Suppose $100 is placed in a savings account at the end of
each month for 50 months. If no further deposits are made,
(a) how much is in the account after seven years, and (b) how
much of this amount is compound interest? Assume that
the savings account pays 9% compounded monthly.
To find out how much is in the account after seven years, we need to calculate the future value of all the monthly deposits. The formula for the future value of an ordinary annuity is:
\[FV = P \times \left(\frac{(1 + r)^n - 1}{r}\right)\]
where:
$FV$ = future value
$P$ = monthly deposit amount = $100
$r$ = monthly interest rate = $\frac{9}{100} \div 12 = \frac{9}{1200}$
$n$ = total number of months = 50
Using this formula, the future value of the monthly deposits is:
\[FV = 100 \times \left(\frac{(1 + \frac{9}{1200})^{50} - 1}{\frac{9}{1200}}\right)\]
Calculating this expression gives us:
\[FV = 100 \times \left(\frac{(1.0075)^{50} - 1}{\frac{9}{1200}}\right)\]
\[FV \approx 6317.31\]
Therefore, after seven years, there will be approximately $\$6317.31$ in the account.
To calculate the compound interest, we need to subtract the total amount of deposits made from the future value. The total amount of deposits made is:
\[Total \, Deposits = \$100 \times 50 = \$5000\]
Therefore, the compound interest is:
\[Compound \, Interest = \$6317.31 - \$5000 = \$1317.31\]
The amount of compound interest in the account is approximately \$1317.31.