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Review of Fractions

Any fraction can be re presented in many ways. Consider the fraction 1/2 . The number 1/2
one part of a whole that is divided into two equal pieces:

1/2 of the above rectangle is shaded. The same rectangle can be divided again so that it is
broken into 4 pieces:

The same area is shaded, but now we have two out of four or 2/4 of the rectangle shaded.
It is clear, then, that 1/2 and 2/4 are equal. Similarly, we could have taken the same rectangle
and cut it into six equal pieces:

We still have exactly half ( 1/2) of the rectangle shaded, but now there are three of the six
equal pieces shaded. So 1/2 and 3/6 are equal. We could cut this rectangle into any number
of equal pieces and then shade half of them, and each one would give us another way to
represent 1/2 . Some other ways are 5/10, 22/44, 500/1000, 37/74. Notice that in each case,
the denominator (that is, the bottom number) is twice as large as the numerator. The
simplest way to write this particular fraction is 1/2 because the numerator and denominator
don’t have any factors in common other than 1. When you are asked to reduce a fraction ,
your goal is to write the fraction this way. To do this, you simply divide the numerator
and denominator by the same number until there are no common factors left. Dividing (or
multiplying) the numerator and denominator of a fraction by the same number does not
change the value of the fraction .

Example 1: Reduce 3/15

Since 3 and 15 are both divisible by 3, we can divide the top and bottom of this fraction to
get 1/5 .

Example 2: Reduce 24/32

Since 24 and 32 are both divisible by 2, we can divide the top and bottom to get 12/16. Notice,
though, that this fraction is not reduced completely: the numerator and denominator still
have factors in common. Since 12 and 16 are both divisible by 4, we can divide the top and
bottom to get 3/4 . 3 and 4 don’t have any factors in common, so we’re done. Of course, 24/32
could be reduced in one step simply by dividing the top and bottom by 8 to get 3/4.

Multiplication and Division of Fractions
Multiplying fractions is the simplest of operations to perform because it is worked exactly
the way one might guess: multiply the tops and multiply the bottoms.

Example 3: Perform the fol lowing multiplications

The symbol ยท represents multiplication.

Notice that this fraction needed to be reduced.

x is treated just like any other number.

Dividing fractions is almost as simple as multiplying fractions, but the ope ration requires
one extra step. First take the reciprocal of the fraction that you are dividing by, and then

Example 4: Perform the following divisions:

Addition and Subtraction of Fractions

Addition and subtraction of fractions takes a little more finesse than multiplication and
division. To add two fractions, the fractions must have the same denominator. We already
know that if we divide the top and bottom of any fraction by the same number the value of
the fraction does not change (we saw this when we reduced fractions). We can also multiply
the top and bottom by any number ( except zero , of course) and it won’t change the value
of the fraction. This fact is used to do fraction addition and subtraction.

Example 5: Perform the following addition and subtraction:

First we get a common denominator by multiplying the numerator and denominator of the
second fraction by 2. Once we have a common denominator, we add the numerators and
keep the denominator.

Subtraction works exactly the same way as addition. Here we had to change both denominators.

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