Comparing Fractions: Definition, Method and Examples

How do you compare fractions?

Comparing fractions means checking their sizes against one another using inequality symbols. Once you understand how to compare two values, you can apply the same rules for ordering fractions in a longer list.

When comparing fractions, the pieces must refer to the same-sized whole. For example, half of a large pizza is physically larger than half of a small pizza, even though both represent $\dfrac{1}{2}$ of their respective pizzas.

We use standard mathematical symbols to state the relationship:

• $>$ means strictly greater than.
• $<$ means strictly less than.
• $=$ means exactly equal to.

Always verify that fractions refer to the same whole before comparing them.

Same denominators

When two fractions have the same denominator, their pieces are the exact same size.

To compare them, you only need to look at the numerators. The fraction with the greater numerator is the larger fraction because it has more of those equal-sized pieces.

Since the denominators in $\dfrac{4}{5}$ and $\dfrac{3}{5}$ are both $5$, the pieces are the same size. Because $4$ is greater than $3$, $\dfrac{4}{5}$ is the larger fraction.

Same numerators

If two fractions have the same numerator, you are comparing the same number of pieces.

However, the pieces are different sizes. A larger denominator means the whole is divided into more pieces, making each individual piece smaller. Therefore, when the numerators are identical, the fraction with the smaller denominator is actually the larger fraction.

In $\dfrac{2}{3}$ and $\dfrac{2}{5}$, both fractions represent two parts. Because thirds are larger pieces than fifths, $\dfrac{2}{3}$ is greater than $\dfrac{2}{5}$.

Compare to benchmarks

When two fractions have different numerators and denominators, you can often compare them to familiar benchmark fractions such as $\dfrac{1}{2}$ or $1$.

For example, to compare $\dfrac{3}{8}$ and $\dfrac{4}{6}$, consider their relationship to $\dfrac{1}{2}$:

• Half of $8$ is $4$, so $\dfrac{4}{8}$ is exactly $\dfrac{1}{2}$. This means $\dfrac{3}{8}$ is less than $\dfrac{1}{2}$.
• Half of $6$ is $3$, so $\dfrac{3}{6}$ is exactly $\dfrac{1}{2}$. This means $\dfrac{4}{6}$ is greater than $\dfrac{1}{2}$.

Because $\dfrac{3}{8}$ is less than half and $\dfrac{4}{6}$ is more than half, you can conclude without further calculation that $\dfrac{3}{8} < \dfrac{4}{6}$.

Compare unlike denominators

If you cannot easily use a benchmark, you must rewrite the fractions so they share a common denominator. This gives the fractions pieces of identical size, allowing direct comparison.

To do this, use equivalent fractions to change the denominator of one or both numbers.

1. Find a common multiple for both denominators.
2. Multiply the numerator and denominator of each fraction by the factor needed to reach that common multiple.
3. Compare the new numerators.

For example, to compare $\dfrac{5}{6}$ and $\dfrac{7}{9}$, find a common multiple of $6$ and $9$. The number $18$ is a multiple of both.

Convert $\dfrac{5}{6}$: multiply the numerator and denominator by $3$ to get $\dfrac{15}{18}$.

Convert $\dfrac{7}{9}$: multiply the numerator and denominator by $2$ to get $\dfrac{14}{18}$.

Now compare $\dfrac{15}{18}$ and $\dfrac{14}{18}$. Since $15$ is greater than $14$, $\dfrac{15}{18}$ is larger. Therefore, $\dfrac{5}{6} > \dfrac{7}{9}$.

Use a number line

Another powerful method is placing the fractions on a number line. This visual approach connects fraction values directly to distance from zero.

Any fraction positioned further to the right on a horizontal number line represents a greater value. By evaluating $\dfrac{1}{4}$ and $\dfrac{2}{3}$, you can visually confirm that $\dfrac{2}{3}$ is further from zero, meaning $\dfrac{1}{4} < \dfrac{2}{3}$.

Worked examples

Review these examples to see fraction comparison strategies applied step by step.

Example 1: Using the same numerator rule

Question: Which is greater: $\dfrac{5}{8}$ or $\dfrac{5}{12}$?

Method:

1. Observe that both fractions have the same numerator, $5$.
2. Compare the denominators. The denominators are $8$ and $12$.
3. Because $8$ is less than $12$, eighths are larger pieces than twelfths.

Answer: $\dfrac{5}{8}$ is greater.

Check: Since $5$ pieces of a larger size are more than $5$ pieces of a smaller size, the logic holds.

Example 2: Finding a common denominator

Question: Compare $\dfrac{3}{4}$ and $\dfrac{7}{10}$ using a common denominator.

Method:

1. Identify a common multiple for $4$ and $10$. The number $20$ is the lowest common multiple.
2. Convert $\dfrac{3}{4}$ by multiplying the numerator and denominator by $5$: $\dfrac{3 \times 5}{4 \times 5} = \dfrac{15}{20}$.
3. Convert $\dfrac{7}{10}$ by multiplying the numerator and denominator by $2$: $\dfrac{7 \times 2}{10 \times 2} = \dfrac{14}{20}$.
4. Compare the new fractions: $\dfrac{15}{20} > \dfrac{14}{20}$.

Answer: $\dfrac{3}{4} > \dfrac{7}{10}$.

Check: Using decimals, $\dfrac{3}{4} = 0.75$ and $\dfrac{7}{10} = 0.70$. Since $0.75 > 0.70$, the answer is correct.

Common mistakes

When analyzing unlike fractions, watch out for these frequent reasoning errors.

Larger denominator always means larger fraction

Many learners mistakenly assume that a larger denominator makes the fraction larger. In fact, a larger denominator means the whole is broken into more pieces, making each individual piece smaller. Always find a common denominator or use a benchmark if the numerators differ.

Adding numerators and denominators

Never add or subtract straight across to compare fractions. Finding that $2+5=7$ and $3+6=9$ does not tell you the relationship between $\dfrac{2}{3}$ and $\dfrac{5}{6}$. You must use equivalent fractions to make the pieces equal in size.