X divided by 7/8 = 3/5

X = ?

So its x / (7/8) = 3/5

x*8/7 = 3/5
8x / 7 = 3/5
8x / 7 - 3/5 = 0
40x / 35 - 21 / 35 = 0
40x - 21 / 35 = 0

If you multiply both sides by 35, then you get:
40x - 21 = 0
40x = 21
x = 21 / 40
x = 0.525

Let's try and substitute that in:
.525 / .875 = ?
.525 / .875 = .6

So x = .525
So x = 21 / 40

SBowman's final answer is correct but I have two comments:

First, the following line is wrong:
40x - 21 / 35 = 0
It should be
40x/35 - 21/35=0

Second, the following method is much simpler:
x/(7/8)=(3/5)
multiply both sides by (7/8)
x=(3/5)*(7/8)
simplify
x=21/40

I agree with asterisk's point, your working made me cringe when I read it SBowman

Ah, found an easier way. Much more effective than my method. :)