Adding Fractions with Different Denominators

To add fractions with different denominators:

1. Find the least common denominator (LCD), the least common multiple of the denominators.
2. Rewrite each fraction with that denominator.
3. Add the numerators and keep the denominator.
4. Simplify the result.

Example: 1/4 + 1/6. The LCD of 4 and 6 is 12, so 1/4 = 3/12 and 1/6 = 2/12. Adding gives 5/12.

Why you need a common denominator

A denominator names the size of the pieces: quarters, sixths, twelfths. You can only add counts of the same-sized piece, just as 3 inches plus 2 feet is not "5 of something" until you convert to one unit. Rewriting 1/4 and 1/6 as twelfths turns them into the same kind of piece, and then you can count them: 3 twelfths plus 2 twelfths is 5 twelfths.

Step 1: finding the LCD

The LCD is the least common multiple of the denominators. List multiples of each until they meet:

• 4: 4, 8, 12, 16…
• 6: 6, 12, 18…

For larger denominators, prime factorization is faster: take every prime factor at its highest power. That method, and how the LCM differs from the GCF, is explained in GCF vs. LCM.

Any common multiple works. You could use 4 × 6 = 24: 6/24 + 4/24 = 10/24. But you would then have to simplify 10/24 to 5/12. Using the least common denominator keeps the numbers small.

Step 2: rewriting each fraction

Ask what the old denominator must be multiplied by to reach the LCD, then multiply the top by the same number:

• 1/4 → 4 × 3 = 12, so 1 × 3 = 3: 3/12
• 1/6 → 6 × 2 = 12, so 1 × 2 = 2: 2/12

Multiplying the top and bottom by the same number does not change the value, because you are multiplying by a form of 1 (3/3, 2/2).

More examples

Three fractions

1/4 + 1/6 + 2/3: the LCD of 4, 6 and 3 is 12.

3/12 + 2/12 + 8/12 = 13/12, an improper fraction, which is 1 1/12.

Subtraction

5/6 − 3/8: the LCD of 6 and 8 is 24.

20/24 − 9/24 = 11/24.

Mixed numbers

2 1/3 + 1 3/4: convert to improper fractions first. 7/3 + 7/4, LCD 12 → 28/12 + 21/12 = 49/12 = 4 1/12.

When one denominator divides the other

1/6 + 1/3: 6 is already a multiple of 3, so the LCD is 6. 1/6 + 2/6 = 3/6 = 1/2, simplified by dividing top and bottom by 3. (Simplifying works exactly like simplifying a ratio: divide by the greatest common factor.)

Common mistakes

• Adding the denominators. 1/4 + 1/6 is not 2/10. The denominator names the piece size and does not get added.
• Changing only the denominator. If 4 becomes 12, the numerator must be multiplied by 3 too.
• Forgetting to simplify. 10/24 is correct but not in lowest terms. Divide by the GCF to finish.

Checking your work with CalcTower

The Fraction Calculator adds, subtracts, multiplies and divides fractions and mixed numbers, and returns the simplified result with its working. For 1/4 + 1/6 it returns 5/12. To find the LCD on its own, for several denominators at once, enter them in the LCM Calculator. For 4, 6 and 3 it returns 12.