Nathaniel wants to buy a new bicycle. His payment options are:
Option 1: Pay $2055.99 cash. He only has $400.00 saved up, so he can take out a loan for the rest from his bank at a rate of 7.25% per annum over 2 years.
Option 2: Take the store payment plan of 12 monthly payments of $180.00.
a) If he chooses Option 1, what will his monthly payments be?
b) If he chooses Option 2, what annual rate of interest will he pay? (in percentage)
To find the monthly payments for Option 1, we need to calculate the loan amount Nathaniel needs to take from the bank.
The difference between the cost of the bicycle ($2055.99) and the amount saved up ($400) is:
Loan amount = $2055.99 - $400 = $1655.99
Now, we need to calculate the total amount Nathaniel needs to pay back to the bank after two years, including the interest. We can use the formula for calculating the future value of a loan:
Future value = Principal + (Principal * Rate * Time)
Principal = Loan amount = $1655.99
Rate = 7.25% per annum = 0.0725
Time = 2 years
Future value = $1655.99 + ($1655.99 * 0.0725 * 2)
Future value = $1655.99 + ($239.78)
Future value = $1895.77
Now, we can find the monthly payments by dividing the total amount by the number of months in two years (24 months):
Monthly payment = Future value / Number of months
Monthly payment = $1895.77 / 24
Monthly payment ≈ $78.99
Therefore, if Nathaniel chooses Option 1, his monthly payments will be approximately $78.99.
To find the annual rate of interest for Option 2, we need to consider the total amount paid over 12 months.
Total amount paid = 12 monthly payments * $180.00
Total amount paid = $2160.00
The interest paid can be calculated by finding the difference between the total amount paid and the initial cost of the bicycle:
Interest paid = Total amount paid - Cost of the bicycle
Interest paid = $2160.00 - $2055.99
Interest paid = $104.01
Now, we can calculate the annual interest rate:
Rate = (Interest paid / Cost of the bicycle) * (100 / Time)
Rate = ($104.01 / $2055.99) * (100 / 2)
Rate ≈ 2.52%
Therefore, if Nathaniel chooses Option 2, he will pay an annual interest rate of approximately 2.52%.
(2055.99-400)(1 + .075)^2 / 12 = 159.48
(2055.99-400)(1 + r) = 12*180
r = 30.4%