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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Step-by-step Solution
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.
Quick Answer
The present value needed to invest now is approximately \$8,446.31.
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