Question

diamond took out a 3 year loan for $2825 at a sports equipment store to be paid back with monthly payments at a 7.8% APR, compounded monthly. if the loan offers no payments for the first 7 months, how much will diamond owe when she begins making payments?

A. $3053.40
B. $2825.00
C. $3567.09
D. $2956.07

Answer 1

To find out how much Diamond will owe after the first 7 months of not making any payments, we will use the formula for compound interest. The formula is:

\[
A = P(1 + \frac{r}{n})^{nt}
\]

Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial loan amount).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested or borrowed for, in years.

In this case:
- \( P = 2825 \)
- \( r = 7.8\% = 0.078 \)
- \( n = 12 \) (compounded monthly)
- \( t = \frac{7}{12} \) (7 months)

Now we plug these values into the formula:

\[
A = 2825(1 + \frac{0.078}{12})^{12 \cdot \frac{7}{12}}
\]

First, calculate \( \frac{0.078}{12} \):

\[
\frac{0.078}{12} \approx 0.0065
\]

Now substitute this back into the equation:

\[
A = 2825(1 + 0.0065)^{7}
\]

Calculating \( 1 + 0.0065 \):

\[
1 + 0.0065 = 1.0065
\]

Now raise this to the power of 7:

\[
1.0065^7 \approx 1.046124
\]

Now multiply this result by the principal amount:

\[
A \approx 2825 \times 1.046124 \approx 2956.07
\]

Thus, the amount Diamond will owe after 7 months is approximately **$2956.07**.

Therefore, the correct answer is **D. $2956.07**.