Subtracting Fractions

Subtracting fractions follows the same core rule as adding them: the denominators must match. Once they do, you subtract the numerators instead of adding them. The extra skill you need here is borrowing when subtracting mixed numbers, which trips up many learners. This page breaks it all down step by step.

Subtracting Fractions with Like Denominators

When denominators are already the same, subtract the numerators and keep the denominator.

Formula:

$\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$

Example 1: Subtract $\frac{5}{8} - \frac{3}{8}$

$\frac{5}{8} - \frac{3}{8} = \frac{5 - 3}{8} = \frac{2}{8} = \frac{1}{4}$

Answer: $\frac{1}{4}$

Example 2: Subtract $\frac{7}{12} - \frac{1}{12}$

$\frac{7}{12} - \frac{1}{12} = \frac{6}{12} = \frac{1}{2}$

Answer: $\frac{1}{2}$

Subtracting Fractions with Unlike Denominators

Just like addition, you must find the Least Common Denominator (LCD) first, convert, then subtract.

Steps:

1. Find the LCD
2. Rewrite each fraction with the LCD
3. Subtract the numerators
4. Simplify if needed

Example 3: Subtract $\frac{3}{4} - \frac{1}{3}$

Step 1: LCD of 4 and 3 is 12.

Step 2: Convert:

$\frac{3}{4} = \frac{9}{12} \qquad \frac{1}{3} = \frac{4}{12}$

Step 3: Subtract:

$\frac{9}{12} - \frac{4}{12} = \frac{5}{12}$

Answer: $\frac{5}{12}$

Example 4: Subtract $\frac{5}{6} - \frac{1}{4}$

Step 1: LCD of 6 and 4 is 12.

Step 2: Convert:

$\frac{5}{6} = \frac{10}{12} \qquad \frac{1}{4} = \frac{3}{12}$

Step 3: Subtract:

$\frac{10}{12} - \frac{3}{12} = \frac{7}{12}$

Answer: $\frac{7}{12}$

Subtracting Mixed Numbers

Subtracting mixed numbers requires you to handle the whole number and fraction parts. There are two common situations.

Case 1: No Borrowing Needed

When the first fraction part is larger than the second, subtract normally.

Example 5: Subtract $5\frac{3}{4} - 2\frac{1}{4}$

Step 1: Subtract whole numbers: $5 - 2 = 3$

Step 2: Subtract fractions: $\frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$

Answer: $3\frac{1}{2}$

Case 2: Borrowing Required

When the first fraction is smaller than the second, you must borrow 1 from the whole number and convert it to a fraction.

Example 6: Subtract $4\frac{1}{6} - 1\frac{5}{6}$

You cannot subtract $\frac{5}{6}$ from $\frac{1}{6}$ directly. Borrow 1 from the 4:

$4\frac{1}{6} = 3 + 1 + \frac{1}{6} = 3 + \frac{6}{6} + \frac{1}{6} = 3\frac{7}{6}$

Now subtract:

$3\frac{7}{6} - 1\frac{5}{6} = 2\frac{2}{6} = 2\frac{1}{3}$

Answer: $2\frac{1}{3}$

Example 7: Subtract $6\frac{1}{4} - 2\frac{2}{3}$

Step 1: Find the LCD of 4 and 3, which is 12. Convert the fractions:

$6\frac{1}{4} = 6\frac{3}{12} \qquad 2\frac{2}{3} = 2\frac{8}{12}$

Step 2: Since $\frac{3}{12} < \frac{8}{12}$, borrow 1 from 6:

$6\frac{3}{12} = 5\frac{15}{12}$

Step 3: Now subtract:

$5\frac{15}{12} - 2\frac{8}{12} = 3\frac{7}{12}$

Answer: $3\frac{7}{12}$

The Borrowing Process at a Glance

Here is a quick reference for how borrowing works:

Real-World Application: Carpentry

A carpenter has a board that is $8\frac{3}{4}$ feet long. She cuts off a piece measuring $3\frac{7}{8}$ feet. How much board is left?

Step 1: Find the LCD of 4 and 8, which is 8. Convert:

$8\frac{3}{4} = 8\frac{6}{8}$

Step 2: Since $\frac{6}{8} < \frac{7}{8}$, borrow 1 from 8:

$8\frac{6}{8} = 7\frac{14}{8}$

Step 3: Subtract:

$7\frac{14}{8} - 3\frac{7}{8} = 4\frac{7}{8} \text{ feet}$

The carpenter has $4\frac{7}{8}$ feet of board remaining. On a tape measure, that is $4$ feet $10\frac{1}{2}$ inches, which she could verify by measuring from the cut end.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Like denominators: subtract numerators and keep the denominator
• Unlike denominators: find the LCD, convert, then subtract
• Borrowing: when the first fraction is smaller, borrow 1 from the whole number and convert it to a fraction with the same denominator
• Always simplify your final answer
• Double-check by adding: your answer plus the number you subtracted should equal the number you started with

Return to Arithmetic for more foundational math topics.