Susan wants to save money to buy a motorcycle. She invests in an ordinary annuity that earns $4.8 \%$ interest, compounded quarterly. Payments will be made at the end of each quarter.
How much money will she need to pay into the annuity each quarter for the annuity to have a total value of $\$ 7000$ after 6 years?
Answer
Final Answer: Susan will need to pay approximately \$253.41 into the annuity each quarter for the annuity to have a total value of \$7000 after 6 years. So, the final answer is \(\boxed{253.41}\).
Step 1 ：Susan wants to save money to buy a motorcycle. She invests in an ordinary annuity that earns 4.8% interest, compounded quarterly. Payments will be made at the end of each quarter. She wants the annuity to have a total value of $7000 after 6 years.
Step 2 ：We need to find out how much money she will need to pay into the annuity each quarter.
Step 3 ：We can use the formula for the future value of an ordinary annuity: \(FV = P \times \frac{(1 + r/n)^{n \times t} - 1}{r/n}\), where \(FV\) is the future value, \(P\) is the payment made each period, \(r\) is the annual interest rate, \(n\) is the number of compounding periods per year, and \(t\) is the number of years.
Step 4 ：We can rearrange this formula to solve for \(P\): \(P = FV \times \frac{r/n}{(1 + r/n)^{n \times t} - 1}\).
Step 5 ：Substituting the given values into the formula, we get: \(P = 7000 \times \frac{0.048/4}{(1 + 0.048/4)^{4 \times 6} - 1}\).
Step 6 ：Solving this equation, we find that \(P \approx 253.41\).
Step 7 ：Final Answer: Susan will need to pay approximately \$253.41 into the annuity each quarter for the annuity to have a total value of \$7000 after 6 years. So, the final answer is \(\boxed{253.41}\).
