Nathaniel want to buy a new bike. 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.
Option2: Take the store payment plan of 12 monthly payments of $180.00.
A.) If he chooses option 1, what will his monthly payment be?
B.) If he chooses Option 2, what annual rate of interest will he pay?
C) Calculate the total cost of each option. Which option should he choose, and why?
Assuming simple interest:
A. Po = 2055.99-400 = 1655.99
P = Po + Po*r*t
P = 1655.99 + 1655.99*0.0725*2
P = 1655.99 + 240.12 = $1896.11
1896.11/24mo = $79.00/Mo.
B. P = 180/mo * 12mo = $2160.
r = (p-Po)/Po = (2160-2055.99)/2055.99 =
0.0506 = 5.06% Per annum
C. Option 1: $1896.11
Option 2: $2160
He should choose option 1, because the
total cost and monthly payments are less,
Everything Henry calculated is correct until question C as the answer doesn't take into account the additional $400 down payment Nathaniel put towards the bike initially.
Total cost for option 1: $1896.11 + 400 = $2296.11
Option 2 is better
A.) If Nathaniel chooses option 1, he needs to take out a loan for the remaining amount.
The amount he needs to borrow is:
Borrowed amount = Total cost of the bike - Amount saved up
Borrowed amount = $2055.99 - $400.00 = $1655.99
Now, we can calculate the monthly payment:
Monthly payment = (Borrowed amount * (1 + (Interest rate / 100)) ^ (Number of years * 12)) / (Number of years * 12)
Interest rate = 7.25% per annum
Number of years = 2
Monthly payment = ($1655.99 * (1 + (7.25 / 100)) ^ (2 * 12)) / (2 * 12)
Monthly payment ≈ $76.97
So, if Nathaniel chooses option 1, his monthly payment will be approximately $76.97.
B.) If Nathaniel chooses option 2, he will be making 12 monthly payments of $180.00.
To calculate the annual interest rate for option 2, we can use the following formula:
Annual interest rate = [(Monthly payment * Number of payments) / Total cost of the bike] * 100
Monthly payment = $180.00
Number of payments = 12
Total cost of the bike = $2055.99
Annual interest rate = [($180.00 * 12) / $2055.99] * 100
Annual interest rate ≈ 10.47%
So, if Nathaniel chooses option 2, the annual rate of interest he will pay is approximately 10.47%.
C) To calculate the total cost of each option, we need to consider the monthly payments for option 1 and the total amount paid for option 2.
For option 1:
Total cost = (Monthly payment * Number of payments) + Amount saved up + Interest paid on the loan
Total cost = ($76.97 * 24) + $400.00 + (Borrowed amount * (Interest rate / 100) * Number of years)
Total cost = ($76.97 * 24) + $400.00 + ($1655.99 * (7.25 / 100) * 2)
Total cost ≈ $2179.28
For option 2:
Total cost = Monthly payment * Number of payments
Total cost = $180.00 * 12
Total cost = $2160.00
Comparing the total costs, option 1 has a total cost of approximately $2179.28 and option 2 has a total cost of $2160.00.
Therefore, Nathaniel should choose option 2 because it has a lower total cost.
To answer these questions, let's break down the information given:
Option 1:
- Nathaniel wants to pay $2055.99 cash.
- He only has $400.00 saved up, so he needs to take out a loan for the rest.
- The loan is for 2 years.
- The interest rate is 7.25% per annum.
Option 2:
- The store offers a payment plan of 12 monthly payments of $180.00.
Now let's calculate the answers:
A) If Nathaniel chooses option 1, what will his monthly payment be?
To calculate the monthly payment for the loan, we need to first find out the amount Nathaniel needs to borrow.
Amount to borrow = Total Cost of Bike - Amount saved up
Amount to borrow = $2055.99 - $400.00
Amount to borrow = $1655.99
To calculate the monthly payment, we can use the loan repayment formula:
Monthly Payment = (Loan Amount * Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate)^(-Number of Months))
Let's plug in the values:
Loan Amount = $1655.99
Number of Months = 2 years * 12 months/year = 24 months
Annual Interest Rate = 7.25%
Monthly Interest Rate = Annual Interest Rate / 12
Monthly Interest Rate = 7.25% / 12 = 0.00604167
Using the loan repayment formula, we get:
Monthly Payment = ($1655.99 * 0.00604167) / (1 - (1 + 0.00604167)^(-24))
Monthly Payment ≈ $78.87
So if Nathaniel chooses option 1, his monthly payment will be approximately $78.87.
B) If Nathaniel chooses Option 2, what annual rate of interest will he pay?
In this case, we already know the monthly payment and the number of payments.
Monthly Payment = $180.00
Number of Payments = 12
Now we can use the loan repayment formula to find the annual interest rate:
Annual Interest Rate = ((Monthly Payment * Number of Payments) / Loan Amount) * 100
Loan Amount = Total Cost of Bike
Loan Amount = $2055.99
Plugging in the values, we get:
Annual Interest Rate = (($180.00 * 12) / $2055.99) * 100
Annual Interest Rate ≈ 10.5%
So if Nathaniel chooses Option 2, he will pay an annual interest rate of approximately 10.5%.
C) Calculate the total cost of each option. Which option should he choose, and why?
To calculate the total cost of each option, we need to consider the amount Nathaniel will pay over the loan period or payment plan.
For Option 1:
Total Cost = Loan Amount + Total Interest Paid
Total Interest Paid = Monthly Payment * Number of Payments - Loan Amount
Plugging in the values, we get:
Total Interest Paid = ($78.87 * 24) - $1655.99
Total Interest Paid ≈ $89.28
Total Cost = $1655.99 + $89.28
Total Cost ≈ $1745.27
For Option 2:
Total Cost = Monthly Payment * Number of Payments
Plugging in the values, we get:
Total Cost = $180.00 * 12
Total Cost = $2160.00
Comparing the total costs:
- Option 1: Total Cost ≈ $1745.27
- Option 2: Total Cost = $2160.00
Option 1 has a lower total cost compared to Option 2, so Nathaniel should choose Option 1.
In summary:
- For option 1, his monthly payment will be approximately $78.87.
- For option 2, he will pay an annual interest rate of approximately 10.5%.
- The total cost for option 1 is approximately $1745.27, while for option 2 it is $2160.00. Nathaniel should choose option 1 because it has a lower total cost.