Use the present value formula to determine the amount to be invested now, or the present value needed. The desired accumulated amount is \( \$ 100,000 \) after 6 years investe 4 in an account with \( 6 \% \) interest compounded quarterly. The amount to be invested now, or the present value needed, is \( \$ \square \) (Round to the nearest cent as needed.)

To find the present value (PV) needed to accumulate a desired amount in the future, we can use the present value formula for compound interest: \[ PV = \frac{FV}{(1 + r/n)^{nt}} \] Where: - \( FV \) is the future value (the desired accumulated amount), - \( r \) is the annual interest rate (as a decimal), - \( n \) is the number of times that interest is compounded per year, - \( t \) is the number of years the money is invested. Given: - \( FV = 100,000 \) - \( r = 0.06 \) (6% as a decimal) - \( n = 4 \) (quarterly compounding) - \( t = 6 \) Now, we can substitute these values into the formula step by step. 1. Calculate \( r/n \): \[ r/n = \frac{0.06}{4} \] 2. Calculate \( nt \): \[ nt = 4 \times 6 \] 3. Substitute these values into the present value formula: \[ PV = \frac{100,000}{(1 + r/n)^{nt}} \] Let's perform these calculations. Calculate the value by following steps: - step0: Calculate: \(4\times 6\) - step1: Multiply the numbers: \(24\) Calculate or simplify the expression \( 0.06/4 \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.06}{4}\) - step1: Convert the expressions: \(\frac{\frac{3}{50}}{4}\) - step2: Multiply by the reciprocal: \(\frac{3}{50}\times \frac{1}{4}\) - step3: Multiply the fractions: \(\frac{3}{50\times 4}\) - step4: Multiply: \(\frac{3}{200}\) Calculate or simplify the expression \( (1 + 0.015)^(24) \). Calculate the value by following steps: - step0: Calculate: \(\left(1+0.015\right)^{24}\) - step1: Add the numbers: \(1.015^{24}\) - step2: Convert the expressions: \(\left(\frac{203}{200}\right)^{24}\) - step3: Use the properties of exponents: \(\frac{203^{24}}{200^{24}}\) Calculate or simplify the expression \( 100000/(1.429503) \). Calculate the value by following steps: - step0: Calculate: \(\frac{100000}{1.429503}\) - step1: Convert the expressions: \(\frac{100000}{\frac{1429503}{1000000}}\) - step2: Multiply by the reciprocal: \(100000\times \frac{1000000}{1429503}\) - step3: Multiply: \(\frac{100000\times 1000000}{1429503}\) - step4: Multiply: \(\frac{100000000000}{1429503}\) Now, let's summarize the calculations step by step: 1. **Calculate \( r/n \)**: \[ r/n = \frac{0.06}{4} = 0.015 \] 2. **Calculate \( nt \)**: \[ nt = 4 \times 6 = 24 \] 3. **Calculate \( (1 + r/n)^{nt} \)**: \[ (1 + 0.015)^{24} \approx 1.429503 \] 4. **Substitute into the present value formula**: \[ PV = \frac{100,000}{1.429503} \approx 69,954.38 \] Thus, the amount to be invested now, or the present value needed, is approximately \( \$ 69,954.38 \). So, the final answer is: The amount to be invested now, or the present value needed, is \( \$ 69,954.38 \).