How do you add fractions with different denominators?

Find the least common denominator (LCD) of both fractions, convert each fraction to an equivalent fraction using that denominator, then add the numerators. Keep the denominator the same and simplify the result. For example, 1/2 + 1/3: the LCD is 6, so convert to 3/6 + 2/6 = 5/6.

How do you subtract fractions with different denominators?

The method is identical to addition. Find the LCD, convert both fractions to equivalent fractions with that denominator, then subtract the numerators. For 3/4 - 1/3: the LCD is 12, so convert to 9/12 - 4/12 = 5/12.

What is the least common denominator?

The least common denominator (LCD) is the smallest number that both denominators divide into evenly. It is the same as the least common multiple (LCM) of the two denominators. For 1/4 and 1/6, the LCD is 12 because 12 is the smallest number divisible by both 4 and 6.

Do you always need to find the LCD, or can you use any common denominator?

You can use any common denominator and still get the correct answer. Using the LCD just means your numbers stay smaller and you may need less simplifying at the end. For 1/4 + 1/6, you could use 24 as the denominator (6/24 + 4/24 = 10/24 = 5/12) and it still works.

How to Add and Subtract Fractions (With Unlike Denominators) | ILoveFractions

Adding fractions with the same denominator is simple: add the numerators, keep the denominator. 2/7 + 3/7 = 5/7. Done.

The part that trips people up is unlike denominators. You cannot just add 1/2 + 1/3 and write 2/5. The denominators are telling you the size of each piece, and you can’t add pieces of different sizes until you give them the same size.

The Method

Step 1: Find the least common denominator (LCD).

The LCD is the smallest number that both denominators divide into evenly. It is the least common multiple (LCM) of the two denominators.

For 1/2 + 1/3:

Multiples of 2: 2, 4, 6, 8, 10…
Multiples of 3: 3, 6, 9, 12…
LCD = 6

Step 2: Convert each fraction to an equivalent fraction using the LCD.

Divide the LCD by each denominator to find the multiplier, then multiply both numerator and denominator by it.

1/2: LCD ÷ 2 = 3, so multiply by 3: 1×3 / 2×3 = 3/6
1/3: LCD ÷ 3 = 2, so multiply by 2: 1×2 / 3×2 = 2/6

Step 3: Add (or subtract) the numerators. Keep the denominator.

3/6 + 2/6 = 5/6

Step 4: Simplify if needed.

5/6 cannot be simplified further. Done.

A Subtraction Example

3/4 - 1/6

LCD of 4 and 6: multiples of 4 are 4, 8, 12; multiples of 6 are 6, 12. LCD = 12.
Convert: 3/4 = 9/12, 1/6 = 2/12
Subtract: 9/12 - 2/12 = 7/12
Check: 7 and 12 share no common factors. Done.

Adding Mixed Numbers

When you have mixed numbers, handle the whole numbers and fractions separately, then combine.

2 1/2 + 1 1/3

Add whole numbers: 2 + 1 = 3
Add fractions: 1/2 + 1/3 = 3/6 + 2/6 = 5/6
Combine: 3 + 5/6 = 3 5/6

If the fractions add up to more than 1, you need to carry. For example, 2 3/4 + 1 2/3:

Whole numbers: 2 + 1 = 3
Fractions: 3/4 + 2/3 = 9/12 + 8/12 = 17/12
Convert 17/12 to a mixed number: 1 5/12
Add to whole number total: 3 + 1 5/12 = 4 5/12

Subtracting Mixed Numbers (With Borrowing)

3 1/4 - 1 3/4

The fraction part 1/4 - 3/4 would go negative, so borrow 1 from the whole number:

Borrow: 3 1/4 becomes 2 5/4 (add 4/4 to 1/4)
Subtract: 2 5/4 - 1 3/4 = 1 2/4 = 1 1/2

Finding the LCD for Harder Numbers

For larger denominators, listing multiples gets slow. Use the formula:

LCD = (a × b) ÷ GCD(a, b)

For 5/12 + 7/18:

GCD(12, 18): factors of 12 are 1, 2, 3, 4, 6, 12; factors of 18 are 1, 2, 3, 6, 9, 18. GCD = 6.
LCD = (12 × 18) ÷ 6 = 216 ÷ 6 = 36
Convert: 5/12 = 15/36, 7/18 = 14/36
Add: 15/36 + 14/36 = 29/36

Use our Fraction Calculator to check any addition or subtraction with full step-by-step working, or our LCM and GCF Calculator if you just need the LCD.