Antonio has just graduated from four years of college. For the last two years, he took out a Stafford loan to pay for his tuition. Each loan had a duration of ten years and interest compounded monthly. Antonio will pay each of them back by making monthly payments, starting as he graduates. Antonio's loans are detailed in the table below.
| Year | Loan Amount ($) | Interest Rate (%) | Subsidized? |
| ------ | --------------- | ----------------- | ----------- |
| Junior | 5,894 | 6.9 | Y |
| Senior | 5,258 | 7.5 | N |
Once all of his loans are paid off, what will Antonio's total lifetime cost be? Round all dollar values to the nearest cent.
a. $16,246.80
b. $17,804,40
c. $7,593.16
d. $9,874.76
Please select the best answer from the choices provided
Answer (1)
Calculate the monthly payment for the Junior year loan using the loan payment formula: P 1 = ( 1 + 0.00575 ) 120 − 1 5894 ⋅ 0.00575 ⋅ ( 1 + 0.00575 ) 120 ≈ 68.13 .
Calculate the monthly payment for the Senior year loan using the loan payment formula: P 2 = ( 1 + 0.00625 ) 120 − 1 5258 ⋅ 0.00625 ⋅ ( 1 + 0.00625 ) 120 ≈ 62.41 .
Calculate the total cost for the Junior year loan: C 1 = 68.13 × 120 = 8175.60 , and for the Senior year loan: C 2 = 62.41 × 120 = 7489.20 .
Calculate the total lifetime cost: C = 8175.60 + 7489.20 = 15664.80 .
Explanation
Understanding the Problem We are given two loans with different amounts, interest rates, and the same duration. We need to calculate the total lifetime cost, which is the sum of all payments made for both loans.
Calculating Monthly Payment for Junior Year Loan First, let's calculate the monthly payment for the Junior year loan. The loan amount is $A_1 = $5894, the annual interest rate is 6.9% , so the monthly interest rate is r 1 = 12 6.9% = 12 0.069 = 0.00575 . The number of payments is n 1 = 10 × 12 = 120 . Using the loan payment formula, the monthly payment is: P 1 = ( 1 + r 1 ) n 1 − 1 A 1 ⋅ r 1 ⋅ ( 1 + r 1 ) n 1 = ( 1 + 0.00575 ) 120 − 1 5894 ⋅ 0.00575 ⋅ ( 1 + 0.00575 ) 120 The result of this calculation is approximately $P_1 = $68.13.
Calculating Monthly Payment for Senior Year Loan Next, let's calculate the monthly payment for the Senior year loan. The loan amount is $A_2 = $5258, the annual interest rate is 7.5% , so the monthly interest rate is r 2 = 12 7.5% = 12 0.075 = 0.00625 . The number of payments is n 2 = 10 × 12 = 120 . Using the loan payment formula, the monthly payment is: P 2 = ( 1 + r 2 ) n 2 − 1 A 2 ⋅ r 2 ⋅ ( 1 + r 2 ) n 2 = ( 1 + 0.00625 ) 120 − 1 5258 ⋅ 0.00625 ⋅ ( 1 + 0.00625 ) 120 The result of this calculation is approximately $P_2 = $62.41.
Calculating Total Cost for Each Loan Now, let's calculate the total cost for each loan. The total cost for the Junior year loan is $C_1 = P_1 \cdot n_1 = 68.13 \times 120 = 8175.60. T h e t o t a l cos t f or t h e S e ni orye a r l o ani s C_2 = P_2 \cdot n_2 = 62.41 \times 120 = $7489.20.
Calculating Total Lifetime Cost Finally, let's calculate the total lifetime cost by adding the total costs of both loans: $C = C_1 + C_2 = 8175.60 + 7489.20 = $15664.80.
Final Answer Therefore, Antonio's total lifetime cost will be approximately $15 , 664.80 .
Examples
Understanding loan payments is crucial in personal finance. For example, when buying a car or a house, you'll likely take out a loan. Knowing how to calculate monthly payments and total costs helps you budget effectively and compare different loan options. This ensures you make informed decisions and avoid overpaying in the long run. By understanding these concepts, you can manage your finances responsibly and achieve your financial goals.
