How to Add, Subtract, Multiply and Divide Fractions

Fractions show up everywhere outside the classroom: half a cup of flour, a board cut to 2 3/8 inches, a three-quarter share of a bill. The four basic operations follow just a few rules, and once you see why each rule works, you will rarely need to look them up again. This guide walks through all four with real examples and points out the mistakes people make most often.

A quick refresher on the parts

A fraction like 3/4 has a numerator (3, the number of parts you have) and a denominator (4, the number of equal parts the whole is split into). A mixed number such as 2 1/3 combines a whole number with a fraction. An improper fraction has a numerator larger than its denominator, like 7/3, which is the same amount as 2 1/3.

To turn a mixed number into an improper fraction, multiply the whole number by the denominator and add the numerator: 2 1/3 = (2 × 3 + 1) / 3 = 7/3. Doing this first makes every operation easier, because you only have to deal with one kind of number.

Adding fractions

You can only add fractions directly when they count the same size of piece, which means they need the same denominator. Thirds and quarters are different-sized pieces, so you first rewrite both in a shared unit.

Example: 2 1/3 + 1 3/4

1. Convert to improper fractions: 2 1/3 = 7/3 and 1 3/4 = 7/4.
2. Find the least common denominator, the smallest number both 3 and 4 divide into: 12.
3. Rewrite each fraction over 12: 7/3 = 28/12 and 7/4 = 21/12.
4. Add the numerators and keep the denominator: 28/12 + 21/12 = 49/12.
5. Check whether it simplifies. 49 and 12 share no common factor, so 49/12 is already in lowest terms.
6. Convert back to a mixed number if you like: 49 ÷ 12 = 4 remainder 1, so the answer is 4 1/12, or about 4.0833 as a decimal.

Subtracting fractions

Subtraction works exactly the same way, with a minus instead of a plus.

Example: 5/6 − 1/4

1. The least common denominator of 6 and 4 is 12.
2. Rewrite: 5/6 = 10/12 and 1/4 = 3/12.
3. Subtract the numerators: 10/12 − 3/12 = 7/12.

You could also use 24 (6 × 4) as the shared denominator. You would get 20/24 − 6/24 = 14/24, which simplifies to 7/12. The answer is the same; the least common denominator just keeps the numbers smaller and saves a simplifying step.

Multiplying fractions

Multiplication is the easiest operation because you do not need a common denominator. Multiply the numerators together and the denominators together.

Example: 3/4 × 2/3

1. Multiply across: (3 × 2) / (4 × 3) = 6/12.
2. Simplify by dividing top and bottom by their greatest common divisor, 6: 1/2.

A useful shortcut is to cancel before you multiply. The 3 on top and the 3 on the bottom cancel, leaving 1/4 × 2/1 = 2/4 = 1/2. Canceling first keeps the numbers small, which matters when the fractions are larger.

To multiply by a whole number, write it as a fraction over 1. For instance, 1/3 × 3 = 1/3 × 3/1 = 3/3 = 1.

Dividing fractions

Dividing by a fraction is the same as multiplying by its reciprocal, which is the fraction flipped upside down. The reason is simple: asking how many 3/4s fit into something is the same as multiplying it by 4/3.

Example: 2 1/2 ÷ 3/4

1. Convert the mixed number: 2 1/2 = 5/2.
2. Flip the second fraction and multiply: 5/2 × 4/3 = 20/6.
3. Simplify by dividing by 2: 10/3.
4. As a mixed number that is 3 1/3.

A real-world reading: if you have 2 1/2 cups of rice and each serving uses 3/4 cup, you get 3 full servings with a third of a serving left over.

Using the Fraction Calculator

The Fraction Calculator does all of this for you and, more usefully, shows the steps. Enter each fraction in the whole, numerator and denominator boxes (leave the whole box empty for a simple fraction), choose the operation and press Calculate. The answer appears in lowest terms, as a mixed number and as a decimal, followed by a numbered list of steps that matches the method above. It is a good way to check homework, or to see where your own working went off track.

Common mistakes

• Adding the denominators. 1/4 + 1/4 is not 2/8. Two quarters make 2/4, which is 1/2. When you add, the denominator only tells you the size of the pieces; it does not change.
• Flipping the wrong fraction. When dividing, you flip the second fraction (the divisor), never the first.
• Forgetting to convert mixed numbers. Multiplying 2 1/2 × 3/4 by handling the whole and fraction parts separately gives the wrong answer. Convert to 5/2 first.
• Losing the sign on negative mixed numbers. -1 1/2 means -(1 + 1/2) = -3/2, not -1 + 1/2. In the calculator, put the minus sign on the whole number only.
• Stopping before simplifying. 20/6 is correct but not finished. Teachers and recipes usually expect 10/3 or 3 1/3.

Good to know

Some fractions have exact decimal forms and some do not. A fraction in lowest terms has a terminating decimal only when its denominator has no prime factors other than 2 and 5. So 1/2 = 0.5 and 3/8 = 0.375 are exact, while 1/3, 7/12 and 49/12 repeat forever. When the decimal repeats, the calculator shows a rounded value, but the fraction itself is exact, so keep working in fractions when precision matters.

FAQ

Why can't I just add the denominators?

Because the denominator describes the size of each piece, not how many pieces you have. Adding 1/4 and 1/4 gives you two pieces that are each a quarter, so the result is 2/4. Adding the denominators would claim the pieces suddenly became eighths, which is half as big.

Do I need the least common denominator, or will any common denominator work?

Any common denominator gives the right answer. Multiplying the two denominators together always works. The least common denominator simply keeps the numbers smaller, so there is less simplifying at the end.

Why does 1/3 show as a rounded decimal?

One third equals 0.3333… with the 3 repeating forever, so any decimal display has to stop somewhere. The fraction 1/3 is the exact value; the decimal is an approximation.

How do I divide a fraction by a whole number?

Write the whole number over 1 and then flip it. For example, 3/4 ÷ 2 = 3/4 × 1/2 = 3/8. In other words, dividing by 2 is the same as taking half.