You can do this one of two ways.
1. Use reason. You have 1M, you want 0.1M which means you want the final solution to be 1/10 the original so dilute the solution by 10. That means dumping, quantitatively, the 1L into a 10L volumetric flask and adding water to the 10 L mark.
2. Use the dilution formula for these.
c1v1 = c2v2
c = concn
v = volume
1L*1M = xL*0.1M
xL = 1/0.1 = 10L final volume.
2. I've answered these before. You can make a more concentrated solution by boiling away the water. You want 1M to become 2M so you must boil away half the water. Use the dilution formula if you wish.
1M*100mL =xmL x 2M
xmL = (1M x 100 mL/2M)
xmL = 50 mL . You have 100 to begin you want the final volume to be 50 mL.
You would dilute the solution using MV=MV. Since you already have 1L of a 1M solution, you need to use the equation to calculate 0.1M. The answer will be 100L solution but you will only need to add 99L of H2O for every 1L of the solution. You will get 100L using the equation, but remember you already have 1L so you only need 99L of H20. You can convert this to mL if needed (i.e. 1mL of 1M solution for very 99mL of H20), and as far as I know, unless you can use distillation to separate 50mL of pure H20 from your 100mL of 2M solution, you will not be able to get a 2M solution from a 1M.
Using MV = MV I get
1M*1L = 0.1M*V and
V = (1*1/0.1) = 10 L and not 100 L.
In addition to the wrong answer, I think it is unwise to talk about taking 1L and adding 99 L to make 100 L. Although it may be close (in fact even VERY close), it gives the impression that all solution volumes are additive and that isn't so. You do this right by "adding to a final volume of 10L or in the case of 100 L to a final volume of 100 L."
Sorry, it should read 10L, not 100L and 9L not 99L. Also, DrBob22 is right you can boil/distill 50mL of water, but only if they do not have the same boiling point, but the answer is to add 9L not 10L.