A Quick Refresher on Adding and Multiplying Fractions
Fraction arithmetic has one genuinely counter-intuitive feature: multiplying is easier than adding, which is the opposite of whole numbers. That reversal is why the rules get muddled. The Fraction Calculator handles all four operations.
Multiplying is the easy one
Multiply the tops, multiply the bottoms. Two-thirds times three-quarters is 6 over 12, which simplifies to one-half. No common denominator, no preparation.
Dividing is multiplying by the flipped second fraction. Two-thirds divided by three-quarters is two-thirds times four-thirds, which is 8 over 9. The rule is "multiply by the reciprocal", and it works because dividing by a fraction asks how many of it fit in - and that is the same as multiplying by its inverse.
The thing that surprises people: multiplying by a fraction less than one makes the answer smaller, and dividing by one makes it bigger. Half of a half is a quarter. A half divided by a quarter is two.
Adding needs the same denominator first
You cannot add thirds to quarters any more than you can add metres to feet. The parts have to be the same size.
So find a common denominator - the lowest common multiple of the two bottoms, which the LCM and GCD Calculator finds directly. For thirds and quarters that is 12. Convert both: two-thirds becomes 8 over 12, three-quarters becomes 9 over 12. Now add the tops: 17 over 12, which is one and five-twelfths.
Multiplying the denominators together always works and sometimes gives a bigger number than necessary - for sixths and fourths it gives 24 when 12 would do. Simplifying at the end sorts it out either way.
Subtracting is identical, with a minus sign.
Mixed numbers
Convert to improper fractions first, every time. One and a half becomes three-halves. Doing arithmetic on the whole part and the fraction part separately works for addition and goes wrong for multiplication, which is where people come unstuck.
Convert back at the end if the answer wants to be a mixed number.
Simplify at the end, and cancel early if you can
Divide top and bottom by their greatest common divisor. 6 over 12 becomes one-half.
With multiplication there is a shortcut: cancel before multiplying. In two-thirds times three-quarters, the 3 on top and the 3 on the bottom cancel, leaving two-quarters, which is one-half. Smaller numbers, same answer, less chance of an error.
Common questions
Why do I need a common denominator to add but not to multiply?
Adding combines quantities, which requires the same-sized parts. Multiplying takes a portion of a portion, which does not.
What is a reciprocal?
The fraction flipped. The reciprocal of three-quarters is four-thirds, and a number times its reciprocal is always 1.
Should the answer be a fraction or a decimal?
A fraction is exact; a decimal may repeat forever. For a third, the fraction is the honest form.
How do I compare two fractions?
Common denominator and compare tops, or cross-multiply.
Got some fractions to work through?