Consumer Mathematics
Finding the periodic payment needed to meet an investment goal
Manuel has always dreamed of opening a café by the seaside. He decides he will save to help open the café by depositing money in an ordinary annuity that earns $2.4 \%$ interest, compounded monthly. Deposits will be made at the end of each month.

How much money will he need to deposit into the annuity each month for the annuity to have a total value of $\$ 25,000$ after 7 years?
Do not round intermediate computations, and round your final answer to the nearest cent. If necessary, refer to the list of financial formulas.
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Step 1 ：Manuel has always dreamed of opening a café by the seaside. He decides he will save to help open the café by depositing money in an ordinary annuity that earns $2.4 \%$ interest, compounded monthly. Deposits will be made at the end of each month.

Step 2 ：He wants to know how much money he will need to deposit into the annuity each month for the annuity to have a total value of $\$ 25,000$ after 7 years.

Step 3 ：We can use the formula for the periodic payment of an ordinary annuity to find this. The formula is \(P = FV \times \frac{r}{n} \div \left( (1 + \frac{r}{n})^{n \times t} - 1 \right)\), where \(FV\) is the future value, \(r\) is the annual interest rate, \(n\) is the number of times interest is compounded per year, and \(t\) is the number of years.

Step 4 ：Substituting the given values into the formula, we get \(P = 25000 \times \frac{0.024}{12} \div \left( (1 + \frac{0.024}{12})^{12 \times 7} - 1 \right)\).

Step 5 ：Solving this expression gives us the monthly deposit amount, which we round to the nearest cent to get \$273.62.

Step 6 ：Final Answer: Manuel will need to deposit approximately \(\boxed{\$273.62}\) into the annuity each month for the annuity to have a total value of \$25,000 after 7 years.