PSAT Math Multiple-Choice Question 670: Answer and Explanation

Question: 670

Two different stock portfolios, A and B, have had no new deposits or withdrawals over a ten-year period and had the same initial amount in the account. If stock portfolio A has grown at an annual rate of x%, if stock portfolio B has grown at an annual rate of y%, and if x > y, what would represent the ratio of the value of portfolio A over that of portfolio B at the end of the ten-year period?

• A.
• B.
• C.
• D.

Correct Answer: A

Explanation:

(A) None of the answer choices has a percentage in it, so convert the percentage to a decimal by dividing by 100:

Suppose P is the initial amount deposited into each portfolio. If portfolio A grows at a rate of x% yearly, the value after 1 year will be given by the expression:

The following year, the amount of money in portfolio A again increases by x%. Therefore, the value after the second year will be given by the following expression:

The third year, the interest will be compounded on the previous value. Therefore, the value will be:

The value after n years is given by:

So after 10 years, the value of portfolio A will be .

Repeat the thought process for portfolio B to arrive at the conclusion that the value of portfolio B after 10 years will be

since both portfolio A and B start out with the same amount initially, P. Therefore, the ratio of the value of portfolio A to the value of portfolio B is:

This matches choice (A).