Morgan Guzman
09/25/2023 · Senior High School

of 8 Step 1 of 1 se the formula for present value of money to calculate the amount you need to invest now in one lump sum in order to have \( \$ 25,000 \) after 10 year. ith an APR of \( 11 \% \) compounded quarterly. Round your answer to the nearest cent, if necessary.

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To calculate the present value of money, we can use the formula: \[ PV = \frac{FV}{(1 + r)^n} \] Where: - \( PV \) is the present value - \( FV \) is the future value - \( r \) is the interest rate per period - \( n \) is the number of periods Given: - Future Value (\( FV \)) = \$25,000 - Annual Percentage Rate (APR) = 11% - Compounded quarterly - Time period = 10 years First, we need to convert the APR to the quarterly interest rate: \[ r = \frac{APR}{4} = \frac{11\%}{4} = 0.0275 \] Next, we substitute the values into the formula and calculate the present value: \[ PV = \frac{25000}{(1 + 0.0275)^{4 \times 10}} \] Let's calculate the present value. Calculate the value by following steps: - step0: Calculate: \(\frac{25000}{\left(1+0.0275\right)^{4\times 10}}\) - step1: Add the numbers: \(\frac{25000}{1.0275^{4\times 10}}\) - step2: Multiply the numbers: \(\frac{25000}{1.0275^{40}}\) - step3: Convert the expressions: \(\frac{25000}{\left(\frac{411}{400}\right)^{40}}\) - step4: Simplify the expression: \(\frac{25000}{\frac{411^{40}}{400^{40}}}\) - step5: Rewrite the expression: \(\frac{25000\times 400^{40}}{411^{40}}\) The present value of money needed to invest now in one lump sum to have \$25,000 after 10 years with an APR of 11% compounded quarterly is approximately \$8,446.31.

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The present value needed to invest now is approximately \$8,446.31.
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