How to Add and Subtract Fractions With Different Denominators

Why the denominators must match

The denominator tells you how big each piece is. Halves and thirds are different sizes, just as adding 2 metres to 3 feet makes no sense until you convert one of them. Adding the numerators and the denominators separately, as in 1/2 + 1/3 = 2/5, treats unlike pieces as if they were alike.

The fix is to rewrite both fractions using the same denominator. Multiplying the top and bottom of a fraction by the same number does not change its value, so 1/2 = 3/6 and 1/3 = 2/6. Now both are counted in sixths: 3/6 + 2/6 = 5/6.

Finding the least common denominator

Any common multiple of the two denominators works, but the smallest one, called the least common denominator (LCD), keeps the numbers small. It is the least common multiple of the denominators, and there are two easy ways to find it.

• List multiples. For 4 and 6: multiples of 4 are 4, 8, 12, 16; multiples of 6 are 6, 12, 18. The first match is 12.
• Use prime factors. 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3. Take each prime the greatest number of times it appears in either: 2 × 2 × 3 × 3 = 36.

If the denominators share no factors (such as 3 and 5), the LCD is simply their product, 15. If one denominator divides the other (such as 4 and 8), the LCD is the larger one.

Adding unlike fractions, step by step

1. Find the LCD of the denominators.
2. Rewrite each fraction with that denominator by multiplying its top and bottom by the same number.
3. Add the numerators and keep the denominator.
4. Simplify the result, and convert to a mixed number if it is improper.

Worked example 1: 3/4 + 5/6. The LCD of 4 and 6 is 12. Then 3/4 = 9/12 and 5/6 = 10/12. Add: 9/12 + 10/12 = 19/12, which is 1 7/12 as a mixed number. As a decimal, 3/4 + 5/6 = 0.75 + 0.8333… ≈ 1.5833, and 19/12 ≈ 1.5833, so the answer checks out.

Worked example 2: 5/12 + 7/18. The LCD is 36. Then 5/12 = 15/36 and 7/18 = 14/36, so the sum is 29/36. The numerator 29 is prime, so the fraction cannot be simplified.

Subtracting and working with mixed numbers

Subtraction follows the same steps, except you subtract the numerators. Worked example: 7/8 − 1/3. The LCD is 24. Then 7/8 = 21/24 and 1/3 = 8/24, so the difference is 13/24.

For mixed numbers, add the whole numbers and the fractions separately, then combine. Worked example: 2 1/3 + 1 3/4. The fractions become 4/12 and 9/12. The wholes add to 3 and the fractions add to 13/12, which is 1 1/12. The total is 3 + 1 1/12 = 4 1/12.

When subtracting mixed numbers and the first fraction is smaller than the second, borrow 1 from the whole number and add it to the fraction. For example, 3 1/4 − 1 1/2 becomes 2 5/4 − 1 2/4 = 1 3/4.

Simplifying the answer

A fraction is in lowest terms when its numerator and denominator share no common factor except 1. To simplify, divide both by their greatest common divisor (GCD). For 18/24, the GCD is 6, so 18/24 = 3/4.

Do the addition or subtraction first and simplify at the end. Simplifying along the way is allowed, but using the LCD from the start keeps every step mechanical and easy to check.

• Never add denominators together.
• Whatever you multiply the bottom by, multiply the top by the same number.
• Check the size: the sum of two positive fractions must be larger than each of them.

To find the common denominator quickly, use the LCD calculator, and to check a full calculation, try the fraction calculator. The simplify fractions calculator reduces any answer to lowest terms.