iwillcrackgmat
On the first of the year, James invested x dollars at Proudstar bank in an account that yields 2% in interest every quarter year. At the end of the year, during which he made no additional deposits or withdrawals, he had y dollars in the account. If James had invested the same amount in an account which pays interest on a yearly basis, what must the interest rate be for James to have y dollars at the end of the year?
A. 2.04%
B. 6.12%
C. 8%
D. 8.25%
E. 10%
We can let James’ initial investment of x be $100. That is, we can say that James invested 100 dollars at Proudstar bank in an account that yields 2% in interest every quarter year. Recall the compound interest formula:
A = P(1 + i)^n
In which P = principal, i = interest rate per period, n = number of periods, and A = amount at the end of n periods.
Since a year has 4 quarters, n = 4 and we have:
A = 100(1 + 0.02)^4
A = 100(1.02)^4
A = 100(1.0824)
A = 108.24
We can see that he will get back $108.24 when the interest is compounded quarterly, and this has to be the value of y (if x = 100). So the question becomes: if James had invested $100 in an account that pays interest on a yearly basis, what must the interest rate be for James to have $108.24 at the end of the year?
We can use the same formula, except now we need to solve for i when n = 1:
108.24 = 100(1 + i)^1
1.0824 = 1 + i
0.0824 = i
8.24% = i
We see that the closest answer choice is D: 8.25%.
Alternate solution:
In this problem, we are given that a certain amount of money is being compounded quarterly with an interest rate of 2%. That is like paying an annual interest rate of 8% (4 x 0.02 = 0.08), but SOMEWHAT BETTER, since the interest is compounded (i.e., interest added on principal plus interest) every quarter of the year. So we are looking for an annual interest rate of slightly more than 8%. The only two answer choices that are more than 8% are 8.25% and 10%. Recall, we’ve said “somewhat better,” so it can’t be 10%. This leaves 8.25% as the most reasonable answer choice and the correct answer choice.
Answer: D