Unit Fraction Examples: 44 Problems with Answers

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

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 fraction with numerator 1, like $\frac{1}{3}$ or $\frac{1}{8}$ , representing exactly one equal part of a whole.

The building blocks of fractions— $\frac{1}{2}$ is one of two equal parts, $\frac{1}{4}$ is one of four.

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 unit fraction is $\tfrac{1}{n}$ — exactly one of the $n$ equal pieces a whole is split into, the building block of all fractions.

Common stuck point: The procedure for unit fraction is the easy part; the trap is thinking 1/8 > 1/4 because 8 > 4. Asking "Does the fraction have a 1 on top, naming exactly one equal part of the whole?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the fraction have a 1 on top, naming exactly one equal part of the whole?

Worked Examples

Example 1

easy

Express

\dfrac{7}{12}

as a sum of distinct unit fractions with denominator at most

12

Answer

\dfrac{7}{12} = \dfrac{1}{2} + \dfrac{1}{12}

First step

A unit fraction has numerator

1

, such as

\dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \ldots

Full solution

2
Start with the largest unit fraction $\leq \dfrac{7}{12}$ : $\dfrac{1}{2} = \dfrac{6}{12}$ . Remainder: $\dfrac{7}{12} - \dfrac{6}{12} = \dfrac{1}{12}$ .
3
So $\dfrac{7}{12} = \dfrac{1}{2} + \dfrac{1}{12}$ . Both denominators are at most $12$ . ✓

Any positive fraction can be written as a sum of distinct unit fractions. The greedy approach — always subtracting the largest possible unit fraction — is a systematic method. Unit fractions were central to ancient Egyptian mathematics.

Example 2

medium

Which is larger,

\dfrac{1}{7}

\dfrac{1}{9}

? Explain using the meaning of unit fractions, then verify with a common denominator.

Example 3

easy

Show that

\dfrac{4}{5}

is four copies of a unit fraction. Name the unit fraction.

Example 4

medium

Express

\dfrac{5}{6}

as a sum of distinct unit fractions.

Example 5

medium

Two friends share a cookie. One takes

\dfrac{1}{2}

, the other takes

\dfrac{1}{4}

. What fraction remains?

Example 6

medium

Compare

\dfrac{1}{5}

and

\dfrac{2}{9}

Example 7

hard

Express

\dfrac{3}{4}

as a sum of two distinct unit fractions.

Example 8

hard

Use the splitting identity

\dfrac{1}{n} = \dfrac{1}{n+1} + \dfrac{1}{n(n+1)}

to split

\dfrac{1}{4}

Example 9

hard

True or false: every positive rational less than

1

can be written as a finite sum of distinct unit fractions.

Example 10

challenge

Find an Egyptian-fraction decomposition of

\dfrac{5}{7}

using distinct unit fractions.

Practice Problems

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

Example 1

easy

Order the unit fractions

\dfrac{1}{3}

\dfrac{1}{10}

\dfrac{1}{5}

\dfrac{1}{2}

from least to greatest.

Example 2

medium

A recipe uses

\dfrac{3}{4}

cup of flour. Express this as a sum of unit fractions using only

\dfrac{1}{2}

and

\dfrac{1}{4}

Example 3

easy

What unit fraction is one of

5

equal parts of a whole?

Example 4

easy

Which is larger:

\frac{1}{4}

\frac{1}{8}

Example 5

easy

How many copies of

\frac{1}{4}

make

\frac{3}{4}

Example 6

easy

Write

\frac{1}{10}

as a decimal.

Example 7

easy

\frac{1}{2}

the same size as

\frac{1}{3}

Example 8

easy

What fraction is one part when a pizza is cut into

6

equal slices?

Example 9

easy

Order the unit fractions least to greatest:

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

Example 10

easy

Two copies of which unit fraction make

\frac{1}{2}

Example 11

medium

Express

\frac{5}{6}

as a sum of unit fractions all equal to

\frac{1}{6}

Example 12

medium

Which is larger and by how much:

\frac{1}{3}

\frac{1}{4}

Example 13

medium

A recipe uses

\frac{1}{8}

cup three times. How much total?

Example 14

medium

Between

\frac{1}{3}

and

\frac{1}{2}

, name a unit fraction that lies between... or explain why none exists.

Example 15

medium

\frac{1}{n} = 0.05

, find

n

Example 16

medium

Which whole has

\frac{1}{4}

equal to

5

items?

Example 17

medium

Order least to greatest:

\frac{1}{100}, \frac{1}{10}, \frac{1}{1000}

Example 18

medium

Express

\frac{2}{3}

as a sum of two different unit fractions.

Example 19

challenge

Show that

\frac{1}{n} - \frac{1}{n+1} = \frac{1}{n(n+1)}

and use it to compute

\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}

Example 20

challenge

For which positive integer denominators

n

\frac{1}{n}

a terminating decimal?

Example 21

challenge

A unit fraction

\frac{1}{n}

satisfies

\frac{1}{6} < \frac{1}{n} < \frac{1}{5}

. Find all integer

n

Example 22

medium

How many unit-fraction pieces of

\frac{1}{12}

are in

\frac{1}{4}

Example 23

easy

What is one of

7

equal parts of a whole, as a unit fraction?

Example 24

easy

Write

\dfrac{5}{8}

as a sum of

\dfrac{1}{8}

Example 25

easy

Order from smallest to largest:

\dfrac{1}{4}, \dfrac{1}{2}, \dfrac{1}{8}, \dfrac{1}{3}

Example 26

easy

What fraction is one of

12

equal slices of a pie?

Example 27

medium

How many

\dfrac{1}{8}

s are in

\dfrac{3}{4}

Example 28

medium

Find a unit fraction between

\dfrac{1}{4}

and

\dfrac{1}{3}

Example 29

medium

Express

\dfrac{2}{3}

as a sum of two distinct unit fractions.

Example 30

medium

A pizza has

8

slices. Maya eats

3

slices. What unit fraction is each slice, and what fraction did she eat?

Example 31

hard

Express

\dfrac{1}{2}

as a sum of two distinct unit fractions.

Example 32

hard

Find three distinct unit fractions that sum to

1

Example 33

hard

Apply the greedy algorithm to start an Egyptian-fraction expansion of

\dfrac{2}{7}

: subtract the largest unit fraction

\le \dfrac{2}{7}

and report the remainder.

Example 34

challenge

Find four distinct unit fractions that sum to

1

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Related Concepts

Fractions Equivalent Fractions Adding Fractions

Background Knowledge

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

fractions