How to add fractions

A fraction describes a part of a whole, written as a numerator (the top number) over a denominator (the bottom number). You can only add or subtract fractions once they share the same denominator. Everything in this guide follows from that one requirement, from the simplest same-denominator case through mixed numbers, multiplication, and division.

Why the denominators have to match

The denominator tells you the size of the pieces you're counting. Quarters and thirds are different sizes, so 1/4 + 1/3 isn't as simple as adding 1 and 1. Before you can combine them, you need to rewrite both fractions in terms of the same size piece, a common denominator, so the numerators are actually counting the same thing.

A useful picture: imagine cutting one pizza into 4 equal slices and an identical pizza into 3 equal slices. A slice from the first pizza and a slice from the second pizza aren't the same size, so you can't just say you have "2 slices." You'd need to cut both pizzas into the same number of pieces, say 12, before the slice counts mean the same thing and can be added together.

Same denominator: the easy case

If two fractions already share a denominator, add the numerators and keep the denominator as is: 2/7 + 3/7 = 5/7. The denominator never changes in this case, because you aren't changing the size of the pieces, only how many of them you have.

Different denominators: worked example

Solve 1/4 + 1/6.

1. Find the least common denominator of 4 and 6. List multiples of each: 4, 8, 12, 16... and 6, 12, 18... The smallest number that appears in both lists is 12.
2. Rewrite each fraction with denominator 12. 1/4 becomes 3/12(multiply top and bottom by 3), and 1/6 becomes 2/12 (multiply top and bottom by 2).
3. Add the numerators: 3/12 + 2/12 = 5/12.
4. Check that 5/12 can't be simplified further: the greatest common divisor of 5 and 12 is 1, so it's already in lowest terms.

Multiplying the top and bottom of a fraction by the same number doesn't change its value, only how it's written. 1/4 and 3/12 represent exactly the same amount. That's the entire trick behind finding a common denominator: you're rewriting the fractions, not changing what they equal.

Worked example: three fractions at once

Solve 1/2 + 1/3 + 1/6.

1. The denominators are 2, 3, and 6. Their least common denominator is 6, since 6 is a multiple of all three.
2. Rewrite each: 1/2 = 3/6, 1/3 = 2/6, and 1/6 is already correct.
3. Add: 3/6 + 2/6 + 1/6 = 6/6.
4. 6/6 simplifies to 1, since a fraction where the numerator equals the denominator is exactly 1 whole.

Subtracting fractions

Subtraction works exactly the same way. Solve 5/6 - 1/4:

1. Common denominator of 6 and 4 is 12.
2. 5/6 becomes 10/12, and 1/4 becomes 3/12.
3. Subtract: 10/12 - 3/12 = 7/12.

As with addition, the denominator stays fixed once it's common; only the numerators are combined, and here they're subtracted rather than added.

Simplifying the result

Once you have a single fraction, check whether the numerator and denominator share a common factor, and divide both by their greatest common divisor if they do. 6/8simplifies to 3/4, because 2 divides evenly into both 6 and 8, and 2 is their greatest common divisor. A fraction is fully simplified once the only number that divides evenly into both the numerator and denominator is 1.

A quick way to check for a common factor without listing every divisor: try 2, 3, 5, and 7 first, since most textbook fractions simplify using one of those. 18/24: both are even, so divide by 2 to get 9/12; both are still divisible by 3, so divide again to get 3/4, which is fully simplified.

Multiplying and dividing fractions

Multiplication and division don't need a common denominator at all, which surprises some people given how central common denominators are to addition and subtraction.

To multiply, multiply the numerators together and the denominators together:2/3 × 3/5 = 6/15, which simplifies to 2/5.

To divide, flip the second fraction and multiply: 2/3 ÷ 3/5 becomes2/3 × 5/3 = 10/9. Flipping the second fraction, called finding its reciprocal, is the standard way to turn a division problem into a multiplication problem.

A helpful way to remember which operation needs a common denominator: addition and subtraction are about combining quantities of the same-sized piece, so the pieces have to match first. Multiplication and division describe scaling, taking a fraction of a fraction, which doesn't depend on the pieces being the same size to begin with.

Mixed numbers

A mixed number like 1 1/2 combines a whole number and a fraction. Before adding or subtracting, convert it to an improper fraction (a fraction where the numerator is larger than the denominator): multiply the whole number by the denominator, add the numerator, and keep the same denominator. 1 1/2 becomes (1 × 2 + 1)/2 = 3/2.

Worked example: add 1 1/2 + 2 1/3.

1. Convert both to improper fractions: 1 1/2 = 3/2, and 2 1/3 = 7/3.
2. Common denominator of 2 and 3 is 6: 3/2 = 9/6, and 7/3 = 14/6.
3. Add: 9/6 + 14/6 = 23/6.
4. Convert back to a mixed number: 23 divided by 6 is 3 with a remainder of 5, so 23/6 = 3 5/6.

To convert an improper fraction back to a mixed number, divide the numerator by the denominator. The whole-number result is the leading number, and the remainder becomes the new numerator over the original denominator.

Worked example: subtracting mixed numbers

Solve 3 1/4 - 1 1/2.

1. Convert both: 3 1/4 = 13/4, and 1 1/2 = 3/2.
2. Common denominator of 4 and 2 is 4: 3/2 = 6/4.
3. Subtract: 13/4 - 6/4 = 7/4.
4. Convert back: 7/4 = 1 3/4.

Converting to improper fractions before subtracting avoids a common trap: trying to subtract the fraction parts separately and running into a fraction part that's too small, which forces an awkward borrowing step from the whole number. Working entirely in improper fractions sidesteps that.

Common mistakes

• Adding numerators and denominators straight across.1/2 + 1/3 is not 2/5. Fractions don't add that way; the denominators have to match first.
• Forgetting to convert the numerator when scaling a fraction. If you multiply the denominator by 3, you have to multiply the numerator by 3 too, or the value of the fraction changes.
• Leaving the answer unsimplified. 4/8 isn't wrong exactly, but most contexts expect the reduced form, 1/2.
• Skipping the mixed-number conversion. Trying to add whole numbers and fraction parts separately works sometimes, but it's easy to mishandle borrowing when subtraction is involved. Converting to improper fractions first avoids that entirely.
• Using a common denominator for multiplication or division. That step is unnecessary and often introduces an avoidable arithmetic error; multiply or find the reciprocal directly instead.

Where this shows up outside of math class

Fractions come up constantly outside school: doubling or halving a recipe that calls for 3/4 cup of an ingredient, figuring out how much of a project is finished when two people have each completed a different fraction of it, or splitting a length of material into unequal pieces measured in fractions of a foot. The same common-denominator process covered here applies directly in each case.

Practice it yourself

Try the fraction calculator for a quick answer, or head tofraction practice for an unlimited set of addition, subtraction, multiplication, and division problems with full step-by-step explanations.