Simplifying Complex Fractions

Question

So I've attempted a few of these and I'm completely lost. I've asked for help, reread my notes multiple times and I'm still having a difficult time. If I could see these done so I see the work and answers for the various problems it should help a little.

Thanks for the help

 

 

 

 

 

 

 

 

 

 

Answer 1

2)

[(k + 1) / (2k)] / [(3k - 1) / (4k)]
= [(4k)(k + 1) / (2k)] / [(4k)(3k - 1) / (4k)]
= [2(k + 1)] / [(3k - 1)]
= (2k + 2) / (3k - 1)

3)

[(3/x) + (3/y)] / [(3/x) - (3/y)]
= 3[(1/x) + (1/y)] / 3[(1/x) - (1/y)]
= [(1/x) + (1/y)] / [(1/x) - (1/y)]
= (xy)[(1/x) + (1/y)] / (xy)[(1/x) - (1/y)]
= [(xy)(1/x) + (xy)(1/y)] / [(xy)(1/x) - (xy)(1/y)]
= (y + x) / (y - x)

4)

[(10x - 5y) / 12] / [(2x - y) / (6y)]
= [5(2x - y) / (6)(2)] / [(2x - y) / (6)(y)]
= [5 / 2] / [1 / y]
= y[5 / 2] / y[1 / y]
= [5y / 2] / 1
= (5y) / 2

5)

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

6)

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

7)

1 / (x^-2 + y^-2)
= 1 / [(1 / x^2) + (1 / y^2)]
= (x^2 y^2) / (x^2 y^2)[(1 / x^2) + (1 / y^2)]
= (x^2 y^2) / [(x^2 y^2)(1 / x^2) + (x^2 y^2)(1 / y^2)]
= (x^2 y^2) / [y^2 + x^2]
= (xy)^2 / (x^2 + y^2)

8)

[x^-1 - y^-1] / [x^-2 - y^-2]
= [(1 / x) - (1 / y)] / [(1 / x^2) - (1 / y^2)]
= (x^2 y^2)[(1 / x) - (1 / y)] / (x^2 y^2)[(1 / x^2) - (1 / y^2)]
= [(x^2 y^2)(1 / x) - (x^2 y^2)(1 / y)] / [(x^2 y^2)(1 / x^2) - (x^2 y^2)(1 / y^2)]
= [(x y^2) - (y x^2)] / [y^2 - x^2]
= [(xy)(y - x)] / [(y + x)(y - x)]
= (xy) / (x + y)

9)

(r^-1 + s^-1)^-1
= 1 / ((1/r) + (1/s))
= (rs) / (rs)((1/r) + (1/s))
= (rs) / ((rs)(1/r) + (rs)(1/s))
= (rs) / (s + r)