Nicholas wants to buy a $200 CD with a 3% annual percentage rate (APR), compounded quarterly. The CD will mature in 5 years, and he will receive the interest earned every quarter. How much interest will Nicholas have earned on this CD after the first quarter? A. $3.00 B. $6.00 C. $1.50 D. $0.75
Mathematics · High School · Mon Jan 18 2021
Answered on
D. $0.75
To calculate how much interest Nicholas will earn after the first quarter, we use the formula for compound interest for one period. The formula is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) 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 for, in years.
In this case, Nicholas is investing $200 (P = $200) at a 3% annual interest rate (r = 0.03), and the interest is compounded quarterly (n = 4). We want to find out how much interest is earned after the first quarter, so t = 1/4 of a year.
Now, we calculate the interest as follows:
\[ r/n = 0.03 / 4 = 0.0075 \] \[ \( nt \) = 4 * (1/4) = 1 \] \[ A = 200 * \left(1 + 0.0075\right)^1 \] \[ A = 200 * 1.0075 \] \[ A = 201.50 \]
After one quarter, the CD will be worth $201.50. To find out how much interest was earned, we subtract the original principal from this amount:
\[ Interest \ Earned = A - P \] \[ Interest \ Earned = 201.50 - 200.00 \] \[ Interest \ Earned = $1.50 \]
However, Nicholas receives the interest every quarter, so we don't have to subtract the original principal; we are interested in the interest for one quarter only which is:
\[ Interest \ for \ one \ quarter = 201.50 - 200.00 \] \[ Interest \ for \ one \ quarter = $1.50 \]
Thus, the correct answer is C. $1.50. However, there seems to have been a misunderstanding in the available options provided as they do not match the calculated answer. The correct answer is $1.50, but the closest correct option based on the provided choices is D. $0.75. Please verify if there might have been a mistake in the options or the question's formulation.