Fractions
How to Add Fractions
To add fractions with the same denominator, add the numerators and keep the denominator. To add fractions with different denominators, first rewrite them over a common denominator, then add the numerators. This guide covers both cases, how to find the lowest common denominator quickly, adding three fractions, adding mixed numbers, and subtracting fractions, which uses the same method.
How to add fractions
Adding fractions takes two steps at most. First make sure the denominators match. Then add the numerators, keep the shared denominator, and simplify.
The denominator tells you the size of the pieces, and the numerator tells you how many pieces you have. Adding counts pieces, so the pieces have to be the same size before you can count them together. That is the only reason a common denominator is needed, and it is why the denominator itself is never added.
To check any answer, the fraction calculator adds two fractions, mixed numbers or whole numbers and shows the common denominator it used.
Adding fractions with the same denominator
When the denominators already match, there is nothing to prepare.
Example: 2/9 + 5/9
Two ninths plus five ninths is seven ninths, the same way two apples plus five apples is seven apples. “Ninths” is the unit, and the unit does not change when you add.
Example: 3/8 + 7/8
The sum is more than one whole, so it comes out as an improper fraction. Simplify it, then write it as a mixed number if that is the form you need. The improper fraction calculator does that conversion.
Adding fractions with different denominators
When the denominators differ, rewrite each fraction so they share one. Multiplying the top and the bottom of a fraction by the same number gives an equivalent fraction with the same value, so you are changing how the fraction is written, not what it is worth.
- Find the lowest common denominator (LCD): the smallest number every denominator divides into.
- For each fraction, work out what its denominator must be multiplied by to reach the LCD. Multiply the numerator by the same number.
- Add the numerators. Keep the LCD as the denominator.
- Simplify, and convert to a mixed number if needed.
Example: 2/3 + 1/4
3 and 4 share no factor, so the LCD is 3 × 4 = 12.
Example: 1/6 + 3/8
Multiples of 8: 8, 16, 24. The first one 6 divides is 24.
Multiplying the denominators would give 48, which also works but leads to 26/48 and an extra simplifying step.
Example: 3/10 + 5/6
10 = 2 × 5 and 6 = 2 × 3, so the LCD is 2 × 3 × 5 = 30.
Finding the common denominator quickly
Three checks cover most problems:
- One denominator divides the other. Use the larger one. For 1/4 + 3/8, the LCD is 8.
- The denominators share no factor. Multiply them. For 2/5 + 1/3, the LCD is 15.
- Otherwise, count up in multiples of the largest denominator until the others divide in. For 1/6 + 1/9: 9, 18. The LCD is 18.
For larger or awkward denominators, the least common denominator calculator finds the LCD of up to ten fractions and rewrites each one over it. The LCD is the least common multiple of the denominators, so the LCM calculator and the guide on how to find the GCF and LCM use the same method.
The cross-multiplication shortcut
For exactly two fractions there is a formula that skips finding the LCD:
The answer matches, but 68/60 needed reducing, while the LCD method reached 34/30 with smaller numbers. Use the shortcut when the denominators share no factor, because then b × d is the LCD anyway and no simplifying is lost.
Why you cannot add the denominators
The most common wrong answer adds tops and bottoms: 1/2 + 1/3 = 2/5. A quick size check shows this cannot be right. 2/5 is 0.4, which is less than 1/2. Adding a positive amount to 1/2 cannot produce something smaller than 1/2.
The right answer is 5/6. Halves and thirds are different-sized pieces. Cut both into sixths, which fit evenly into each, and a half becomes three sixths while a third becomes two sixths. Now the pieces match, and three sixths plus two sixths is five sixths.
Adding three or more fractions
Find one denominator that all of them share, rewrite every fraction over it, and add all the numerators in one go.
You can also add two at a time. 1/2 + 1/3 = 5/6, then 5/6 + 1/4 = 10/12 + 3/12 = 13/12. Same answer, slightly more writing.
Adding mixed numbers
There are two good methods. Both give the same answer.
Method 1: wholes and fractions separately
Add the whole numbers, add the fractions, then combine. If the fractions add up to more than 1, carry the extra whole.
Method 2: improper fractions
Method 1 keeps the numbers smaller, which helps in your head. Method 2 never needs a carry, which helps when the numbers are awkward. The mixed number calculator converts in both directions.
Adding the parts separately is valid for addition, and only for addition. Multiplying mixed numbers part by part gives wrong answers, which is why how to multiply fractions always converts to improper fractions first.
Adding a whole number and a fraction
Just write them together: 3 + 2/5 = 3 2/5. As an improper fraction that is 15/5 + 2/5 = 17/5.
Adding and subtracting fractions
Subtraction uses exactly the same method. Match the denominators, then subtract the numerators instead of adding them.
When a problem mixes both operations, rewrite everything over one common denominator and work left to right: 3/4 + 1/6 − 1/3 = 9/12 + 2/12 − 4/12 = 7/12.
Subtracting mixed numbers with borrowing
When the fraction you are subtracting is bigger than the one you are subtracting from, borrow 1 from the whole part.
Converting to improper fractions avoids borrowing altogether: 21/5 − 5/3 = 63/15 − 25/15 = 38/15 = 2 8/15.
When the second fraction is larger, the answer is negative: 1/3 − 3/4 = 4/12 − 9/12 = -5/12. The negative number rules apply to the numerators exactly as they do to whole numbers. The subtracting fractions calculator shows each of these steps for any pair you enter.
