Respuesta :
B(n) = A(1 + i)^n - (P/i)[(1 + i)^n - 1]
where B is the balance after n payments are made, i is the monthly interest rate, P is the monthly payment and A is the initial amount of loan.
We require B(n) = 0...i.e. balance of 0 after n months.
so, 0 = A(1 + i)^n - (P/i)[(1 + i)^n - 1]
Then, with some algebraic juggling we get:
n = -[log(1 - (Ai/P)]/log(1 + i)
Now, payment is at the beginning of the month, so A = $754.43 - $150 => $604.43
Also, i = (13.6/100)/12 => 0.136/12 per month
i.e. n = -[log(1 - (604.43)(0.136/12)/150)]/log(1 + 0.136/12)
so, n = 4.15 months...i.e. 4 payments + remainder
b) Now we have A = $754.43 - $300 = $454.43 so,
n = -[log(1 - (454.43)(0.136/12)/300)]/log(1 + 0.136/12)
so, n = 1.54 months...i.e. 1 payment + remainder
where B is the balance after n payments are made, i is the monthly interest rate, P is the monthly payment and A is the initial amount of loan.
We require B(n) = 0...i.e. balance of 0 after n months.
so, 0 = A(1 + i)^n - (P/i)[(1 + i)^n - 1]
Then, with some algebraic juggling we get:
n = -[log(1 - (Ai/P)]/log(1 + i)
Now, payment is at the beginning of the month, so A = $754.43 - $150 => $604.43
Also, i = (13.6/100)/12 => 0.136/12 per month
i.e. n = -[log(1 - (604.43)(0.136/12)/150)]/log(1 + 0.136/12)
so, n = 4.15 months...i.e. 4 payments + remainder
b) Now we have A = $754.43 - $300 = $454.43 so,
n = -[log(1 - (454.43)(0.136/12)/300)]/log(1 + 0.136/12)
so, n = 1.54 months...i.e. 1 payment + remainder
The number of months it will take to pay off the credit card is 4
How to calculate the remaining balance if a constant payment is being done against a loan amount?
Suppose that we have:
- Present value of amount to be paid = [tex]P_v[/tex]
- The rate of interest per unit time = r
- The counts of units of time = n
- The payment amount that is being done in each unit of time once = P
- The remaining balance after n units of time to be paid = Future value = [tex]F_v[/tex]
Then, we get:
[tex]F_v = P_V (1+r)^n - P\left[ \dfrac{(1+r)^n - 1}{r} \right ][/tex]
For this case, we're specified that:
Payments are done on monthly basis, so unit of time is in months.
The initial balance of credit card = [tex]P_v[/tex] = $754.43
The monthly payment that will be done = P = $300
The rate of interest = 13.6% APR. Since unit of time is in months, and APR is interest for 1 year, so r = 13.6%/12 (as there are 12 months in 1 year) = 1.133% = 0.0113 (converted percent to decimal) approx.
n = to find out.
Since we need to pay off the balance, so the future value is , or [tex]F_v[/tex] = 0
Thus, using the formula for future value, we get:
[tex]F_v = P_V (1+r)^n - P\left[ \dfrac{(1+r)^n - 1}{r} \right ][/tex]
[tex]0 \approx 754.43 (1+0.0113)^n - 300\left[ \dfrac{(1+0.0113)^n - 1}{0.0113} \right ]\\2.5148(1+0.0113)^n = \dfrac{(1+0.0113)^n - 1}{0.0113} \\\\0.03345 = 1 - 1/(1+0.0113)^n\\\\(1+0.0113)^n = \dfrac{1}{1 - 0.03345}\\n = \log_{1.0113}\left(\dfrac{1}{0.96655}\right) \approx 3.028[/tex]
Since payment is being done on the beggining of each month, so the .028 after 3 month will be done on the beggining of fourth month.
Thus, the number of months it will take to pay off the credit card is 4
Learn more about monthly payment here:
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