Gene's Art Gallery is notoriously known as a slow-payer. The firm currently needs to borrow $27,500 and only one company will deal with them. The terms of the loan call for daily payments of $100. The first payment is due today. The interest rate is 21.9%, compounded daily. What is the time period of this loan? (Assume 365 days in a year)
assuming payments at the end of the day.
100(1 - 1.0006^-n)/.0006 = 27500
1 - 1.0006^-n = .165
1.006^-n = .835
log both sides and use log rules
-n(log 1.0006) = log .835
-n = -300.6
300 full daily payments of $100 plus a partial payment on the 301st day.
To find the time period of the loan, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = Total amount repaid
P = Principal loan amount
r = Annual interest rate (as a decimal)
n = Number of times interest is compounded per year
t = Time period in years
We need to find the time period, t. Let's rearrange the formula to solve for t:
t = (ln(A/P)) / (n * ln(1 + r/n))
Given the principal loan amount, P = $27,500, the daily payment, A = $100, and the annual interest rate, r = 21.9% = 0.219. The interest is compounded daily, so n = 365.
Substituting the values into the formula:
t = (ln(100/27,500)) / (365 * ln(1 + 0.219/365))
Calculating this expression:
t ≈ -0.5502
The result is negative because the amount repaid is less than the principal loan amount. In this case, it indicates that the loan will never be fully repaid.
However, let's assume that the daily payments continue until the full loan amount is satisfied. To find the number of days it would take to repay the loan completely, we can rearrange the formula:
t = (ln(A/P)) / (n * ln(1 + r/n))
Substituting the values:
t = (ln(27,500/100)) / (365 * ln(1 + 0.219/365))
Calculating this expression:
t ≈ 41.6888
Therefore, it would take approximately 41.69 days to repay the loan completely.
To find the time period of the loan, we need to determine how many days it will take for Gene's Art Gallery to repay the loan.
First, let's calculate the daily interest rate. The annual interest rate is 21.9%, so the daily interest rate is 21.9% divided by 365 (the number of days in a year).
Daily interest rate = 21.9% / 365 = 0.006%
Next, we can determine how many days it will take to repay the loan by dividing the loan amount by the daily payment:
Number of days = Loan amount / Daily payment
= $27,500 / $100
= 275
Therefore, it will take Gene's Art Gallery 275 days to repay the loan.
Note: In this calculation, we assume that the daily interest is calculated based on a 365-day year. Keep in mind that interest rates can be calculated differently by different lenders, so always review the loan agreement to ensure accuracy.