APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

(Interest Rate Word Problems)

1. To solve an exponential or logarithmic word problems, convert the narrative to an equation and solve the equation.

Example 3: You are trying to decide whether to deposit your $1,000 in Bank A, Bank B, Bank C, Bank D, or Bank E. You want to leave your money in the bank for 5 years and you want to maximize your ending balance.
Bank A pays 15% simple interest; Bank B pays 14% interest compounded monthly; Bank C pays 13.3% interest compounded quarterly; Bank D pays 13% interest compounded continuously; and Bank E pays 13.5% interest compounded weekly.
Which bank pays the most and which bank pays the least? In which bank will you deposit your money? What is the effective interest rate of each of the banks?

Explanation and Solution:

Bank A: Simple interest means that there is no compounding. At the end of the five years, you will have your initial $1,000 plus the interest earned on the $1,000. The interest is calculated by multiplying the interest rate, 15%, by the number of years, 5. Therefore, the interest earned is . You will have

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

or

which gives

hence

This means that although the bank is quoting a high interest rate of 15%, in reality, the effective interest rate is only 11.84%

Bank B: This bank pays 14% interested compounded monthly. This means that the interest is paid every month on the balance in the account. Or, the interest is paid 12 times a year for 5 years. Your balance will be

or

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

or

which gives

hence

This means that although the bank is quoting a 14% interest rate, in reality, the effective interest rate is a little higher at 14.93.%

Bank C: This bank pays 13.3% interest compounded quarterly. This means that the interest is paid every quarter on the balance in the account. Or, the interest is paid 4 times a year for 5 years. Your balance will be

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

which gives

This means that although the banks is quoting a 13.3% interest rate, in reality, the effective interest rate is a little higher at 13.98%

Bank D: This bank pays 13% interested compounded continuously. This means that the interest is calculated and paid continuously on the balance in the account. Your balance will be

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

which gives

This means that although the banks is quoting a 13% interest rate, in reality, the effective interest rate is a little higher at 13.88%

Bank E: This bank pays 13.5% interested compounded weekly. This means that the interest is paid every week on the balance in the account. Or, the interest is paid 52 times a year for 5 years. Your balance will be

or

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

which gives

This means that although the banks is quoting a 13.5% interest rate, in reality, the effective interest rate is a little higher at 14.43%

To recap: The effective interest rates of Banks A, B, C, D, and E are: 11.84%, 14.93%, 13.98%, 13.88%, and 14.43%, respectively. It appears that Bank B pays the most and Bank A pays the least.
Bank B: This bank pays 14% interested compounded monthly. This means that the interest is paid every month on the balance in the account. Or, the interest is paid 12 times a year for 5 years. Your balance will be

or

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

or

which gives

hence

This means that although the bank is quoting a 14% interest rate, in reality, the effective interest rate is a little higher at 14.93.%

Bank C: This bank pays 13.3% interest compounded quarterly. This means that the interest is paid every quarter on the balance in the account. Or, the interest is paid 4 times a year for 5 years. Your balance will be

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

which gives

This means that although the banks is quoting a 13.3% interest rate, in reality, the effective interest rate is a little higher at 13.98%

Bank D: This bank pays 13% interested compounded continuously. This means that the interest is calculated and paid continuously on the balance in the account. Your balance will be

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

which gives

This means that although the banks is quoting a 13% interest rate, in reality, the effective interest rate is a little higher at 13.88%

Bank E: This bank pays 13.5% interested compounded weekly. This means that the interest is paid every week on the balance in the account. Or, the interest is paid 52 times a year for 5 years. Your balance will be

or

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

which gives

This means that although the banks is quoting a 13.5% interest rate, in reality, the effective interest rate is a little higher at 14.43%

To recap: The effective interest rates of Banks A, B, C, D, and E are: 11.84%, 14.93%, 13.98%, 13.88%, and 14.43%, respectively. It appears that Bank B pays the most and Bank A pays the least.
Bank C: This bank pays 13.3% interest compounded quarterly. This means that the interest is paid every quarter on the balance in the account. Or, the interest is paid 4 times a year for 5 years. Your balance will be

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

which gives

This means that although the banks is quoting a 13.3% interest rate, in reality, the effective interest rate is a little higher at 13.98%

Bank D: This bank pays 13% interested compounded continuously. This means that the interest is calculated and paid continuously on the balance in the account. Your balance will be

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

which gives

This means that although the banks is quoting a 13% interest rate, in reality, the effective interest rate is a little higher at 13.88%

Bank E: This bank pays 13.5% interested compounded weekly. This means that the interest is paid every week on the balance in the account. Or, the interest is paid 52 times a year for 5 years. Your balance will be

or

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

which gives

This means that although the banks is quoting a 13.5% interest rate, in reality, the effective interest rate is a little higher at 14.43%

To recap: The effective interest rates of Banks A, B, C, D, and E are: 11.84%, 14.93%, 13.98%, 13.88%, and 14.43%, respectively. It appears that Bank B pays the most and Bank A pays the least.
Bank D: This bank pays 13% interested compounded continuously. This means that the interest is calculated and paid continuously on the balance in the account. Your balance will be

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

which gives

This means that although the banks is quoting a 13% interest rate, in reality, the effective interest rate is a little higher at 13.88%

Bank E: This bank pays 13.5% interested compounded weekly. This means that the interest is paid every week on the balance in the account. Or, the interest is paid 52 times a year for 5 years. Your balance will be

or

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

which gives

This means that although the banks is quoting a 13.5% interest rate, in reality, the effective interest rate is a little higher at 14.43%

To recap: The effective interest rates of Banks A, B, C, D, and E are: 11.84%, 14.93%, 13.98%, 13.88%, and 14.43%, respectively. It appears that Bank B pays the most and Bank A pays the least.
Bank E: This bank pays 13.5% interested compounded weekly. This means that the interest is paid every week on the balance in the account. Or, the interest is paid 52 times a year for 5 years. Your balance will be

or

at the end of the five years.
The effective interest rate is the interest rate, compounded yearly, that would give you the same balance in five years

which gives

This means that although the banks is quoting a 13.5% interest rate, in reality, the effective interest rate is a little higher at 14.43%

To recap: The effective interest rates of Banks A, B, C, D, and E are: 11.84%, 14.93%, 13.98%, 13.88%, and 14.43%, respectively. It appears that Bank B pays the most and Bank A pays the least.