you borrow $5,000 from your parents to purchase a used car. The arrangements of the loan are such that you make payments of $250 per month toward the balance plus 1% interest on the unpaid balance from the previous month. (a) Find the first year monthly payments and the unpaid balance after each month. (b) Find the total amount of interest paid over the term of the loan.
create a table (amortization table)
time-interest-payment-balance
now ---- 0 ------0 ------ 5000.00
1 ---- 50.00 ---300.00 - 4475.00
2 ---- 44.75 ---294.75 - 4225.00
3 ---- 42.25 ---292.25 - 3975.00
4 ---- 39.75 ---289.75 - 3725.00
5 ---- 37.25 ---287.25 - 3475.00
etc
Since the monthly payment varies and is the total of
250 + interest, the monthly balance is reduced by exactly $250 , and it will take 20 months to pay it off.
This is not a usual procedure, the payments usually are the same for each month, and the calculations do not fit into one of the four basic compound interest formulas.
To solve this problem, we need to use the given information and apply it step-by-step. Let's break it down:
(a) First, let's calculate the monthly payments and unpaid balance for the first year.
1. Balance at the beginning: $5,000
2. Monthly payment: $250
3. Interest rate: 1% per month
Month 1:
Payment: $250
Interest on the previous balance: 1% of $5,000 = $50
Principal payment (Payment - Interest): $250 - $50 = $200
Remaining balance: $5,000 - $200 = $4,800
Month 2:
Payment: $250
Interest on the previous balance: 1% of $4,800 = $48
Principal payment: $250 - $48 = $202
Remaining balance: $4,800 - $202 = $4,598
Month 3:
Payment: $250
Interest on the previous balance: 1% of $4,598 = $45.98 (round to $46)
Principal payment: $250 - $46 = $204
Remaining balance: $4,598 - $204 = $4,394
Continuing this process, we can calculate the monthly payments and unpaid balance for the remaining months of the first year.
Month 4:
Payment: $250
Interest on the previous balance: 1% of $4,394 = $43.94 (round to $44)
Principal payment: $250 - $44 = $206
Remaining balance: $4,394 - $206 = $4,188
Month 5:
Payment: $250
Interest on the previous balance: 1% of $4,188 = $41.88 (round to $42)
Principal payment: $250 - $42 = $208
Remaining balance: $4,188 - $208 = $3,980
Month 6:
Payment: $250
Interest on the previous balance: 1% of $3,980 = $39.80 (round to $40)
Principal payment: $250 - $40 = $210
Remaining balance: $3,980 - $210 = $3,770
Month 7:
Payment: $250
Interest on the previous balance: 1% of $3,770 = $37.70 (round to $38)
Principal payment: $250 - $38 = $212
Remaining balance: $3,770 - $212 = $3,558
Month 8:
Payment: $250
Interest on the previous balance: 1% of $3,558 = $35.58 (round to $36)
Principal payment: $250 - $36 = $214
Remaining balance: $3,558 - $214 = $3,344
Month 9:
Payment: $250
Interest on the previous balance: 1% of $3,344 = $33.44 (round to $33)
Principal payment: $250 - $33 = $217
Remaining balance: $3,344 - $217 = $3,127
Month 10:
Payment: $250
Interest on the previous balance: 1% of $3,127 = $31.27 (round to $31)
Principal payment: $250 - $31 = $219
Remaining balance: $3,127 - $219 = $2,908
Month 11:
Payment: $250
Interest on the previous balance: 1% of $2,908 = $29.08 (round to $29)
Principal payment: $250 - $29 = $221
Remaining balance: $2,908 - $221 = $2,687
Month 12:
Payment: $250
Interest on the previous balance: 1% of $2,687 = $26.87 (round to $27)
Principal payment: $250 - $27 = $223
Remaining balance: $2,687 - $223 = $2,464
Therefore, the first-year monthly payments and unpaid balance are as follows:
Month 1: Payment $250, Remaining balance: $4,800
Month 2: Payment $250, Remaining balance: $4,598
Month 3: Payment $250, Remaining balance: $4,394
Month 4: Payment $250, Remaining balance: $4,188
Month 5: Payment $250, Remaining balance: $3,980
Month 6: Payment $250, Remaining balance: $3,770
Month 7: Payment $250, Remaining balance: $3,558
Month 8: Payment $250, Remaining balance: $3,344
Month 9: Payment $250, Remaining balance: $3,127
Month 10: Payment $250, Remaining balance: $2,908
Month 11: Payment $250, Remaining balance: $2,687
Month 12: Payment $250, Remaining balance: $2,464
(b) To find the total amount of interest paid over the term of the loan, we need to sum up the interest paid each month.
Total interest paid:
Month 1: $50
Month 2: $48
Month 3: $46
Month 4: $44
Month 5: $42
Month 6: $40
Month 7: $38
Month 8: $36
Month 9: $33
Month 10: $31
Month 11: $29
Month 12: $27
Total interest paid = $50 + $48 + $46 + $44 + $42 + $40 + $38 + $36 + $33 + $31 + $29 + $27 = $494
Therefore, the total amount of interest paid over the term of the loan is $494.
To calculate the first-year monthly payments and the unpaid balance after each month, we'll break the problem down into steps.
Step 1: Calculate the interest for the first month.
To calculate the interest, we need to determine the unpaid balance from the previous month. Since it's the first month, the unpaid balance is $5,000. The interest for the first month is 1% of $5,000, which is $50.
Step 2: Calculate the payment for the first month.
The payment for the first month is the monthly payment ($250) plus the interest ($50), which gives us a total payment of $300.
Step 3: Calculate the unpaid balance after the first month.
To calculate the unpaid balance after the first month, subtract the payment ($300) from the previous month's unpaid balance ($5,000), which gives us $4,700.
Step 4: Repeat steps 1-3 for the subsequent months.
Using the result from Step 3 as the unpaid balance for the next month, we can calculate the interest and payment for the second month. Repeat this process for each subsequent month.
Month 2:
Unpaid balance = $4,700
Interest for the second month: 1% of $4,700 = $47
Payment for the second month: $250 + $47 = $297
Unpaid balance after the second month: $4,700 - $297 = $4,403
Month 3:
Unpaid balance = $4,403
Interest for the third month: 1% of $4,403 = $44.03
Payment for the third month: $250 + $44.03 = $294.03
Unpaid balance after the third month: $4,403 - $294.03 = $4,108.97
Continue this process for the remaining months of the first year.
For part (b) - finding the total amount of interest paid over the term of the loan:
To calculate the total interest paid over the term of the loan, sum up the interest paid for each month.
For example, to find the interest paid in the first year, add up the interest paid in each of the twelve months.
Interest paid in the first month = $50
Interest paid in the second month = $47
Interest paid in the third month = $44.03
...
Interest paid in the twelfth month = $5.49
Add up these amounts to find the total interest paid over the first year, and repeat the process for subsequent years, if necessary.
Once you have calculated the interest paid over the entire term, you can get the answer to part (b) of the question.