Questions: Find the accumulated value of an investment of 10,000 for 3 years at an interest rate of 1.25% if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly d. compounded continuously. a. What is the accumulated value if the money is compounded semiannually? 10,380.91 (Round to the nearest cent as needed.) b. What is the accumulated value if the money is compounded quarterly? 10,381.51 (Round to the nearest cent as needed.) c. What is the accumulated value if the money is compounded monthly? 10,381.92 (Round to the nearest cent as needed.) d. What is the accumulated value if the money is compounded continuously? 10,382.12 (Round to the nearest cent as needed.)

Find the accumulated value of an investment of 10,000 for 3 years at an interest rate of 1.25% if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly d. compounded continuously. a. What is the accumulated value if the money is compounded semiannually? 10,380.91 (Round to the nearest cent as needed.) b. What is the accumulated value if the money is compounded quarterly? 10,381.51 (Round to the nearest cent as needed.) c. What is the accumulated value if the money is compounded monthly? 10,381.92 (Round to the nearest cent as needed.) d. What is the accumulated value if the money is compounded continuously? 10,382.12 (Round to the nearest cent as needed.)
Transcript text: Points: 0 of 5 Find the accumulated value of an investment of $\$ 10,000$ for 3 years at an interest rate of $1.25 \%$ if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly d. compounded continuously. (i) Click the icon to view some finance formulas. a. What is the accumulated value if the money is compounded semiannually? \$ 10,380.91 ${ }^{\top}$ (Round to the nearest cent as needed.) b. What is the accumulated value if the money is compounded quarterly? \$ 10,381.51 (Round to the nearest cent as needed.) c. What is the accumulated value if the money is compounded monthly? \$ 10,381.92 (Round to the nearest cent as needed.) d. What is the accumulated value if the money is compounded continuously? \$ 10,382.12 (Round to the nearest cent as needed.)
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Solution

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Solution Steps

To find the accumulated value of an investment with compound interest, we use the formula for compound interest: \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where \( A \) is the accumulated amount, \( P \) is the principal amount, \( r \) is the annual interest rate (as a decimal), \( n \) is the number of times interest is compounded per year, and \( t \) is the time in years. For continuous compounding, we use the formula \( A = Pe^{rt} \).

a. For semiannual compounding, \( n = 2 \). b. For quarterly compounding, \( n = 4 \). c. For monthly compounding, \( n = 12 \).

Step 1: Accumulated Value Compounded Semiannually

To find the accumulated value when the investment is compounded semiannually, we use the formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Substituting the values \( P = 10000 \), \( r = 0.0125 \), \( n = 2 \), and \( t = 3 \): \[ A_{\text{semiannual}} = 10000 \left(1 + \frac{0.0125}{2}\right)^{2 \cdot 3} = 10000 \left(1 + 0.00625\right)^{6} \approx 10380.9084 \]

Step 2: Accumulated Value Compounded Quarterly

For quarterly compounding, we again use the compound interest formula with \( n = 4 \): \[ A_{\text{quarterly}} = 10000 \left(1 + \frac{0.0125}{4}\right)^{4 \cdot 3} = 10000 \left(1 + 0.003125\right)^{12} \approx 10381.5129 \]

Step 3: Accumulated Value Compounded Monthly

For monthly compounding, we set \( n = 12 \): \[ A_{\text{monthly}} = 10000 \left(1 + \frac{0.0125}{12}\right)^{12 \cdot 3} = 10000 \left(1 + 0.00104167\right)^{36} \approx 10381.9173 \]

Final Answer

The accumulated values for the different compounding methods are:

  • Compounded Semiannually: \( A_{\text{semiannual}} \approx 10380.91 \)
  • Compounded Quarterly: \( A_{\text{quarterly}} \approx 10381.51 \)
  • Compounded Monthly: \( A_{\text{monthly}} \approx 10381.92 \)

Thus, the final answers are: \[ \boxed{A_{\text{semiannual}} \approx 10380.91} \] \[ \boxed{A_{\text{quarterly}} \approx 10381.51} \] \[ \boxed{A_{\text{monthly}} \approx 10381.92} \]

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