Unit Fractions: Definition, Method and Examples

What is a unit fraction?

A unit fraction (sometimes called a one-part fraction or a one over n fraction) is the building block of all other fractions. The word unit means one.

A unit fraction has a numerator of 1 and a positive integer denominator.

The denominator tells us how many equal parts the whole is divided into. The numerator tells us we have exactly one of those equal parts. For example, in the fraction $\dfrac{1}{6}$, the whole is divided into $6$ equal pieces, and the unit fraction represents just $1$ of those pieces.

Every standard fraction is built by adding unit fractions together. For instance, $\dfrac{3}{4}$ is made by adding three separate $\dfrac{1}{4}$ fractions together.

Identify unit and non-unit fractions

To determine if a fraction is a unit fraction, check its numerator.

• Unit fractions: The numerator is exactly $1$. Examples include $\dfrac{1}{2}$, $\dfrac{1}{8}$, and $\dfrac{1}{100}$. They are always proper fractions because the numerator is smaller than the denominator.
• Non-unit fractions: The numerator is any whole number other than $1$. Examples include $\dfrac{2}{3}$, $\dfrac{5}{8}$, and $\dfrac{7}{4}$.

Unit fractions in visual models

We can use visual fraction models to see how one whole can be divided into different equal parts. A fraction strip is a rectangular bar that represents one whole.

When we divide the strip into more equal pieces, the denominator increases. Because the same whole is shared among more pieces, each individual piece becomes smaller.

Looking at the models, notice that one part out of two ($\dfrac{1}{2}$) takes up much more space than one part out of five ($\dfrac{1}{5}$).

Unit fractions on a number line

We can also position unit fractions between $0$ and $1$ on a number line. Placing fractions on a number line helps us visualize their value as numbers rather than just shapes.

Because every unit fraction is greater than zero but less than one whole, they all live on the segment between $0$ and $1$. The larger the denominator, the closer the unit fraction is to $0$.

This number line confirms what the fraction strips showed: $\dfrac{1}{10}$ is much smaller than $\dfrac{1}{2}$.

Compare unit fractions

When comparing two unit fractions, the relationship between their sizes can feel reversed compared to whole numbers.

The greater the denominator, the smaller the unit fraction.

Imagine sharing a pizza. If you share it between $3$ people, you get a larger slice than if you share it between $8$ people. Therefore, one third is greater than one eighth ($\dfrac{1}{3} > \dfrac{1}{8}$).

When putting unit fractions in order from smallest to largest, you list them in descending order of their denominators: $\dfrac{1}{12}$, $\dfrac{1}{8}$, $\dfrac{1}{4}$, $\dfrac{1}{2}$.

Fractions of quantities

We can also find a unit fraction of a whole number, such as the number of objects in a group or a measurement. This requires dividing the total quantity by the denominator.

To find $\dfrac{1}{5}$ of $15$ items, we split the $15$ items into $5$ equal groups. The number of items in one single group is our answer.

Since $15 \div 5 = 3$, we know that $\dfrac{1}{5}$ of $15$ equals $3$. Learning how to calculate fractions of a whole and a set will let you solve these problems for any type of fraction.

Worked examples

Example 1: Identifying a unit fraction

Question: Which of the following fractions is a unit fraction: $\dfrac{4}{1}$, $\dfrac{1}{9}$, or $\dfrac{7}{7}$?

Method:

1. Recall that a unit fraction must have exactly $1$ as the numerator.
2. Examine the numerators of the given fractions.
3. $\dfrac{4}{1}$ has a numerator of $4$.
4. $\dfrac{1}{9}$ has a numerator of $1$.
5. $\dfrac{7}{7}$ has a numerator of $7$.

Answer: The unit fraction is $\dfrac{1}{9}$.

Check: The denominator is $9$, which is a positive whole number, and the numerator is exactly $1$.

Example 2: Comparing sizes of unit fractions

Question: Order the fractions $\dfrac{1}{6}$, $\dfrac{1}{3}$, $\dfrac{1}{10}$, and $\dfrac{1}{2}$ from smallest to largest.

Method:

1. Recognize that all four numbers are unit fractions.
2. Remember the rule: the larger the denominator, the smaller the fraction's value.
3. Compare the denominators: $10 > 6 > 3 > 2$.
4. Order the fractions inversely to their denominators. The largest denominator ($10$) belongs to the smallest fraction ($\dfrac{1}{10}$).

Answer: The fractions in order from smallest to largest are $\dfrac{1}{10}$, $\dfrac{1}{6}$, $\dfrac{1}{3}$, $\dfrac{1}{2}$.

Check: Visualize the sizes on a number line. $\dfrac{1}{10}$ is closest to $0$, and $\dfrac{1}{2}$ is exactly in the middle.

Example 3: Working backward from a unit fraction

Question: If $\dfrac{1}{4}$ of a ribbon is $12$ centimeters long, what is the total length of the ribbon?

Method:

1. Identify the unit fraction and its corresponding value: $\dfrac{1}{4}$ of the length is $12$ centimeters.
2. Understand that the denominator $4$ means the whole ribbon is made up of $4$ equal parts.
3. If one part is $12$ centimeters, multiply the value of one part by the total number of parts to find the whole.
4. Calculate $12 \times 4 = 48$.

Answer: The total length of the ribbon is $48$ centimeters.

Check: Verify by finding $\dfrac{1}{4}$ of the answer. $48 \div 4 = 12$, which matches the original problem.

Frequently asked questions

Is 1 over 1 a unit fraction?

Yes, $\dfrac{1}{1}$ is technically a unit fraction because its numerator is $1$ and its denominator is a positive whole number. It represents one whole. However, it is most commonly written simply as the whole number $1$.

Can a unit fraction have a negative denominator?

No. By definition, a standard unit fraction represents dividing a whole into a positive number of equal parts. The denominator must be a positive integer.

How do you add unit fractions?

If the denominators are the same, you add the numerators while keeping the denominator exactly the same. For example, $\dfrac{1}{5} + \dfrac{1}{5} = \dfrac{2}{5}$. If the denominators are different, you must find a common denominator before adding.

Are all unit fractions proper fractions?

Every unit fraction with a denominator greater than $1$ is a proper fraction because the numerator ($1$) is less than the denominator. The only exception is $\dfrac{1}{1}$, which is an improper fraction because the numerator is equal to the denominator.