Problem Solving

AAPL wrote:Manhattan Prep

Jeremiah invests his savings of $120,000 by dividing it between two interest-earning accounts. He puts 3/4 of his savings in an account that earns lower interest and 1/4 of his savings in an account that earns higher interest. He has no other accounts that earn interest and he makes $3,636 in interest by the end of the year. If one account earns 2 percent annual interest, and both accounts are compounded semiannually, what percent interest does the other account earn?

A. 3
B. 4
C. 5
D. 6
E. 7

OA D

Since all the answer choices are greater than 2, we can see that the account that earns 2 percent annual interest is the one that earns the lower interest (let's call it account A), and thus it has 120,000 x 3/4 = $90,000 (before earning any interest). And the one that earns higher interest (let's call it account B) has $30,000 (before earning any interest).

Since the interest is compounded semiannually, account A will earn:

90,000 x 0.01 = $900 for the first half of the year, and

(90,000 + 900) x 0.01 = 90,900 x 0.01 = $909 for the second half of the year.

So the total amount of interest account A earns is 900 + 909 = $1809. That means account B earns 3636 - 1809 = $1827. We see that account B only has 1/3 of the amount of account A, yet it earns about the same amount in interest. It must mean that the interest rate of account B must be 3 times that of account A; that is, account B must be earning 6 percent annual interest. Let's verify that's the case:

If account B earns 6 percent annual interest, it will earn:

30,000 x 0.03 = $900 for the first half of the year, and

(30,000 + 900) x 0.03 = 30,900 x 0.03 = $927 for the second half of the year.

So the total amount of interest account B earns is 900 + 927 = $1827, which is exactly the amount of interest it earns for the year. So its interest rate must be 6 percent.

Answer: D

AAPL wrote:Manhattan Prep

Jeremiah invests his savings of $120,000 by dividing it between two interest-earning accounts. He puts 3/4 of his savings in an account that earns lower interest and 1/4 of his savings in an account that earns higher interest. He has no other accounts that earn interest and he makes $3,636 in interest by the end of the year. If one account earns 2 percent annual interest, and both accounts are compounded semiannually, what percent interest does the other account earn?

A. 3
B. 4
C. 5
D. 6
E. 7

OA D

Since all the answer choices are greater than 2, we can see that the account that earns 2 percent annual interest is the one that earns the lower interest (let's call it account A), and thus it has 120,000 x 3/4 = $90,000 (before earning any interest). And the one that earns higher interest (let's call it account B) has $30,000 (before earning any interest).

Since the interest is compounded semiannually, account A will earn:

90,000 x 0.01 = $900 for the first half of the year, and

(90,000 + 900) x 0.01 = 90,900 x 0.01 = $909 for the second half of the year.

So the total amount of interest account A earns is 900 + 909 = $1809. That means account B earns 3636 - 1809 = $1827. We see that account B only has 1/3 of the amount of account A, yet it earns about the same amount in interest. It must mean that the interest rate of account B must be 3 times that of account A; that is, account B must be earning 6 percent annual interest. Let's verify that's the case:

If account B earns 6 percent annual interest, it will earn:

30,000 x 0.03 = $900 for the first half of the year, and

(30,000 + 900) x 0.03 = 30,900 x 0.03 = $927 for the second half of the year.

So the total amount of interest account B earns is 900 + 927 = $1827, which is exactly the amount of interest it earns for the year. So its interest rate must be 6 percent.

Answer: D