Math Problem Statement
Lois received a 9-year subsidized student loan of $39,000 at an annual interest rate of 3.625%. Determine her monthly payment on the loan after she graduates in 3 years.
Solution
To determine Lois's monthly payment after she graduates in 3 years, we'll assume that the loan is structured such that she starts making payments only after graduation. Let's break down the process in the following steps:
Step 1: Calculate the loan balance at the time of repayment.
Since Lois will not start repaying the loan until after 3 years, we need to find the loan balance after this period. Interest will accumulate during these 3 years, assuming she is not making any payments during this time.
The formula for the future value of a loan with compound interest is:
Where:
- is the principal loan amount.
- is the monthly interest rate.
- is the total number of months over which the loan accrues interest before she starts repayment.
So:
Step 2: Calculate the monthly payment
Once we have the total loan balance after 3 years, Lois will begin repaying it over the remaining 6 years. The formula for calculating the monthly payment on an amortized loan is:
Where:
- is the loan balance after 3 years (from Step 1).
- is the monthly interest rate.
- is the number of months over which the loan is repaid.
Let me compute these steps for you.After 3 years of accrued interest, the loan balance will be approximately $43,473.33.
Lois will need to make a monthly payment of approximately $672.75 over the remaining 6 years to pay off the loan.
Would you like more details on any part of this process, or do you have further questions?
Here are 5 related questions you might consider:
- How does the loan balance change with a higher interest rate?
- What would the monthly payment be if the repayment period was extended to 10 years?
- How is the loan affected if she makes partial payments during the first 3 years?
- What is the total amount paid over the life of the loan?
- How does compound interest differ from simple interest in this scenario?
Tip: When calculating interest on a loan, remember that small changes in the interest rate or loan term can lead to significant differences in the total amount paid.
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Math Problem Analysis
Mathematical Concepts
Loan Amortization
Interest Rates
Compound Interest
Formulas
Future value of a loan with compound interest: A = P × (1 + r)^t
Monthly payment on an amortized loan: M = (A × r) / (1 - (1 + r)^(-n))
Theorems
Compound Interest Formula
Amortization Formula
Suitable Grade Level
Grades 11-12
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