Honestly, this problem has less to do with chemistry and more to do with math. The question apparently views that a 10 amino-acid length peptide consists of coupling amino acids independently 10 times. The coupling reaction is treated to be independent to avoid having to deal with complicated intermediate steps that may affect the resulting yield.
Now the actual math gets complicated and that's not surprising because NextStep exams involve insanely complicated math and difficult approximations... which in the age of calculators is basically unnecessary. I used calculators regularly in math, physics and engineering courses because the classes cared less about calculations (leave the calculations to computers and calculators) and focused more about the concepts.
0.95^10 isn't something you will see on AAMC practice tests or on test day, but it can be done by hand using the following approach:
0.95^10 = (0.95)^(2*5) = (0.95^2)^5 (remember: (a^b)^c = a^b * a^b * ... a^b c times or a^(bc))
Solve for 0.95^2 first. This can be done as follows:
0.95^2 -->
(0.90 + 0.05)^2 -->
0.90^2 + 2*0.90*0.05 + 0.05^2 (remember: (a+b)^2 = a^2 + 2ab + b^2)-->
0.81 + 0.09 + 0.0025 -->
0.9025
0.9025 can be approximated as 0.90 (it's a bit greater than 0.90).
0.9^2 = 0.81 which is a bit greater than 0.80. 0.9^4 is therefore a bit greater than 0.80*0.80 or 0.64.
0.9^5 >= 0.9 * 0.64 = (1-0.1)*(0.64) = 0.64 - 0.1*0.64 = 0.64 - 0.064 = 0.576
What this ultimately means is 0.9^5 > 0.576. Which means 0.9025^5 > 0.576. Which therefore means 0.95^10 > 0.576. The approximations weren't significantly off since I was just rounding down 0.9025 slightly down to 0.90 and 0.81 slightly down to 0.80.
Which means the correct answer is slightly higher than 0.576 or answer choice B): 0.60.
Important Takeaway: The important thing on doing well on the MCAT is understanding the concepts and not wasting time on the calculations. If the calculation is hard or time-consuming, take a best guess, mark the question and move on. Don't waste too much time on a difficult question when you can use that time efficiently on knocking out easier questions.
