Simplifying Complex Fractions?

Question

x - x+2/ 4 over 3/4-5/2p


3/x + 3/y over 3/x -3/y



2/s - 3/t over 4t^2 - 9s^2 / st

Answer 1

When asking questions like this it is best to use a lot of parentheses. Now it is not totally clear what you mean :)

... [x - (x+2)/4] / [3/4 - 5/(2p)]

You may multiply both parts of the fraction with any number except zero. If you choose that number to be the denominator of one of the fractions in numerator or denominator, that fraction will disappear. Such denominators are 4 and 2p. We take care of both by multiplying by 4p.

The numerator becomes
... 4p * [x - (x+2)/4] = 4px - p(x+2)
This can be simplified further:
... 4px - p(x+2) = 4px - px - 2p = 3px - 2p.

The denominator becomes
... 3p - 10.

So the answer to the first question is
... (3px - 2p) / (3p - 10).

====================================

... (3/x + 3/y) / (3/x - 3/y)

The denominators inside the fraction are x and y, so we multiply by xy. That gives
... (3y + 3x) / (3y - 3x).

Note that all terms in the fraction contain 3 as a factor. We get rid of it by dividing everything by 3:
... (y + x) / (y - x).

=====================

... (2/s - 3/t) / [(4t^2 - 9s^2) / st]

(At least, I think that this is what you mean.) In any case, the numerators are s, t and st. We multiply by st:
... (2t - 3s) / (4t^2 - 9s^2)

This is a special situation. It is not obvious, but the numerator is a factor of the denominator. Here is the trick: the difference of two squares can be factored as a product of a sum and a difference. In this case, 4t^2 is the square of 2t and 9s^2 is the square of 3s. The sum and the difference are 2t + 3s and 2t - 3s. Therefore
... 4t^2 - 9s^2 = (2t + 3s) (2t - 3s).

Plug that in the fraction:
... (2t - 3s) / [(2t + 3s) (2t - 3s)]

The factors (2t - 3s) cancel, so the final answer is
... 1 / (2t + 3s).

-- who'd have thunk that!

Answer 2

(1) x - x+2/ 4 over 3/4-5/2p

I assume this is:

[ x - (x+2)/4 ] / [(3/4) - 5/(2p) ]

I'd multiply everything by 4p to remove all denominators within the larger fraction:

[ x - (x+2)/4 ] / [3/4 - 5/2p ]
[ 4px - p(x+2) ] / [3p - 10]

Then multiply out the p(x+2) and combine like terms:
[ 4px - p(x+2) ] / [3p - 10]
[ 4px - px - 2p ] / [3p - 10]
[ 3px - 2p ] / [ 3p - 10 ]

(2) 3/x + 3/y over 3/x -3/y

[ 3/x + 3/y ] / [ 3/x - 3/y ]

Multiply everything by xy to eliminate denominators, divide by 3 to eliminate all the 3's:

[ 3/x + 3/y ] / [ 3/x - 3/y ]
[ 3xy/3x + 3xy/3y ] / [ 3xy/3x - 3xy/3y ]
[ y + x ] / [ y - x ]

(3) 2/s - 3/t over 4t^2 - 9s^2 / st

[ 2/s - 3/t ] / [ (4t^2 - 9s^2) / st ]

Multiply everything by st to eliminate the denominators, then factor the difference of squares:

[ 2/s - 3/t ] / [ (4t^2 - 9s^2) / st ]
[ 2st/s - 3st/t ] / [ (4t^2 - 9s^2)st / st ]
[ 2t - 3s ] / [ 4t^2 - 9s^2 ]
[ 2t - 3s ] / [ (2t + 3s)(2t - 3s) ]
1 / (2t + 3s)

Answer 3

For parentheses help see

http://www.purplemath.com/modules/mathtext.htm

Your type of problem is there.