What is a counter example for: If the name of the month begins with a J, then it is a summer month.
2 answers:
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Part 1:
After payment of $300, remaining balance = $2,348.62 - $300 = $2,048.62.
Interest accrued is given by:
Had it been $600 was paid, remaining balance = $2,348.62 - $600 = $1748.62. Interest accrued is given by:
Difference in interest accrued = $14.94 - $12.75 = $2.19
Part 2:
The present value of an annuity is given by:
![PV= \frac{P\left[1-\left(1+ \frac{r}{12} \right)^{-12n}\right]}{ \frac{r}{12} }](https://tex.z-dn.net/?f=PV%3D%20%5Cfrac%7BP%5Cleft%5B1-%5Cleft%281%2B%20%5Cfrac%7Br%7D%7B12%7D%20%5Cright%29%5E%7B-12n%7D%5Cright%5D%7D%7B%20%5Cfrac%7Br%7D%7B12%7D%20%7D)
Where PV is the amount to be repaid, P is the equal monthly payment, r is the annual interest rate and n is the number of years.
Thus,
![2348.62= \frac{600\left[1-\left(1+ \frac{0.0875}{12}\right)^{-12n}\right]}{\frac{0.0875}{12}} \\ \\ \Rightarrow 1-(1+0.007292)^{-12n}= \frac{2348.62\times0.0875}{12\times600} =0.028542 \\ \\ \Rightarrow(1.007292)^{-12n}=1-0.028542=0.971458 \\ \\ \Rightarrow \log(1.007292)^{-12n}=\log0.971458 \\ \\ \Rightarrow-12n\log1.007292=\log0.971458 \\ \\ \Rightarrow-12n= \frac{\log0.971458}{\log1.007292} =-3.985559 \\ \\ \Rightarrow n= \frac{-3.985559}{-12} =0.332130](https://tex.z-dn.net/?f=2348.62%3D%20%5Cfrac%7B600%5Cleft%5B1-%5Cleft%281%2B%20%5Cfrac%7B0.0875%7D%7B12%7D%5Cright%29%5E%7B-12n%7D%5Cright%5D%7D%7B%5Cfrac%7B0.0875%7D%7B12%7D%7D%20%20%5C%5C%20%20%5C%5C%20%5CRightarrow%201-%281%2B0.007292%29%5E%7B-12n%7D%3D%20%5Cfrac%7B2348.62%5Ctimes0.0875%7D%7B12%5Ctimes600%7D%20%3D0.028542%20%5C%5C%20%20%5C%5C%20%5CRightarrow%281.007292%29%5E%7B-12n%7D%3D1-0.028542%3D0.971458%20%5C%5C%20%20%5C%5C%20%5CRightarrow%20%5Clog%281.007292%29%5E%7B-12n%7D%3D%5Clog0.971458%20%5C%5C%20%20%5C%5C%20%5CRightarrow-12n%5Clog1.007292%3D%5Clog0.971458%20%5C%5C%20%20%5C%5C%20%5CRightarrow-12n%3D%20%5Cfrac%7B%5Clog0.971458%7D%7B%5Clog1.007292%7D%20%3D-3.985559%20%5C%5C%20%20%5C%5C%20%5CRightarrow%20n%3D%20%5Cfrac%7B-3.985559%7D%7B-12%7D%20%3D0.332130)
Therefore, the number of months it will take to pay of the debt is 3.99 months which is approximately 4 months.
After payment of $300, remaining balance = $2,348.62 - $300 = $2,048.62.
Interest accrued is given by:
Had it been $600 was paid, remaining balance = $2,348.62 - $600 = $1748.62. Interest accrued is given by:
Difference in interest accrued = $14.94 - $12.75 = $2.19
Part 2:
The present value of an annuity is given by:
Where PV is the amount to be repaid, P is the equal monthly payment, r is the annual interest rate and n is the number of years.
Thus,
Therefore, the number of months it will take to pay of the debt is 3.99 months which is approximately 4 months.