Adding and Subtracting Fractions

Fractions with Like Denominators

When adding or subtracting fractions with the same denominator, we simply add or subtract the numerators and keep the denominator the same.

Examples:

• 1/4 + 2/4 = 3/4
• 5/6 - 2/6 = 3/6 = 1/2 (simplified)
• 3/8 + 4/8 = 7/8

Fractions with Unlike Denominators

You cannot add 1/2 + 1/3 and write 2/5. The denominator tells you the size of each piece, and pieces of different sizes cannot be counted together until you cut them to a common size. That is the whole reason for the extra steps below.

When adding or subtracting fractions with different denominators, we need to:

1. Find a common denominator
2. Convert each fraction to an equivalent fraction with the common denominator
3. Add or subtract the numerators
4. Simplify the result if possible

Example: 1/3 + 1/4

1. Find common denominator: 12
2. Convert fractions:
   • 1/3 = 4/12
   • 1/4 = 3/12
3. Add numerators: 4/12 + 3/12 = 7/12

Finding Common Denominators

The smallest number both denominators divide into evenly is called the least common denominator, or LCD. It is the same thing as the least common multiple of the two denominators. There are two main methods to find a common denominator:

1. Least Common Multiple (LCM)

The LCM of two numbers is the smallest number that is a multiple of both numbers. This is the preferred method as it results in simpler calculations.

Example: Find LCM of 4 and 6

• Multiples of 4: 4, 8, 12, 16, 20, ...
• Multiples of 6: 6, 12, 18, 24, ...
• LCM = 12

2. Product Method

Multiply the denominators together. This always works but may result in larger numbers that need more simplification.

Example: 1/3 + 1/4

• Common denominator = 3 × 4 = 12
• 1/3 = 4/12
• 1/4 = 3/12
• 4/12 + 3/12 = 7/12

3. The GCD Shortcut for Larger Denominators

Listing multiples gets slow once the denominators are big. Use this formula instead: LCD = (a × b) ÷ GCD(a, b).

Example: 5/12 + 7/18

• GCD of 12 and 18 is 6
• LCD = (12 × 18) ÷ 6 = 216 ÷ 6 = 36
• Convert: 5/12 = 15/36 and 7/18 = 14/36
• Add: 15/36 + 14/36 = 29/36

Our LCM and GCF Calculator will find the LCD for you if you only need that step, and the Fraction Calculator shows the full working for any addition or subtraction.

Mixed Numbers

When adding or subtracting mixed numbers:

1. Add or subtract the whole numbers
2. Add or subtract the fractions
3. If the fraction result is improper, convert to a mixed number and add to the whole number

Example: 2 1/3 + 1 1/4

1. Add whole numbers: 2 + 1 = 3
2. Add fractions: 1/3 + 1/4 = 4/12 + 3/12 = 7/12
3. Final answer: 3 7/12

When the Fractions Add Up to More Than 1 (Carrying)

Example: 2 3/4 + 1 2/3

1. Whole numbers: 2 + 1 = 3
2. Fractions: 3/4 + 2/3 = 9/12 + 8/12 = 17/12
3. 17/12 is improper, so convert it: 17/12 = 1 5/12
4. Carry that whole into the total: 3 + 1 5/12 = 4 5/12

When the Fraction Part Would Go Negative (Borrowing)

Example: 3 1/4 - 1 3/4. Subtracting 3/4 from 1/4 goes negative, so borrow 1 from the whole number first, exactly like borrowing in column subtraction.

1. Borrow: 3 1/4 becomes 2 5/4, because you add 4/4 to the 1/4
2. Subtract: 2 5/4 - 1 3/4 = 1 2/4
3. Simplify: 1 2/4 = 1 1/2

Practice Tips

• Always simplify your final answer
• Check if your answer makes sense (e.g., is it reasonable?)
• When subtracting mixed numbers, check whether the fraction part goes negative; if it does, borrow
• Use visual models to help understand the process
• Practice with both proper and improper fractions

Frequently Asked Questions