Final answer:
The amount due on a $16,120 loan at 5% interest, compounded annually, will reach $18,300 in approximately 3 years. The future value of the loan is calculated using the compound interest formula, and the correct answer is option A.
Step-by-step explanation:
The question asks about the time it will take for a loan to reach a certain amount when compounded annually. To find out after how many years a loan of $16,120 at 5% interest, compounded annually, would grow to become $18,300, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual interest rate (decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for
In this case, we have:
A = 18,300
P = 16,120
r = 0.05 (5% as a decimal)
n = 1 (since it is compounded annually)
t = unknown (this is what we are solving for)
Rearranging the formula to solve for t, we have:
t = log(A/P) / (n * log(1 + r/n))
Plugging in our values:
t = log(18,300/16,120) / (1 * log(1 + 0.05/1))
t = log(1.135) / log(1.05)
t ≈ 2.54 years
Since we round to the nearest year, the amount due will reach $18,300 in approximately 3 years, making option A the correct answer.