I don't know where you're not understanding so I'll break it down really simply...
You dissolved a compound with the formula XClx with X being the other atom and x being the charge of that atom (like AlCl3 = Al3+ and Cl1-...X is Al and x is 3+)
You know that Ag+ and Cl- react in a 1:1 ratio to form AgCl (s):
Ag+(aq) + Cl-(aq) --> AgCl (s)
So from here, it's all stoichiometry:
20ml of Ag+ x .15 mol/1000 ml of Ag+ = .003 mols of Ag+
.003 mols of Ag+ == .003 mols of Cl- since they're in a 1:1 ration
That means there was .003 mols of Cl- in that 10ml solution of the XCl
.003 mols of Cl-/10ml solution = __ moles of Cl-/200ml solution?
Simple proportion: 200*.003/10
So now you know that there is .06 moles of Cl- in the entire solution.
Let's backtrack a little...
If you have 1 mole of H2O compound, you have 2 moles of H and 1 mole of O
If you have 2 moles of CH4, you have 2 moles of C and 8 moles of H
Understand?
In other words A moles of XyYx = A*y moles of X and A*x moles of Y
*Sorry if that's confusing. I don't know how else to put it.
So if back to the problem.
You have .02 moles of the compound (A) and you have .06 moles of Cl-
.02 * what # gives you .06? 3! So the compound has to be [blahblah]Cl3
This is what they were getting to with the ratio part.