The outstanding balance on Bill's credit card account is 3690 dollars. The bank issuing the credit card is charging 21 percent of interest per year compounded monthly. If Bill decides to pay off his balance in equal monthly installments at the end of each month for the next 17 months, how much will be his monthly payment?
Use the compound interest formula:
P=$3690
i=0.21/12=0.0175
n=17
M=amount of monthly payment
Amount after 17 months (without payment)
= A(1+i)^n
Value of his payments at the same interest rate
=M+M(1+i)+M(1+i)^2+...+M(1+i)^(n-1)
=M((1+i)^n-1)/(1+i-1)
=M(1+i)^n/i
Equate the two:
M((1+i)^n-1)/i = A(1+i)^n
=>
M=A(1+i)^n*i/((1+i)^n-1)
=3690(1.0175)^17*0.0175/(1.0175^17-1)
=$252.824
Correction in an intermediate step (only):
...
=M((1+i)^n-1)/(1+i-1)
=M((1+i)^n-1)/i
To determine the monthly payment required to pay off Bill's outstanding credit card balance over 17 months, we can use the formula for the present value of an annuity.
The formula for the present value of an annuity is given as:
P = PMT × (1 - (1 + r)^(-n)) / r
Where:
P = Present value (outstanding balance)
PMT = Monthly payment
r = Monthly interest rate
n = Number of months
First, we need to calculate the monthly interest rate. The annual interest rate of 21 percent is compounded monthly, so we divide it by 12 to get the monthly interest rate. In this case, the monthly interest rate (r) would be: 21% / 12 = 0.0175.
Next, we substitute the given values into the formula and solve for PMT (Monthly payment):
3690 = PMT × (1 - (1 + 0.0175)^(-17)) / 0.0175
To solve this equation, perform the following steps:
1. Add 1 to the monthly interest rate: 1 + 0.0175 = 1.0175
2. Raise the result to the power of the negative number of months: 1.0175^(-17)
3. Subtract the result from 1: 1 - 1.0175^(-17)
4. Multiply this result by the monthly payment (PMT): PMT * (1 - 1.0175^(-17))
5. Divide the outstanding balance (3690) by the above value to solve for PMT: 3690 / (1 - 1.0175^(-17)) = 230.78 (approx.)
Therefore, Bill's monthly payment would be approximately $230.78.