Like and Unlike Fractions

What are like and unlike fractions?

Fractions are classified as like or unlike based entirely on their bottom numbers, or denominators.

Fractions with common denominators are grouped together because their fractional pieces represent the exact same proportion of a whole.

When the denominators do not match, the fractions belong to different families and their pieces are fundamentally different sizes.

The numerators, which tell us how many parts we have, do not play a role in determining whether a set of fractions is like or unlike.

Identify like fractions

Same-denominator fractions are easy to identify because you only need to check the bottom number of each fraction.

If a group of fractions all share the exact same denominator, they are like fractions.

For example, $\dfrac{1}{7}$, $\dfrac{3}{7}$, and $\dfrac{6}{7}$ are all like fractions. Every whole is cut into exactly $7$ equal parts, making it easy to see which fraction represents the largest amount.

Because the pieces are identical in size, we can add, subtract, and compare these amounts smoothly without changing their format.

Identify unlike fractions

Different-denominator fractions are easy to spot because their bottom numbers do not match.

If you are given $\dfrac{1}{2}$, $\dfrac{2}{3}$, and $\dfrac{3}{5}$, they are unlike fractions. Each fraction is built from pieces of an entirely different size. Halves are larger than thirds, and thirds are larger than fifths.

Whenever a set of fractions contains even one fraction with a different denominator, the entire set is considered unlike.

Why the denominator matters

The denominator defines the standard of measurement. Comparing fractions mathematically is only straightforward when both fractions use the same standard.

Imagine trying to compare $3$ meters to $3$ inches. Even though the number is the same, the units represent entirely different lengths. Fractions work the same way. You cannot easily add $\dfrac{1}{3}$ and $\dfrac{1}{4}$ together because you are adding two completely different sizes of pieces.

Converting unlike fractions into like fractions creates a shared standard, allowing you to count and combine the total number of parts accurately.

Change unlike fractions to like fractions

To convert unlike fractions into like fractions, you must find a common multiple for their denominators.

1. Find the lowest common denominator (LCM) of the bottom numbers.
2. Multiply the numerator and the denominator of each fraction by the factor needed to reach the common denominator.
3. Write the new equivalent fractions, which are now like fractions.

Once converted, you can add, subtract, and compare the numerators easily because the pieces are now exactly the same size.

Worked examples

Example 1: Identifying fraction types

Question: Are the fractions $\dfrac{5}{9}$ and $\dfrac{7}{9}$ like or unlike fractions?

Method:

1. Look at the denominators of both fractions.
2. Both fractions share the exact same denominator of $9$.

Answer: They are like fractions.

Check: Because they share a denominator of $9$, they represent parts of the same size, which perfectly meets the definition of like fractions.

Example 2: Grouping a set of fractions

Question: Is the set of fractions $\left\{ \dfrac{2}{5}, \dfrac{3}{5}, \dfrac{4}{10} \right\}$ like or unlike?

Method:

1. Check all the denominators in the set.
2. The denominators are $5$, $5$, and $10$.
3. Because $10$ is different from $5$, the fractions do not all share the same denominator.

Answer: The set is composed of unlike fractions.

Check: Even though $\dfrac{4}{10}$ is equivalent to $\dfrac{2}{5}$, we classify fractions based entirely on their written denominators, not their values.

Example 3: Converting unlike to like fractions

Question: Convert $\dfrac{2}{3}$ and $\dfrac{3}{8}$ into like fractions.

Method:

1. Identify the denominators: $3$ and $8$.
2. Find the lowest common multiple of $3$ and $8$. Since they share no common factors other than $1$, the LCM is $3 \times 8 = 24$.
3. Convert $\dfrac{2}{3}$ by multiplying both parts by $8$: $\dfrac{2 \times 8}{3 \times 8} = \dfrac{16}{24}$.
   
4. Convert $\dfrac{3}{8}$ by multiplying both parts by $3$: $\dfrac{3 \times 3}{8 \times 3} = \dfrac{9}{24}$.

Answer: The like fractions are $\dfrac{16}{24}$ and $\dfrac{9}{24}$.

Check: When simplifying fractions, dividing $\dfrac{16}{24}$ by $8$ returns $\dfrac{2}{3}$, and dividing $\dfrac{9}{24}$ by $3$ returns $\dfrac{3}{8}$. The conversion is correct.

Common mistakes

A frequent misconception is assuming that fractions with the same numerator are like fractions. For example, $\dfrac{3}{4}$ and $\dfrac{3}{5}$ have the same top number, but they are absolutely unlike fractions. Having the same number of parts does not mean the parts are the same size.

Another common mistake is confusing equivalence with classification. The fractions $\dfrac{1}{2}$ and $\dfrac{4}{8}$ represent the exact same amount, but because their written denominators are different, they are technically unlike fractions until one is converted to match the other.

Frequently asked questions

Can like fractions have different numerators?

Yes. In fact, they almost always do. The defining rule for like fractions relies entirely on the denominators matching. The numerators can be any number.

Are whole numbers considered like fractions?

Whole numbers can be written as like fractions if they are given the same denominator. For example, the whole numbers $4$ and $7$ can be written as $\dfrac{4}{1}$ and $\dfrac{7}{1}$, making them like fractions.