Subtracting Fractions: Definition, Method and Examples

How do you subtract fractions?

To calculate the difference between two fractions, you must first determine whether their denominators match. Fractions represent pieces of a whole, and subtraction is only straightforward when those pieces are exactly the same size.

When denominators are identical, the sizes match, and you can simply find the difference between the numerators. When denominators are different, the fractional parts differ in size. You must convert them into a uniform size before subtracting them.

Subtract fractions with the same denominator

Fractions with matching bottom numbers are called like fractions. Because the denominators are identical, the parts being subtracted represent the same size, allowing you to subtract immediately.

Method:

1. Subtract the second numerator from the first numerator.
2. Keep the common denominator exactly the same.
3. Simplify the resulting fraction to its lowest terms if possible.

Always leave the common denominator unchanged. Never subtract the denominators.

Subtract fractions with different denominators

Fractions with different bottom numbers are called unlike fractions. Before you can subtract them, you must convert them into equivalent fractions that share a common denominator.

Method:

1. Find the lowest common multiple of the two denominators. This becomes your common denominator.
2. Multiply the numerator and denominator of each fraction by the factor needed to reach the common denominator.
3. Subtract the new numerators and keep the common denominator unchanged.

Subtract a fraction from a whole number

To subtract a fraction from a whole number, express the whole number as an equivalent fraction by writing it over a denominator of $1$.

Once both numbers are written in fractional form, find a common denominator just as you would with any other unlike fractions, then subtract.

Check and simplify the difference

After finding the difference, your final answer might not be in its simplest form. The process of simplifying fractions involves finding the greatest common factor shared by the numerator and the denominator.

Divide both parts of the fraction by this factor to reduce it to its lowest terms. If your result is an improper fraction, where the numerator is larger than the denominator, you can also rewrite it as a mixed number.

Visual worked examples

Applying these steps to fraction word problems and standard equations helps build a strong understanding of how fraction subtraction behaves in different practical contexts.

Example 1: Subtracting like fractions

Question: A baker has $\dfrac{7}{8}$ of a bag of flour. They use $\dfrac{3}{8}$ of the bag for a recipe. What fraction of the bag remains?

Method:

1. Confirm the denominators are the same. Both are $8$.
2. Subtract the numerators: $7 - 3 = 4$.
3. Keep the denominator: $\dfrac{4}{8}$.
4. Simplify by dividing the numerator and denominator by $4$.

Answer: $\dfrac{1}{2}$ of the bag remains.

Check: Add the difference back to the subtracted amount to verify the original total: $\dfrac{1}{2} + \dfrac{3}{8} = \dfrac{4}{8} + \dfrac{3}{8} = \dfrac{7}{8}$.

Example 2: Subtracting unlike fractions

Question: Calculate $\dfrac{5}{6} - \dfrac{1}{4}$.

Method:

1. Find the lowest common multiple of $6$ and $4$, which is $12$.
2. Convert $\dfrac{5}{6}$: multiply the numerator and denominator by $2$ to get $\dfrac{10}{12}$.
3. Convert $\dfrac{1}{4}$: multiply the numerator and denominator by $3$ to get $\dfrac{3}{12}$.
4. Subtract the converted numerators: $10 - 3 = 7$.

Answer: $\dfrac{7}{12}$.

Check: Verify using addition: $\dfrac{7}{12} + \dfrac{1}{4} = \dfrac{7}{12} + \dfrac{3}{12} = \dfrac{10}{12}$, which simplifies to $\dfrac{5}{6}$.

Example 3: Subtracting from a whole number

Question: Evaluate $4 - \dfrac{2}{3}$.

Method:

1. Write the whole number as a fraction: $\dfrac{4}{1}$.
2. Find the common denominator for $1$ and $3$, which is $3$.
3. Convert $\dfrac{4}{1}$: multiply the numerator and denominator by $3$ to get $\dfrac{12}{3}$.
4. Subtract the fractions: $\dfrac{12}{3} - \dfrac{2}{3} = \dfrac{10}{3}$.
5. Convert the improper fraction to a mixed number if required.

Answer: $\dfrac{10}{3}$ or $3\dfrac{1}{3}$.

Check: Evaluate using a regrouping method: rewrite $4$ as $3 + \dfrac{3}{3}$. Then evaluate $3\dfrac{3}{3} - \dfrac{2}{3} = 3\dfrac{1}{3}$.

Common mistakes

A frequent error is subtracting both the numerators and the denominators across the equation. This mistake ignores the mathematical meaning of the denominator, which simply names the size of the fractional parts, not a quantity that can be subtracted.

Always establish a common denominator first, and never subtract the denominators from each other.

Frequently asked questions

Can you subtract an improper fraction?

Yes. The exact same rules apply when subtracting an improper fraction. Ensure the denominators match, subtract the numerators, and simplify the final result.

Do you always have to use the lowest common denominator?

No. Any common multiple of the denominators will work. For example, simply multiplying the two denominators together will always give you a valid common denominator. However, using the lowest common multiple keeps the numbers smaller and usually requires less simplification at the end.

How does this compare to fraction addition?

The steps are largely identical to adding fractions. In both operations, you must establish a common denominator and keep that denominator unchanged in your answer. The only difference is whether you add or subtract the numerators.