Fraction Calculator: Add, Subtract, Multiply and Divide

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How to use this tool

Enter two fractions and pick an operation. Leave the whole number boxes at zero for plain fractions, or fill them in for mixed numbers like 2 and three quarters. The answer comes back reduced, with the improper form and the decimal alongside.

The green box shows the actual working: which common denominator was used, what each fraction was rewritten as, and what cancelled at the end. That is the part worth reading if you are trying to learn the method rather than just get past a question.

Why adding needs a common denominator and multiplying does not

A fraction is a count of equal pieces. The denominator says what size the pieces are and the numerator says how many you have.

You cannot add halves to thirds directly for the same reason you cannot add three apples to two oranges and get five of anything useful. Rewrite both as sixths, which is 3/6 and 2/6, and now they are the same size of piece, so they add to 5/6. That is the entire purpose of the common denominator.

Multiplying is different, because you are not combining counts, you are taking a fraction of a fraction. Two thirds of three quarters means cutting three quarters into three parts and keeping two. The arithmetic is just top times top and bottom times bottom, and no rewriting is needed.

Why dividing flips the second fraction

Dividing by a half is the same as multiplying by two, and that is not a trick, it is what division means. Asking how many halves fit into one is asking one divided by a half, and the answer is two.

So dividing by any fraction is multiplying by its reciprocal, the same fraction upside down. A half divided by a quarter becomes a half times four over one, which is two. Sense check it: four quarters fit into one, so two quarters fit into a half, so the answer of two is right.

A worked check you can repeat on paper

Take 3/8 plus 5/16. The denominators are 8 and 16, and 16 is already a multiple of 8, so the lowest common denominator is 16 rather than 128. Rewrite 3/8 as 6/16, then add: 6 plus 5 is 11, giving 11/16. Nothing cancels because 11 is prime, so that is the answer.

Note the shortcut there. Multiplying the denominators would have given 128 and worked perfectly well, but the answer would have arrived as 88/128 needing a cancel by 8. Using the lowest common denominator saves that step.

Now 2/3 divided by 4/9. Flip the second to 9/4 and multiply: 2 times 9 is 18, 3 times 4 is 12, giving 18/12. Both divide by 6, so the answer is 3/2, or one and a half. Check it: 4/9 is a bit under a half, and 2/3 is a bit over a half, so the answer should be a bit over one. It is.

Mistakes that catch people out

Adding the denominators. A half plus a third is not two fifths. If that were right, a half plus a half would be two quarters, which is a half, and clearly adding two halves gives one.

Forgetting to convert a mixed number first. Two and a half is 5/2, not 2 and 1 over 2 treated separately. Multiplying mixed numbers without converting them gives nonsense.

Flipping the wrong fraction when dividing. The reciprocal goes on the second one, the divisor. Flipping the first gives the wrong answer and the sense check will catch it.

Stopping before the answer is reduced. 18/12 is correct but not finished. If the top and bottom share any factor, keep going.

Common fraction calculations

The calculations people look up most, each opening the tool ready to go with its own address.

How this was checked

All four operations were tested against known results: 1/2 plus 1/3 gives 5/6, 3/4 minus 1/2 gives 1/4, 2/3 times 3/4 gives 1/2, and 1/2 divided by 1/4 gives 2. Reduction was checked separately, so that 1/4 plus 1/4 returns 1/2 rather than 2/8, and 2/3 divided by 4/9 returns 3/2 rather than 18/12.

The awkward cases were checked too. A result of zero, from 5/6 minus 5/6, and a negative result, from 1/4 minus 1/2 giving minus a quarter, both come back correctly signed and reduced. A zero denominator is refused rather than producing an infinity, and dividing by a fraction that works out to zero is refused for the same reason.

Related Calculators and Reference

• Decimal to Fraction Converter for going the other way, from a decimal into the simplest fraction.
• Prime Factorisation Calculator for the common factors that do the cancelling here.
• Ratio and Proportion Calculator for fractions used as mixes and scales.
• Percentage Calculator for the same quantities expressed out of a hundred.
• Length Converter for turning a fractional inch into millimetres.
• Quadratic Formula Calculator: roots of ax² + bx + c = 0 with the steps.
• Long Division Calculator: turn a fraction into a decimal by dividing, step by step.
• GCF Calculator: reduce a fraction by the greatest common factor of its top and bottom.
• LCM Calculator: find the least common denominator before adding or subtracting fractions.