Hi there. I'm happy to give my 2¢ here.
What is the source of this question. I feel as if some vital information is missing. As it stands, it's a very difficult question.
Prompt: Someone invested at annual interest rate r, compound quarterly. What's the value of r?
Call the principal P. Change the APR into a fraction: r/100. Divide this by four, for compounding quarterly: r/400. Add one to create a multiplier: (1 + r/400). The total amount is multiplied by this amount each quarter. For example, in one year:
A = P(1 + r/400)^4
I'm going to start with Statement #2, which seems a bit simpler. (
There's no law saying you have to do the DS statements in order all the time.)
Statement #2: The interest earned in the second quarter is $4 more than that in the first quarter.
OK, this allows us to set up an equation of sorts.
Amount after Q1 = P(1 + r/400)
Interest after Q1 = Pr/400
Amount after Q2 = P(1 + r/400)^2
Interest after Q2 = P(1 + r/400)^2 - P(1 + r/400) = P[(1 + 2r/400 + (r^2)/16000) - (1 + r/400)]
= P(r/400 + (r^2)/16000)
Interest after Q2 = Interest after Q2 + $4
P(r/400 + (r^2)/16000) = Pr/400 + 4
P(r^2)/16000 = 4
This equation has two variables, P and r, and so cannot be solved for unique values of either. Statement #2 is
insufficient.
Statement #1: At the end of the year, the amount of the interest is between $800 and $850.
Amount after Q4 = P(1 + r/400)^4
(
I will refrain from expanding the quartic. To the best my knowledge, expanding a quartic is well beyond the math you need to know for the GMAT.)
Interest after one year = P(1 + r/400)^4 - P
Statement #1 says:
800 < P(1 + r/400)^4 - P < 850
This is an inequality of two variables, which doesn't allow us to conclude diddly-squat. Statement #1 is wildly
insufficient.
Statements #1 & #2 combined: Basically, now we have two unknowns, P and r, one equation, and one inequality.
In general, when you have two unknowns, you need two independent equations to solve for them. One equation and one inequality is not enough to solve. Combined, the statements are still
insufficient.
As it stands, the answer is
E.
If we were given any more information --- for example, knowing that r had to be an integer --- then there's the possibility that we could solve for this. As it stands, though, we can conclude nothing.
Does all this make sense? Please let me know if you have any questions.
Mike
