Given a math problem with context and a question and 5 answer choices, the task is to provide the correct answer choice based on the problem. You must choose one of the given answer choices by letter: a, b, c, d, or e; anything else is invalid.

Input: Consider Input: Problem: if a / b = 4 / 3, then ( 3 a + 2 b. / ( 3 a - 2 b. = ?
Options: a. 6, b. 3, c. 5, d. - 1, e. none

Output: b

Input: Consider Input: Problem: if a and b are the roots of the equation x 2 - 9 x + 9 = 0, then the value of a 2 + b 2 is :
Options: a. 63, b. 24, c. 17, d. 6, e. 5

Output: a

Input: Consider Input: Problem: the events a and b are independent, the probability that event a occurs is greater than 0, and the probability that event a occurs is twice the probability that event b occurs. the probability that at least one of events a and b occurs is 8 times the probability that both events a and b occur. what is the probability that event a occurs ?
Options: a. 1 / 12, b. 1 / 8, c. 1 / 6, d. 1 / 3, e. 2 / 3
Output: d