

9

Find an equivalent fraction to 2/3 with denominator 9.

Example 5

easy

Simplify

\frac{8}{12}

Simplify 8/12 — both bars shade the same fraction of the whole.

Example 6

easy

Simplify

\frac{15}{20}

Simplify 15/20 — compare the shaded areas.

Example 7

easy

Find an equivalent fraction to

\frac{3}{5}

with numerator

9

Find an equivalent fraction to 3/5 with numerator 9.

Example 8

easy

Are

\frac{3}{4}

and

\frac{9}{16}

equivalent?

Compare the shaded regions — are 3/4 and 9/16 equivalent?

Example 9

easy

\frac{7}{14}

in simplest form?

Is 7/14 in simplest form? Compare it to 1/2.

Example 10

easy

Find an equivalent fraction to

\frac{4}{6}

with smaller numbers.

Find the equivalent fraction to 4/6 with smaller numbers.

Example 11

medium

Fill in the blank:

\frac{5}{8} = \frac{?}{24}

5/8 of the bar is shaded. Fill in the blank: 5/8 = ?/24.

Example 12

medium

Simplify

\frac{24}{36}

Example 13

medium

Order from smallest to largest:

\frac{2}{3}, \frac{3}{4}, \frac{5}{8}

Example 14

medium

Find an equivalent fraction to

\frac{3}{7}

with denominator between

20

and

30

Example 15

medium

Which is larger:

\frac{7}{12}

\frac{3}{5}

Example 16

medium

Reduce

\frac{45}{60}

Example 17

medium

\frac{x}{12} = \frac{5}{6}

, find

x

If x/12 = 5/6, how many parts are shaded in the right bar?

Example 18

medium

Find an equivalent fraction to

\frac{6}{8}

with even larger numerator and denominator.

Find an equivalent fraction to 6/8 with a larger numerator and denominator.

Example 19

medium

Which fraction is in simplest form:

\frac{6}{10}

\frac{9}{16}

, or

\frac{8}{12}

Example 20

challenge

Find the fraction equivalent to

\frac{2}{5}

whose numerator and denominator differ by exactly

6

Find the fraction equivalent to 2/5 whose numerator and denominator differ by 6.

Example 21

challenge

\frac{a}{b}

in simplest form has

a + b = 12

, what are all possible fractions less than

1

Example 22

challenge

True or false: if

\frac{a}{b}

and

\frac{c}{d}

are equivalent, then

\frac{a+c}{b+d}

is also equivalent. Justify.

Example 23

easy

Find an equivalent fraction to

\frac{1}{3}

with denominator

12

Find the equivalent fraction to 1/3 with denominator 12.

Example 24

easy

Simplify

\frac{10}{15}

Simplify 10/15 — compare the shaded areas.

Example 25

easy

Are

\frac{2}{5}

and

\frac{8}{20}

equivalent?

Are 2/5 and 8/20 equivalent? Compare the shaded regions.

Example 26

easy

Simplify

\frac{9}{12}

Simplify 9/12 — the shaded areas are equal.

Example 27

easy

Find an equivalent fraction to

\frac{2}{7}

with numerator

10

Example 28

easy

\frac{4}{5}

in simplest form?

4 out of 5 parts are shaded. Is 4/5 already in simplest form?

Example 29

medium

Fill in:

\frac{7}{9} = \frac{?}{45}

Example 30

medium

Simplify

\frac{42}{56}

Example 31

medium

Are

\frac{12}{18}

and

\frac{10}{15}

equivalent?

Are 12/18 and 10/15 equivalent? Compare the shaded regions.

Example 32

medium

Order from least to greatest by finding equivalents:

\frac{2}{3}, \frac{5}{6}, \frac{3}{4}

Example 33

medium

Solve: if

\frac{x}{20} = \frac{3}{4}

, find

x

If x/20 = 3/4, how many parts are shaded in the right bar?

Example 34

medium

Simplify

\frac{18}{30}

Example 35

medium

Find two fractions equivalent to

\frac{2}{3}

with denominator less than 20.

Two fractions equivalent to 2/3 with denominator less than 20.

Example 36

medium

Which is greater:

\frac{3}{8}

\frac{5}{12}

Example 37

hard

A recipe uses

\frac{2}{3}

cup of sugar. Express this as twelfths.

Express 2/3 cup of sugar as twelfths — the shaded areas are equal.

Example 38

hard

Find the fraction equivalent to

\frac{3}{5}

whose numerator and denominator add to 32.

Find the fraction equivalent to 3/5 whose numerator and denominator add to 32.

Example 39

hard

Solve:

\frac{x}{x+3} = \frac{2}{5}

Example 40

hard

Find all positive integer fractions equivalent to

\frac{4}{6}

whose denominator is at most 30.

Example 41

hard

\frac{2x}{3x+5} = \frac{2}{4}

, find

x

Example 42

challenge

For positive integers, how many fractions equivalent to

\frac{1}{2}

have a denominator that is a perfect square less than 100?

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Related Concepts

Fractions Multiplication Adding Fractions Simplification

Background Knowledge

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

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