How to Add and Subtract Fractions with Different Denominators
To add or subtract fractions with different denominators, find a common denominator (ideally the least common denominator, or LCD), rewrite both fractions over it, then add or subtract the numerators. The denominator itself never changes during that last step.
This comes up constantly outside a math class: doubling a recipe that calls for 1/4 cup and 1/6 cup of something, cutting a board when two measurements are marked in different fractions of an inch, or figuring out how much time is left when a task took 2 1/3 hours out of a planned 4. In every case, the fractions can’t be combined until they’re speaking the same denominator.
Step-by-step method
- Find the LCD of the two denominators. This is the least common multiple (LCM) of the two numbers, the smallest number both denominators divide into evenly.
- Convert each fraction to an equivalent fraction with that denominator. Multiply the numerator and denominator of each fraction by whatever factor gets it to the LCD.
- Add or subtract the numerators, and keep the denominator as is. Once both fractions share a denominator, only the top numbers change.
- Simplify the result. Divide the numerator and denominator by their greatest common divisor (GCD) to reduce the fraction to lowest terms.
That’s the whole method. The two examples below show it with real numbers, one addition and one subtraction involving mixed numbers.
Finding the LCD when it’s not obvious
For small denominators like 4 and 6, most people can just see that 12 works. For bigger numbers, two reliable ways to find the LCD without guessing:
List the multiples. Write out multiples of each denominator until one appears in both lists. For 8 and 12: multiples of 8 are 8, 16, 24, 32; multiples of 12 are 12, 24, 36. The first match is 24, so that’s the LCD.
Use prime factorization. Break each denominator into prime factors, then take the highest power of every prime that appears. For 8 and 12: 8 = 2³, and 12 = 2² × 3. Take 2³ (the higher power of 2) and 3¹, and multiply: 2³ × 3 = 24. Same answer, and it scales better once the numbers get large or there are more than two of them.
Either method gets you to the same place. Listing multiples is faster for small numbers you can do in your head; prime factorization is more reliable once the denominators stop being friendly.
Worked example 1: adding 1/4 + 1/6
The denominators are 4 and 6. The LCD is 12, since 12 is the smallest number both 4 and 6 divide into evenly.
- 1/4 = 3/12 (multiply top and bottom by 3)
- 1/6 = 2/12 (multiply top and bottom by 2)
Add the numerators: 3/12 + 2/12 = 5/12. Since 5 and 12 share no common factor other than 1, the fraction is already in lowest terms. The answer is 5/12.
Worked example 2: subtracting mixed numbers, 2 1/3 − 1 3/4
Mixed numbers need one extra step before you touch the denominators: convert each one to an improper fraction.
- 2 1/3 = (2 × 3 + 1) / 3 = 7/3
- 1 3/4 = (1 × 4 + 3) / 4 = 7/4
Now find the LCD of 3 and 4, which is 12, and convert both fractions:
- 7/3 = 28/12
- 7/4 = 21/12
Subtract: 28/12 − 21/12 = 7/12. That’s the answer, already in lowest terms.
As a sanity check, convert both mixed numbers to decimals first: 2 1/3 ≈ 2.3333 and 1 3/4 = 1.75. The difference is 0.58333, and 7/12 also works out to 0.58333 as a decimal. The two methods agree, so the fraction answer holds up.
Converting to improper fractions before subtracting matters more than it looks. If you tried to subtract the fractional parts on their own, 1/3 − 3/4, you’d get a negative fraction, because 3/4 is bigger than 1/3. You’d then have to borrow a whole unit from the 2 and convert it into extra thirds before the subtraction works, the same kind of borrowing you’d do subtracting 21 from 32 in regular arithmetic. Converting both numbers to improper fractions first sidesteps all of that: there’s exactly one subtraction to do, and it’s always a straightforward one.
Common denominators at a glance
| Denominators | LCD |
|---|---|
| 2 and 3 | 6 |
| 3 and 4 | 12 |
| 4 and 6 | 12 |
| 5 and 10 | 10 |
| 6 and 9 | 18 |
| 8 and 12 | 24 |
Notice that the LCD isn’t always the product of the two numbers. 4 and 6 multiply to 24, but their LCD is only 12, since both share a factor of 2. The same happens with 5 and 10: the LCD is just 10, because 5 already divides evenly into it.
Add or subtract your own fractions
Common mistakes
- Adding or subtracting the denominators directly. 1/2 + 1/3 is not 2/5. Adding numerators and denominators separately ignores what a fraction actually represents; the correct answer, once both fractions share the denominator 6, is 3/6 + 2/6 = 5/6.
- Forgetting to convert mixed numbers to improper fractions first. Trying to combine whole numbers and fractional parts separately works some of the time, but it breaks the moment a subtraction requires borrowing, and it’s easy to make an error that way.
- Forgetting to simplify the final answer. An answer like 6/8 is correct but not finished; it should be reduced to 3/4 by dividing top and bottom by their GCD, 2.
- Defaulting to the product of the denominators instead of the true LCD. Multiplying 4 × 6 to get 24 as a common denominator for 1/4 and 1/6 works, and you’d get 6/24 + 4/24 = 10/24, but that’s more simplifying at the end than if you’d used the LCD of 12 in the first place. It’s not wrong, just extra work.
FAQ
What if the two fractions already have the same denominator? Then you can skip the LCD step entirely. Just add or subtract the numerators and keep the denominator, for example 3/8 + 2/8 = 5/8.
Do I need to convert mixed numbers before subtracting them? Yes, especially when the second number’s fractional part is larger than the first’s. Convert both to improper fractions before subtracting, or you’ll end up with a negative fractional part and have to borrow from the whole number to fix it.
Is there a faster method than finding the least common denominator? Cross-multiplication is quicker to set up: a/b ± c/d = (ad ± cb) / bd. It skips the step of finding the LCM, but the resulting denominator, bd, is not always the smallest one possible, so you may need an extra simplification step at the end that the LCD method would have avoided.
How do I turn my fraction answer into a decimal? Divide the numerator by the denominator. For 5/12, that’s 5 ÷ 12 = 0.41666…, a repeating decimal. Useful when you need to enter the value into a calculator or spreadsheet that doesn’t accept fractions directly, or when you’re comparing a fraction against a measurement already given in decimal form.