Complex Fractions

There are two different ways to look at complex fractions. For the first way they don't really need to be anything new. You just regard the big fraction line as a division. As far as order of operations, there are automatic parentheses around the stuff in the numerator and the stuff in the denominator, so you do all the operations in the numerator and do all the operations in the denominator, and then divide the results, and remember that dividing is just multiplying by the reciprocal. (See Rational Numbers for a review of all of the fractions operations.)

Example:

Simplify the complex fraction.

Solution:

Explanation:

To add the top fraction we need to find the LCM of 2 and 3 and use that as a common denominator. That would be 6, so we write each fraction as an equivalent one that has a denominator of 6. Then we add up the numerators. To add the bottom fractions we need to find the LCM of 6 and 12 to use as our common denominator. The LCM of 6 and 12 is 12, so 12 is our common denominator. For this we leave the second fraction alone and multiply the first fraction top and bottom by 2. Then we add the numerators.

After completing both of the addition then it is time to divide, and to divide we multiply by the reciprocal. In the multiplying there is some cross canceling. We can cancel the 6 with the 12 leaving a 2 in the place of the 12, and then we can multiply what is left to get the final answer.

For many kinds of problems, however, there is another method that works better. The trick is to treat the big fraction as a fraction and remember that multiplying the top and bottom of a fraction by the same thing gives you an equivalent fraction. The other thing you need to know for this method is that whenever you multiply a fraction by a whole number that is a multiple of the denominator, the denominator will cross cancel out and you get a whole number for your answer. For example if you multiply 12 times 1/3 you get

, so, here is what you do. Take all the fractions that occur in the problem, top or bottom, and find their least common multiple. Then put parentheses around the top and bottom and multiply this least common multiple by the top and bottom of the big fraction. when you distribute this in to each term, because it is a multiple of each of the denominators, all of the denominators will cross cancel out, and you will get all whole numbers and no longer have a complex fraction. Then you just have to reduce your fraction to lowest terms.

Here is how we would do the above problem using this method.

With a little practice you can get so that you won't have to write out the step with all the canceling, because the canceling will always give you whole numbers. You can just divide and multiply along the way imagining that you have the number distributed in. So here you could say, "the two cancels with the 12 to give me 6 and 6 times 1 is 6, the 3 cancels with the 12 to give me 4 and 4 times 2 is 8, the 6 cancels with the 12 to give me 2 and 2 times 5 is 10, and the 12 cancels with the 12 to give me nothing, so I am just left with the 7", and then just write down the 6+8 and the 10+7.

You can also use these methods for algebraic expressions (see Rational Expressions).

Example:

Simplify the expression.

Method 1:

y+x

=y+x

Explanation:

This is the straight forward way without using any tricks. We just regard the big fraction line as a division, so first we must perform the addition in the numerator. To do that we use a common denominator of xy and multiply the first one top and bottom by y and the second one top and bottom by x to get them both over that denominator. Then once we have a common denominator we add the numerators. After that we can do the division by multiplying by the reciprocal. The xy's cancel out leaving us just y+x.

Method 2:

Explanation:

Here we look at all of the denominators in the problem and find the LCM. In this case it is xy, so we multiply top and bottom of the big fraction by this. This time I haven't written out all of the cancellation, but I think you can follow it if you try. Just imagine the xy next to each of the terms. We can't really write the cancellations, because we need them for the other term, but in the first term the x's cancel and we are left with y, and in the second term the y's cancel and we are left with x. In the bottom the whole xy's cancel out leaving just 1.

We can even use this method when the top or bottom doesn't have any fractions. In that case the multiplication where there are no fraction is just ordinary multiplication, and all we really need to do is write it. (See Rational Expressions.)

Example:

Simplify the expression.

x+h

Solution:

Explanation:

This time I'm only showing the second method. The need to simplify an expression like this will come up in Calculus. Again we multiply by the LCM of all of the denominators. Since the denominator of the big fraction here doesn't have any denominators, it has no effect, so we are just looking for the LCM of x+h and x, which is (x+h)x, since these two are like prime numbers and have no common factors. Don't let the fact that they both have x's fool you for that, because of the h being added. They have no more in common than x and y do. So we multiply top and bottom by that product. In the bottom where there are no fractions, the multiplying is done just by writing it. In the top, just as before, all of the denominators cancel out. For the first term the x+h's cancel and for the second term the x's cancel, so for the first term we are left with x, and for the second term we are left with x+h. Then when we subtract them, the x's subtract to 0 and go away, leaving just -h. Then when we cancel the h's, only a factor of -1 is left in the top.

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