Conditional Probability

Let A and B be events such that P(A)=0.2, P(B)=0.3, and P(A|B)=0.1. What is P(A|B^c)? (i.e. what is B complement?)

Answer is 0.243, but I have no idea how to get it. Thanks.

Here is my attempt:
P(A^c and B)=P(A^c|B)P(B) =0.9*0.3=0.27
Not sure how to get P(A and B), but I know that I have to use Bayes's rule.
P(a) = p(a,b) + p(a,~b)
p(a) = p(a|b)*p(b) + p(a|~b)*p(~b)
[p(a) - p(a|b)*p(b)] / p(~b) = p(a|~b)

p(a|~b) = (.2-.1*.3)/(1-.3) = 0.2429