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 × 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 × 3 could be thought of as 1 × 2 × 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!!]