Multiplying and Dividing Fractions

For instance:

When it's possible, you reduce the fraction by cancelling off common factors; that is, you cross out any factors from one side of the fraction line that are duplicated on the other side of the line. In the example above, however, nothing reduces, because 8 and 45 have no factors in common.

If you're not sure whether anything can be cancelled off, you can always factor the numerator and denominator, and check for any duplicated factors:

Nothing is duplicated between the top and the bottom, so nothing cancels.

Often, though, something will cancel:

• Simplify

You don't have to factor everything completely, of course. You can use shorthand to get the same answer:

...which leaves me with , as before.

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How do you divide by a fraction?

Dividing fractions is just about as easy as multiplying them; there's just one extra step. When you divide by a fraction, the first thing you do is "flip-n-multiply". That is, you take the second fraction, flip it upside-down (that is, you "find the reciprocal"), and then you multiply the first fraction by this flipped fraction.

• Simplify

(Why does flip-n-multiply work?)

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• Simplify

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• Simplify

Note: When the inputs are mixed numbers, as in the last example above, the book (or your instructor, or the grader) usually expects a mixed number as the output, too. So, if your answer is an improper fraction, you'll need to convert it back to mixed-number form. Don't forget this step!

Next, we move on to the much-more-difficult addition and subtraction of fractions...