Final answer:
To calculate the time to settle a $30,000 loan at 4.50% interest with $1,000 monthly payments, we use the Present Value of an annuity formula, requiring a financial calculator or software to solve for the number of months, which must then be rounded up to the nearest month.
Step-by-step explanation:
To determine how long it will take to settle a loan of $30,000 at 4.50% interest compounded monthly, with payments of $1,000 at the end of every month, we need to use the formula for the Present Value of an annuity:
PV = R * [(1 - (1+i)^-n) / i]
Where:
- PV is the present value or the amount of the loan.
- R is the monthly payment.
- i is the monthly interest rate (annual rate divided by 12).
- n is the number of months.
We can solve for n (number of months) once we have PV, R, and i. However, this cannot be directly solved through algebraic means because it is a form of the transcendental equation. Instead, a financial calculator or software that can handle such calculations is needed to find the precise duration.
Given that making higher payments can significantly decrease both the time and the total interest paid on a loan, as shown in the provided examples, it's essential to calculate the exact number of months accurately, rounding up to the nearest month for practical payment scheduling purposes.