Last Updated on May 10, 2023
GMAT OFFICIAL GUIDE DS
Solution:
We need to determine whether 5^k is less than 1,000.
Statement One Alone:
5^(k+1) > 3,000
We can simplify the inequality 5^(k+1) > 3,000.
(5^k)(5^1) > 3,000
5^k > 600
We only know that 5^k > 600, but we can’t tell whether 5^k < 1000 or not. For example, if k = 4, then 5^4 = 625 is less than 1,000. However, if k = 5, then 5^5 = 3,125 is greater than 1,000. Thus, we do not have enough information to answer the question. We can eliminate answer choices A and D.
Statement Two Alone:
5^(k-1) = 5^k – 500
We see that statement two presents a fairly difficult equation to solve, so our main goal is to determine whether we can come up with a value for k. If we can do this, then we know that we will have sufficient information to answer the question.
5^(k-1) = 5^k – 500
(5^k)(5^-1)=5^k – 500
(5^k)(5^-1) – 5^k = – 500
Noting that we have a common factor of 5^k on the left side of the equation, we factor it out.
(5^k)(5^-1 – 1) = -500
(5^k)(1/5 – 1) = -500
(5^k)(-4/5) = -500
At this point, it is obvious that we are able to determine a specific value for 5^k. We could stop at this point, but we show the remaining steps:
(5^k)(-4/5) = -500
5^k = (-500)(-5/4)
5^k = 2500/4 = 625
Since we know 5^k = 625, 5^k is less than 1,000. Thus, statement two alone is sufficient to answer the question.
Answer: B
