How to Add Fractions

What is the formula for adding fractions?

The adding fraction formula is a/b + c/d = (a × d + b × c) ÷ (b × d). In any fraction, the top number is the numerator, which counts the parts you have, and the bottom number is the denominator, which says how many equal parts make a whole. The formula works by rewriting both fractions in the same size of part, so the numerators can be added directly.

The fraction calculator uses this formula and then reduces the result. For 1/4 + 1/6, the formula gives (1 × 6 + 1 × 4) ÷ (4 × 6) = 10/24, which simplifies to 5/12.

The table shows the rule for each of the four operations, and every result comes from our calculator.

How do you add fractions with different denominators?

Find the least common denominator (LCD), convert each fraction to it, add the numerators and simplify. The LCD is the smallest number both denominators divide into evenly, so it is the same as their least common multiple, which the LCM calculator will find for you.

Worked example: add 3/4 + 1/6.

• Find the LCD: the least common multiple of 4 and 6 is 12.
• Convert 3/4: 12 ÷ 4 = 3, so multiply top and bottom by 3 to get 9/12.
• Convert 1/6: 12 ÷ 6 = 2, so multiply top and bottom by 2 to get 2/12.
• Add the numerators and keep the denominator: 9/12 + 2/12 = 11/12.
• Simplify if you can: 11 and 12 share no factor above 1, so 11/12 is final.

What if the fractions already have the same denominator?

Add the numerators and keep the denominator: 3/8 + 2/8 = 5/8. The denominator stays the same because it names the size of the parts, and adding three eighths to two eighths gives five of those same parts.

This is also why you can't add across. 1/2 + 1/3 is not 2/5, which is less than the 1/2 you started with. Using the LCD of 6, the correct sum is 3/6 + 2/6 = 5/6.

A sum can come out larger than 1. For instance, 5/6 + 1/4 = 10/12 + 3/12 = 13/12, which is 1 1/12. A recipe that calls for 3/4 cup of flour and then another 2/3 cup needs 9/12 + 8/12 = 17/12, or 1 5/12 cups.

How do you subtract fractions?

Subtract fractions the same way you add them: convert to a common denominator, then subtract the numerators. For 5/6 − 1/4, the LCD is 12, so 10/12 − 3/12 = 7/12.

If the second fraction is larger, the answer is negative. For example, 1/2 − 3/4 = 2/4 − 3/4 = −1/4.

How do you multiply fractions?

Multiply the numerators together and the denominators together, with no common denominator needed. For 2/3 × 3/4, the result is (2 × 3) ÷ (3 × 4) = 6/12, which simplifies to 1/2.

You can save work by canceling a factor shared between any top and any bottom number before you multiply. In 2/3 × 3/4, the 3s cancel and the 2 reduces against the 4, leaving 1/2 straight away.

To multiply by a whole number, write it over 1. For 3 × 2/5, that is 3/1 × 2/5 = 6/5, or 1 1/5.

How do you divide fractions?

Keep the first fraction, change division to multiplication and flip the second fraction. The flipped fraction is called the reciprocal. For 2/3 ÷ 3/4, the reciprocal of 3/4 is 4/3, so 2/3 × 4/3 = 8/9.

Division asks how many of one amount fit into another. For 3/4 ÷ 3/8, the result is 3/4 × 8/3 = 2, which matches the fact that two 3/8 pieces make 6/8, or 3/4. You can't divide by a fraction whose value is zero, and a fraction can't have 0 as its denominator, so our calculator returns an error in those cases.

How do you simplify a fraction?

Divide the numerator and denominator by their greatest common factor (GCF), the largest whole number that divides both evenly. A fraction is in simplest form when that factor is 1. The GCF calculator gives 6 as the GCF of 18 and 24, so 18/24 = (18 ÷ 6) ÷ (24 ÷ 6) = 3/4.

Simplifying changes the numbers but not the amount. The pies below show that 1/2, 2/4 and 4/8 cover the same share of a whole, and the reverse of this idea is what lets you convert fractions to a common denominator. The Simplify Fractions tool on the fraction calculator page does this reduction for you and shows the GCF it used.

What is the formula for mixed numbers?

A mixed number converts to an improper fraction with (whole × denominator + numerator) ÷ denominator. A mixed number combines a whole number and a fraction, such as 2 1/2, and an improper fraction has a numerator at least as large as its denominator, such as 5/2. So 2 1/2 = (2 × 2 + 1) ÷ 2 = 5/2.

To go back, divide the numerator by the denominator. The quotient is the whole number and the remainder is the new numerator: 17 ÷ 4 = 4 remainder 1, so 17/4 = 4 1/4.

Once they are improper fractions, the four operations work as usual. The Mixed Numbers tool on the fraction calculator page produced each result below.

Can you add mixed numbers without converting them?

Yes. For addition and subtraction, you can handle the whole numbers and fractions separately, then combine them. For 2 1/2 + 1 3/4, the whole numbers give 3 and the fractions give 2/4 + 3/4 = 5/4, or 1 1/4. Together that is 3 + 1 1/4 = 4 1/4, the same answer as before.

This shortcut doesn't work for multiplication. 1 1/2 × 2 2/3 is 4, while multiplying the parts separately gives 1 × 2 and 1/2 × 2/3, or 2 1/3. Convert to improper fractions for multiplication and division.

What mistakes should you avoid with fractions?

Most fraction errors come from applying the rule for one operation to another.

• Adding denominators: 1/2 + 1/3 is 5/6, not 2/5.
• Finding a common denominator to multiply. It isn't wrong, but it adds work you don't need.
• Flipping the wrong fraction when dividing. Only the second fraction, the divisor, gets flipped.
• Changing the denominator without changing the numerator. Whatever you multiply the bottom by, multiply the top by too.
• Multiplying mixed numbers part by part instead of converting them first.
• Treating a negative mixed number as a sum of opposite signs. −2 3/4 means −(2 + 3/4), which is −11/4.