Math 2000 - Solutions 7

Fall 2000

Unit 5A

6d. Setting P = $15,000, APR = 0.078, Y = 15, and n = 12 (for monthly compounding), the accumulated balance is

8. With P = $500, APR = 0.045, and n = 12, the compound interest formula gives the following balance after Y = 1 year.

The annual yield is the percentage increase in the balance in one year. We find that

The account has increased its value by 4.6% in one year.

20. Let’s look at the first five years. For Bernard’s plan, we set P = $1600, APR = 0.04, n = 1, and Y = 5. The accumulated balance is

For Carla’s plan, we set P = $1400, APR = 0.05, n = 365, and Y = 5. The accumulated balance is

Over a five-year period, Bernard has the better plan because of his larger initial deposit.

How about a 20-year period? For Bernard’s plan, we set P = $1600, APR = 0.04, n = 1, and Y = 20. The accumulated balance is

For Carla’s plan, we set P = $1400, APR = 0.05, n = 365, and Y = 20. The accumulated balance is

Over a 20-year period, Carla has the best plan, because of the higher interest rate.

24a. In this problem, we must find the initial deposit P that results in an accumulated balance of $100,000. We have n = 365 (daily compounding), Y = 18, and APR = 0.06. The initial deposit P must satisfy

We can solve this directly for P by dividing both sides of the equation by 2.9444. The required initial deposit is

An initial deposit of $33,963 is needed to accumulate $100,000 in 18 years with daily compounding.

Unit 5B

2. In this case, we set PMT = $50, APR = 0.0825, n = 12 payments per year, and Y = 40 years. Note that the monthly interest rate is i = APR/n. The accumulated balance at age 65 will be

The total amount deposited into the account over the 40-year period is

Thus the IRA has earned about $160,000 in interest.

6. We use the savings plan formula. For Polly’s investment plan, we set PMT = $50, APR = 0.06, and n = 12 payments per year. The accumulated balance after Y = 10 years will be

Polly’s total payments are

For Quint’s investment plan, we set PMT = $40, APR = 0.065, and n = 12 payment per year. The accumulated balance after Y = 10 years will be

Quint’s total payments are

Quint deposits less money into his savings plan than Polly, and he accumulates less money over the 10-year period (even though he has a slightly higher APR). There is another way to make the comparison. Quint’s total deposit ($4800) was increased by 40% over the 10-year period, while Polly’s total deposit was increased by about 36% over the same period. In a relative sense, one could argue that Quint did better.