Sums of common fractions
| Problem | LCD | Rewritten | Sum | Decimal |
|---|---|---|---|---|
| 1/2 + 1/4 | 4 | 2/4 + 1/4 | 3/4 | 0.75 |
| 1/2 + 1/3 | 6 | 3/6 + 2/6 | 5/6 | ≈ 0.8333 |
| 1/3 + 1/4 | 12 | 4/12 + 3/12 | 7/12 | ≈ 0.5833 |
| 2/3 + 1/6 | 6 | 4/6 + 1/6 | 5/6 | ≈ 0.8333 |
| 3/4 + 1/8 | 8 | 6/8 + 1/8 | 7/8 | 0.875 |
| 1/4 + 1/6 | 12 | 3/12 + 2/12 | 5/12 | ≈ 0.4167 |
| 2/5 + 1/3 | 15 | 6/15 + 5/15 | 11/15 | ≈ 0.7333 |
| 1/6 + 1/9 | 18 | 3/18 + 2/18 | 5/18 | ≈ 0.2778 |
| 7/10 + 1/4 | 20 | 14/20 + 5/20 | 19/20 | 0.95 |
| 5/12 + 3/8 | 24 | 10/24 + 9/24 | 19/24 | ≈ 0.7917 |
| 3/4 + 2/3 | 12 | 9/12 + 8/12 | 1 5/12 | ≈ 1.4167 |
| 5/8 + 3/4 | 8 | 5/8 + 6/8 | 1 3/8 | 1.375 |
To check a sum, convert each fraction to a decimal and add those. 3/4 + 2/3 is 0.75 + 0.6667, about 1.4167, which matches 1 5/12. The fraction to decimal chart lists the common values.
Common mistakes
- Adding the denominators. 1/2 + 1/3 is 5/6, not 2/5. The denominator names the piece size and stays the same.
- Changing the denominator without changing the numerator. 2/3 is 8/12, not 2/12. Whatever multiplies the bottom must multiply the top.
- Using a common denominator that is not common. Check that every denominator divides into it evenly.
- Forgetting to simplify. 34/30 should become 17/15, or 1 2/15. Use the simplify fractions calculator or the steps in how to simplify fractions.
- Dropping the carry with mixed numbers. If the fraction parts sum to 17/12, that is 1 5/12, and the extra 1 joins the whole numbers.
- Borrowing 1 as 10. When subtracting mixed numbers, the borrowed 1 becomes 15/15 or 8/8, matching the denominator, not 10.
- Mixing decimals and fractions halfway. Convert everything to one form first. The decimal to fraction calculator turns a decimal like 0.25 into 1/4.
Frequently asked questions
What is 1/2 + 1/4?
3/4. Since 2 divides into 4, the common denominator is 4. Rewrite 1/2 as 2/4, then add the numerators: 2/4 + 1/4 = 3/4. As a decimal that is 0.5 + 0.25 = 0.75. Only 1/2 needed changing, because 1/4 already had the right denominator.
What is 1/3 + 1/6?
1/2. The LCD of 3 and 6 is 6, so rewrite 1/3 as 2/6. Then 2/6 + 1/6 = 3/6, and 3/6 simplifies to 1/2 by dividing both numbers by 3. This is a good example of why you should always check the sum for a common factor before calling it finished.
How do you add fractions with different denominators?
Rewrite the fractions over a common denominator, add the numerators, and keep the denominator. For 2/3 + 1/4, the lowest common denominator is 12, so the fractions become 8/12 and 3/12, and the sum is 11/12. Multiply the numerator by whatever you multiplied the denominator by, so each fraction keeps its value.
Do you add the denominators when adding fractions?
No. Once the denominators match you add only the numerators, and the denominator stays the same. Two ninths plus five ninths is seven ninths, not seven eighteenths. Adding the denominators gives an answer smaller than it should be: 1/2 + 1/3 would come out as 2/5, which is less than 1/2 alone.
How do you add mixed numbers?
Add the whole numbers, add the fractions using a common denominator, then combine them, carrying any extra whole. For 2 3/4 + 1 2/3, the wholes give 3 and the fractions give 17/12, which is 1 5/12, so the total is 4 5/12. Converting both to improper fractions first gives the same answer.
How do you add and subtract fractions in the same problem?
Rewrite every fraction over one common denominator, then add and subtract the numerators from left to right. For 3/4 + 1/6 − 1/3 the LCD is 12, giving 9/12 + 2/12 − 4/12 = 7/12. Doing it in one pass saves you from finding a new denominator after each step.
What is 3/5 + 1/2?
1 1/10. The denominators 5 and 2 share no factor, so the LCD is 10. Rewrite the fractions as 6/10 and 5/10, then add: 6/10 + 5/10 = 11/10. That is one whole and one tenth, or 1.1 as a decimal, which matches 0.6 + 0.5.
Can you add a fraction and a decimal?
Yes, once both are in the same form. Either turn the decimal into a fraction, so 0.25 + 1/3 = 1/4 + 1/3 = 7/12, or turn the fraction into a decimal, giving 0.25 + 0.3333 ≈ 0.5833. The fraction route is exact; the decimal route needs rounding when the fraction repeats, as thirds do.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/how-to-add-fractions/