Danny Joe borrows $10,500 from the bank at 11 percent annually compounded interest to be repaid in six equal annual installments. The interest paid in the first year is: $_________.
(Please calculate the arithmetic solution and show your work)
Pt = Po*(1+r)^n.
r = 11%/100% = 0.11 = APR expressed as a decimal.
n = 1 comp./yr * 1yr = 1 compounding
period.
Pt = $10500*(1.11)^1 = $11655.
I = Pt-Po = 11,655-10,500 = $1155.00.
Correct
To calculate the interest paid in the first year, we need to find the interest on the principal amount of $10,500 during the first year.
The formula to calculate compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment/loan amount
r = annual interest rate (in decimal)
n = number of times that interest is compounded per year
t = number of years
In this case, the principal (P) is $10,500, the annual interest rate (r) is 11%, the number of times compounded per year (n) is 1, and the total number of years (t) is 1 because we are only interested in the interest paid during the first year.
Plugging the values into the formula gives us:
A = 10,500(1 + 0.11/1)^(1*1)
A = 10,500(1 + 0.11)^1
A = 10,500(1.11)^1
A = 10,500 * 1.11
A = 11,655
To find the interest paid in the first year, we subtract the principal amount from the future value:
Interest paid in first year = Future value - Principal
Interest paid in first year = 11,655 - 10,500
Interest paid in first year = $1,155
Therefore, the interest paid in the first year is $1,155.
To calculate the interest paid in the first year, we need to know the formula for compound interest and the formula for calculating the equal annual installment.
The compound interest formula is:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the loan
P = the principal amount (loan amount)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
Using this formula, we can find the future value of Danny Joe's loan after 6 years.
Now, let's calculate the equal annual installment (EAI) using the formula:
EAI = P / ((1 - (1 + r/n)^(-nt)) / (r/n))
Once we have the EAI, we can find the interest paid in the first year by multiplying the EAI by the interest rate.
Given:
P = $10,500
r = 11% = 0.11 (as a decimal)
n = 1 (annually compounded)
t = 6
Now, let's calculate the future value of the loan after 6 years:
A = 10,500(1 + 0.11/1)^(1*6)
A = 10,500(1.11)^6
A ≈ 10,500 * 1.771561 = 18,502.395
Next, we can calculate the EAI:
EAI = 10,500 / ((1 - (1 + 0.11/1)^(-1*6)) / (0.11/1))
EAI = 10,500 / ((1 - (1.11)^(-6)) / (0.11))
EAI ≈ 10,500 / (1 - 0.593847) / 0.11
EAI ≈ 10,500 / 0.406153 / 0.11
EAI ≈ 10,500 / 36.9326652 ≈ 283.963
Finally, let's calculate the interest paid in the first year:
Interest Paid in First Year = EAI * r
Interest Paid in First Year ≈ 283.963 * 0.11 ≈ $31.24
Therefore, the interest paid in the first year is approximately $31.24.