# Fractions

## Equivalent fractions

If the top and bottom of any fraction are multiplied by the
same number, the resulting fraction looks different, but it is
equal to the original fraction. Finding equivalent fractions is
used to reduce fractions, to find common denominators, and for
many other things later on in math.

are all equivalent fractions.

## Reducing Fractions

This process is used to express fractions in simpler form.
Break down the numerator and denominator into their factors (e.g.
6 breaks down into 3 Ã— 2 ). If any factors appear in both the
numerator and denominator, cancel them out.

(a pair of 2’s and a pair of
3’s cancel out)

## Cancellation

Consider the following multiplication problem.

The answer needs to be reduced. To reduce it we would spread
out the numerator and denominator into their factors and cross
out the ones that appear in both numerator and denominator.

Notice that factoring the result undoes the work we just did
in multiplying the original numbers. We would have been better
off to reduce first and multiply only the factors that survive .
Let’s do the same problem again, reducing before
multiplying:

A pair of 2’s and a pair of 3’s cancel out, leaving
only a 2 in the denominator.

Notice that everything in the numerator cancels out. When this
happens, what is left behind is not a 0, but rather a 1. That is
because 1 is always a “hidden” factor in every number.
It’s not written down unless it is needed. In this problem
the numerator 2 Ã— 3 could be thought of as 1 Ã— 2 Ã— 3, so when
the 2 and 3 both cancel, what is left is the 1.

**Example: **

Set up the problem as before, cancel the factors that appear
in both numerator and denominator (in this case only the
5’s), then multiply the surviving factors.

**[Warning: Do not cancel in division problems until you
have converted them to multiplication problems!!] **