Correct Answer: A) Less than 5 years
The key to this problem is understanding how interest affects the time required to repay a loan.
Even though the loan balance is cut in half, the repayment time will be less than half the original 10 years because less interest accumulates.
Step 1: Identify the information given.
Original loan = $48,000
New loan = $24,000
Annual interest rate = 12%
Monthly payment = $689
Interest compounds monthly.
Monthly interest rate = 12% / 12 = 1% = 0.01
Step 2: Calculate how long it takes to repay $24,000.
We use the loan repayment formula:
n = -ln(1 - Pr/M) / ln(1 + r)
Where:
n = Number of monthly payments
P = Loan balance ($24,000)
r = Monthly interest rate (0.01)
M = Monthly payment ($689)
Step 3: Substitute the numbers.
n = -ln(1 - (24,000 × 0.01)/689) / ln(1.01)
n = -ln(1 - 240/689) / ln(1.01)
n = -ln(0.651669) / ln(1.01)
n ≈ 43.04 months
Step 4: Convert months into years.
43.04 / 12 ≈ 3.59 years
This means the loan will be paid off in approximately 3 years and 7 months, with a smaller final payment in the 44th month.
Final Answer: A) Less than 5 years
Why Isn't the Answer 5 Years?
It might seem logical that cutting the loan in half would cut the repayment time from 10 years to 5 years.
However, interest changes the calculation.
For the original $48,000 loan:
First month's interest = $48,000 × 0.01 = $480
Amount of first payment reducing principal = $689 - $480 = $209
For the new $24,000 loan:
First month's interest = $24,000 × 0.01 = $240
Amount of first payment reducing principal = $689 - $240 = $449
Notice that with the smaller loan, $449 of the first payment reduces the principal instead of just $209.
That means the smaller loan gets paid off much faster.
Easy Way to Remember:
A smaller loan balance means less interest is charged each month. If you keep making the same monthly payment, more money goes toward reducing the principal, allowing you to pay off the loan faster.
Therefore, the correct choice is A) Less than 5 years.
