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Subtracting Fractions — Definition, Formula & Examples

Subtracting fractions is finding the difference between two fractions. When the fractions share the same denominator, you subtract the numerators and keep the denominator; when they have different denominators, you first rewrite them with a common denominator.

For fractions ab\frac{a}{b} and cd\frac{c}{d} where b≠0b \neq 0 and d≠0d \neq 0, the difference is defined as ab−cd=ad−bcbd\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}. In practice, using the least common denominator rather than the product bdbd keeps numbers smaller and simplifies the result.

Key Formula

ab−cd=ad−bcbd\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}
Where:
  • aa = Numerator of the first fraction
  • bb = Denominator of the first fraction (cannot be 0)
  • cc = Numerator of the second fraction
  • dd = Denominator of the second fraction (cannot be 0)

How It Works

If both fractions already have the same denominator, simply subtract the top numbers (numerators) and write the result over the shared bottom number (denominator). If the denominators are different, find the least common denominator (LCD), convert each fraction so it uses that denominator, then subtract the numerators. After subtracting, always check whether the answer can be simplified by dividing the numerator and denominator by their greatest common factor. When subtracting mixed numbers, you can either convert them to improper fractions first or subtract the whole-number parts and fraction parts separately, borrowing if needed.

Worked Example

Problem: Subtract: 3/4 − 1/6
Find the LCD: The denominators are 4 and 6. The least common denominator is 12.
LCD of 4 and 6=12\text{LCD of } 4 \text{ and } 6 = 12
Rewrite each fraction: Multiply the numerator and denominator of each fraction so both have a denominator of 12.
34=3×34×3=912,16=1×26×2=212\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}, \quad \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}
Subtract the numerators: Keep the denominator 12 and subtract the numerators.
912−212=712\frac{9}{12} - \frac{2}{12} = \frac{7}{12}
Simplify if possible: 7 and 12 share no common factor other than 1, so the fraction is already in simplest form.
712\frac{7}{12}
Answer: 3/4 − 1/6 = 7/12

Another Example

Problem: Subtract: 5/8 − 1/8
Check the denominators: Both fractions already have the same denominator, 8.
Subtract the numerators: Subtract the top numbers and keep the denominator.
58−18=5−18=48\frac{5}{8} - \frac{1}{8} = \frac{5 - 1}{8} = \frac{4}{8}
Simplify: Divide numerator and denominator by their greatest common factor, 4.
48=12\frac{4}{8} = \frac{1}{2}
Answer: 5/8 − 1/8 = 1/2

Why It Matters

Subtracting fractions appears throughout elementary and middle-school math, from word problems about recipes and measurements to working with algebraic expressions in pre-algebra. Carpenters, bakers, and nurses routinely subtract fractional amounts on the job. Mastering this skill also builds the foundation for subtracting rational expressions in algebra courses.

Common Mistakes

Mistake: Subtracting the denominators along with the numerators, for example writing 5/8 − 1/8 = 4/0.
Correction: Only the numerators change. The denominator stays the same when both fractions already share one: 5/8 − 1/8 = 4/8.
Mistake: Using different denominators without converting first, such as writing 3/4 − 1/6 = 2/4.
Correction: You must rewrite both fractions with a common denominator before subtracting. Convert 3/4 to 9/12 and 1/6 to 2/12, then subtract to get 7/12.

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