Fraction Arithmetic Step by Step

A fraction such as 3/4 has a numerator on top (3, how many parts you have) and a denominator underneath (4, how many equal parts make a whole). Everything in fraction arithmetic comes down to keeping those parts consistent.

A fraction is also just a division written another way: 3/4 means 3 ÷ 4. That is why the denominator can never be zero, and why fractions bigger than 1, such as 9/4, are perfectly valid. They are called improper fractions.

Adding and subtracting: find a common denominator

You can only add or subtract fractions when the pieces are the same size, meaning the denominators match. If they do not, rewrite each fraction with a common denominator, often the product or the lowest common multiple of the two denominators.

Example 1: 1/2 + 1/3. A common denominator is 6. 1/2 = 3/6 and 1/3 = 2/6, so the sum is 3/6 + 2/6 = 5/6.

Example 2: 3/4 − 1/6. The lowest common denominator is 12. 3/4 = 9/12 and 1/6 = 2/12, so the difference is 9/12 − 2/12 = 7/12.

Only the numerators are added or subtracted. The denominator stays the same. The most frequent mistake is adding both tops and both bottoms: 1/2 + 1/3 is not 2/5. A quick reality check shows why: 2/5 is less than a half, yet you added a third to a half.

Tip: if the denominators share a factor, use the lowest common multiple instead of their product. For 1/4 + 1/6, the common denominator 12 is smaller than 24 and keeps the numbers small: 3/12 + 2/12 = 5/12.

Multiplying fractions

Multiplication is easier: multiply the numerators together and the denominators together. No common denominator needed.

Example: 2/3 × 9/10 = 18/30. Simplify by dividing top and bottom by 6 to get 3/5. You can also cancel first: the 9 and the 3 share a factor of 3, and the 2 and the 10 share a factor of 2, giving (1 × 3) ÷ (1 × 5) = 3/5 directly.

Taking a fraction of a quantity is the same operation: 3/5 of 40 is 40 ÷ 5 × 3 = 24. A whole number can be written over 1 (4 = 4/1), so 4 ÷ 2/3 = 4/1 × 3/2 = 12/2 = 6.

Multiplying by a fraction smaller than 1 makes a number smaller. A recipe that needs 3/4 of a cup, scaled by 2/3, needs 3/4 × 2/3 = 6/12 = 1/2 cup.

Dividing fractions

To divide by a fraction, flip it (take the reciprocal) and multiply.

Example: 3/5 ÷ 9/10 = 3/5 × 10/9 = 30/45. The highest common factor of 30 and 45 is 15, so the answer is 2/3.

Why does flipping work? Dividing by 9/10 asks how many 9/10s fit into a number, and multiplying by 10/9 gives exactly that answer. In practice, remember: keep the first fraction, flip the second, then multiply.

Simplifying fractions

A fraction is in its simplest form when the top and bottom share no common factor except 1. Divide both by their greatest common factor (GCF).

List or find the factors of both numbers. For 18 and 48 the common factors are 1, 2, 3 and 6.
Pick the greatest, which is 6.
Divide: 18 ÷ 6 = 3 and 48 ÷ 6 = 8.

So 18/48 simplifies to 3/8. If you are not sure of the GCF, divide by any common factor and repeat until nothing more divides evenly. Quick tests help: even numbers divide by 2, a number whose digits add up to a multiple of 3 divides by 3, and a number ending in 0 or 5 divides by 5. The fraction calculator always returns the simplified result.

Mixed numbers

A mixed number such as 2 1/4 combines a whole number with a fraction. For arithmetic, convert it to an improper fraction first: multiply the whole number by the denominator and add the numerator. 2 1/4 = (2 × 4 + 1)/4 = 9/4.

Example: 2 1/4 + 1 5/6.

Convert: 2 1/4 = 9/4 and 1 5/6 = 11/6.
Common denominator 12: 9/4 = 27/12 and 11/6 = 22/12.
Add: 27/12 + 22/12 = 49/12.
Convert back: 49 ÷ 12 = 4 remainder 1, so 4 1/12.

Subtraction works the same way. For 5 1/3 − 2 3/4: convert to 16/3 − 11/4, use denominator 12 to get 64/12 − 33/12 = 31/12, and convert back: 31 ÷ 12 = 2 remainder 7, so the answer is 2 7/12.

Converting fractions to decimals

Divide the numerator by the denominator. This is where the basic calculator is handy.

Fraction	Division	Decimal
3/8	3 ÷ 8	0.375
1/3	1 ÷ 3	0.333… (repeating)
7/12	7 ÷ 12	0.58333… (repeating)

Some fractions give a decimal that stops, and others repeat forever. A fraction in simplest form stops only if its denominator has no prime factors other than 2 and 5, which is why 3/8 stops and 1/3 does not. You can also go the other way: 0.375 = 375/1000, and dividing both parts by 125 gives 3/8. As a check on any answer, convert to decimals: 2/3 × 9/10 is about 0.667 × 0.9 = 0.6, which matches 3/5. To turn a decimal into a percentage, multiply by 100: 3/8 is 37.5%, as covered in the percentage guide.

FAQ

How do I know if two fractions are equal?
Simplify both, or cross-multiply. 2/3 and 6/9 are equal because 2 × 9 = 18 and 3 × 6 = 18.

Which is bigger, 3/4 or 5/8?
Use a common denominator of 8: 3/4 = 6/8, which is larger than 5/8.

Can the denominator be zero?
No. Division by zero is undefined, so a fraction with a zero denominator has no value.

Open the calculator: Fraction Calculator →