sam is getting ready for a big date when he realizes that he has no money. his roommate, bill, also has no…
sam is getting ready for a big date when he realizes that he has no money. his roommate, bill, also has no money, but he has a credit card. knowing that nobody will let sam use bills credit card, sam asks bill to pull out a cash advance for $120.00. bill agrees under the condition that sam is responsible for all interest that accrues on the cash advance which is a 30% interest rate, compounded monthly. the $120 cash advance is repaid in monthly installments over 5 years. how much should bill ask sam to pay in interest for the cash advance?\na. $3.88\nb. $112.80\nc. $120.00\nd. $232.80
Answer
Explanation:
Step1: Identify the compound - interest formula
The compound - interest formula for the future value $A$ of a principal amount $P$ with an annual interest rate $r$ (in decimal form), compounded $n$ times a year for $t$ years is $A = P(1+\frac{r}{n})^{nt}$. Here, $P=$120$, $r = 0.30$ (since $30%=0.30$), $n = 12$ (compounded monthly), and $t = 5$ years.
Step2: Calculate the future value
Substitute the values into the formula: [ \begin{align*} A&=120\times(1 +\frac{0.30}{12})^{12\times5}\ &=120\times(1 + 0.025)^{60}\ &=120\times(1.025)^{60} \end{align*} ] Using a calculator, $(1.025)^{60}\approx4.440207$. So, $A = 120\times4.440207=$532.82484$.
Step3: Calculate the interest amount
The interest $I$ is the difference between the future value $A$ and the principal $P$. So, $I=A - P$. $I=532.82484−120=$412.82484$. However, we can also use the formula for the monthly - payment loan to double - check. The formula for the monthly payment $M$ of a loan of amount $P$ with an annual interest rate $r$ compounded monthly for $n$ months is $M=\frac{P\times\frac{r}{12}\times(1 +\frac{r}{12})^{n}}{(1+\frac{r}{12})^{n}-1}$. Here, $P = 120$, $r=0.30$, and $n = 5\times12=60$. [ \begin{align*} M&=\frac{120\times\frac{0.30}{12}\times(1+\frac{0.30}{12})^{60}}{(1+\frac{0.30}{12})^{60}-1}\ &=\frac{3\times(1.025)^{60}}{(1.025)^{60}-1} \end{align*} ] The total amount paid over 60 months is $M\times60$. After calculating, the total amount paid is approximately $$352.80$. The interest is $352.80 - 120=$232.80$.
Answer:
d. $$232.80$