Simplifying Rational Expressions...?

Question

Can anyone help me with simplifying these problems?

1) ((1/x)+3)/((1/x)-5)

2) ((2/y)+(1/2y))/(y+(y/2))

3) (x/(x-y))/((x^2)/(x^2-y^2))

Thanks for any help!

Answer 1

1. ((1+3x)/x)/((1-5x)/x)
(1+3x)/(1-5x)

2. ((4+1)/2y)/((2y+y)/2)
= (5/2y) / (3y/2)
= 5/3y^2

3. (x/(x-y))/((x^2)/(x+y)(x-y))
= (x(x+y)) / x^2
= (x+y) / x

Answer 2

(1/x + 3) / (1/x -5)

Multiply top and bottom by x

1 + 3x / 1 - 5x

(2/y + 1/2y) / (y + y/2)

Multiply top and bottom by 2

(4/y +1/y) / (2y + y)

(5/y) / (3y)

Multiply top and bottom by y

5 / 3y^2

(x/ x +y) / (x^2/ x^2- y^2)

Factor the bottom of the bottom fraction into
(x + y)(x - y)

(x/x-y) / (x^2 /(x-y)(x=y)

Multiply top and bottom by (x-y)(x+y)

(x (x+y) / x^2

x^2 + xy / x^2