DAT Destroyer Common Ion Error?

[charcot bouchard]

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The question in question (no pun intended) asks:

The Ksp of PbCl2 is 1.6 x 10^-5 at 25 degrees C. What is the solubility of PbCl2 in 0.01M KCl?

First, let's find the molar solubility of PbCl2 with no common ion effect:

Ksp = 1.6 x 10^-5 = x(2x)^2
1.6 x 10^-5 = 4x^3
x = 0.0158

Now, if we solve the question asked using 0.01M Cl- as the common ion:

Ksp = 1.6 x 10^-5 = x(0.01)^2
1.6 x 10^-5 = 0.001x
x = 0.16

0.16 is the correct answer given in DAT Destroyer. My question is: why is the molar solubility of PbCl2 so much higher when you start with Cl- in solution? Is that not supposed to deter PbCl2 from dissolving? Doesn't this problem contradict what you would expect if you understand the common ion effect?

 

[malikhind]

When I put (1.6 x 10^-5)/0.001 in my calculator I get x=0.016 which is fairly close to it's solubility in pure water. I assume it's close because the [KCl] is relatively low. However it still increases which doesn't make sense...

I found a similar problem on khan academy however KCl has a higher concentration.

Edit: I did an exact calculation without canceling out the 2x.
(so 1.6x10^-5= 0.0001x + 0.04x^2 + 4x^3). You get an answer of x=0.01273, which is lower and makes sense. So I guess since it's such a low concentration, assuming x is small causes a bigger error in the answer than with a higher concentration of KCl.