Unit Fraction Examples in Math

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: Every fraction is made of unit fractions: \frac{3}{4} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4}.

Common stuck point: \frac{1}{8} is smaller than \frac{1}{4}, even though 8 > 4.

Sense of Study hint: Draw a rectangle and cut it into the number of equal parts shown in the denominator. Shade just one piece — that's your unit fraction.

Worked Examples

Example 1

easy
Express \dfrac{7}{12} as a sum of distinct unit fractions with denominator at most 12.

Solution

  1. 1
    A unit fraction has numerator 1, such as \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, \ldots
  2. 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. 3
    So \dfrac{7}{12} = \dfrac{1}{2} + \dfrac{1}{12}. Both denominators are at most 12. ✓

Answer

\dfrac{7}{12} = \dfrac{1}{2} + \dfrac{1}{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} or \dfrac{1}{9}? Explain using the meaning of unit fractions, then verify with a common denominator.

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}.
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Background Knowledge

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

fractions