Can someone help with this question??

I'm having trouble figuring out how to start with some parts of these probability questions, like part A 
I tried to used P(pass test&non survivor)/ P(non survivor) to get P(passtest|nonsurvivor) 
But when I checked the solution I saw that they used the unconditional probability of P(pass test) = P(pass test|survivor)* P(survivor) + P(pass test| non survivor)*P(non survivor). 
I understand why they used that but I don't understand why the method I chose was wrong, can someone please clarify? How do you know which formula to use in these situations. Thanks, (Question is below) 



You have developed a set of criteria for evaluating distressed credits. Companies that do not receive a passing

score are classed as likely to go bankrupt within 12 months. You gathered the following information when

validating the criteria:

•

Forty percent of the companies to which the test is administered will go bankrupt within 12 months: P

(nonsunivor) = 0.40.

Fifty-five percent of the companies to which the test is administered pass it: P(pass test)= 0.55.

• The probability that a company will pass the test given that it will subsequently survive 12 months, is 0.85:

P(pass test|survivor) =0.85.

A. What is P (pass test|nonsurvivor)?

B. Using Bayes' formula, calculate the probability that a company is a survivor, given that it passes the test;

that is, calculate P(survivor|pass test).

C. What is the probability that a company is a nonsurvivor given that it fails the test?

D. Is this test effective?