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Find both the compound amount and the compound interest an \( \$ 9900 \) at \( 2 \% \), compounded semiannually, for 3 years. a) Find the values using the compound interest table. Click the icon to view the compound interest table. b) Find the values using your TVM Solver. a) Use the Compound Interest Table. Round answers to the nearest cent. What is the compound amount? \( \$ \square \) What is the compound interest? \( \$ \square \) b) Use the Compound Interest Formula or our TVM Solver. Round answers to the aearest cent. What is the compound amount? \( \$ \square \) What is the compound interest? \( \$ \square \)

Ask by Norton Reed. in the United States
Mar 14,2025

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Answer

a) Compound amount: $10,509.05 Compound interest: $609.05 b) Compound amount: $10,509.05 Compound interest: $609.05

Solution

To find the compound amount and compound interest for $9900 at 2% compounded semiannually for 3 years, we can use both the compound interest table and the TVM Solver. a) Using the Compound Interest Table: The compound interest table provides the values for the compound amount and compound interest for different interest rates and compounding periods. b) Using the TVM Solver: The TVM Solver is a tool that can be used to calculate the compound amount and compound interest for a given principal amount, interest rate, compounding frequency, and time period. Let's calculate the values using both methods. Calculate the value by following steps: - step0: Calculate: \(9900\left(1+\frac{0.02}{2}\right)^{6}\) - step1: Divide the terms: \(9900\left(1+\frac{1}{100}\right)^{6}\) - step2: Add the numbers: \(9900\left(\frac{101}{100}\right)^{6}\) - step3: Simplify: \(9900\times \frac{101^{6}}{100^{6}}\) - step4: Rewrite the expression: \(100\times 99\times \frac{101^{6}}{100^{6}}\) - step5: Reduce the numbers: \(99\times \frac{101^{6}}{100^{5}}\) - step6: Multiply: \(\frac{99\times 101^{6}}{100^{5}}\) a) Using the Compound Interest Table: The compound amount is $10,509.05. The compound interest is $609.05. b) Using the TVM Solver: The compound amount is $10,509.05. The compound interest is $609.05.

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To find the compound amount and compound interest for an investment of \( \$9900 \) at \( 2\% \) compounded semiannually for 3 years, let's go through it step by step. ### a) Using the Compound Interest Table: The formula to compute the compound amount \( A \) using the compound interest table is: \[ A = P(1 + \frac{r}{n})^{nt} \] Where: - \( P = 9900 \) (the principal amount) - \( r = 0.02 \) (the annual interest rate as a decimal) - \( n = 2 \) (the number of times interest is compounded per year) - \( t = 3 \) (the number of years) First, we calculate \( nt \): \[ nt = 2 \times 3 = 6 \] Now, find \( (1 + \frac{r}{n}) \): \[ 1 + \frac{0.02}{2} = 1 + 0.01 = 1.01 \] Now substitute into the formula: \[ A = 9900 \times (1.01)^{6} \] Calculating \( (1.01)^{6} \) gives approximately \( 1.06152 \). Now, calculate \( A \): \[ A \approx 9900 \times 1.06152 \approx 10400.13 \] So, the compound amount is approximately \( \$10400.13 \). To find the compound interest: \[ \text{Compound Interest} = A - P = 10400.13 - 9900 = 500.13 \] So, the compound interest is approximately \( \$500.13 \). ### b) Using the Compound Interest Formula or TVM Solver: Using the same formula as above: \[ A = 9900(1 + \frac{0.02}{2})^{6} \Rightarrow A = 9900(1.01)^{6} \] As calculated before, \( A \approx 10400.13 \). Confirming the compound interest: \[ \text{Compound Interest} = A - P = 10400.13 - 9900 \approx 500.13 \] Hence, the answers remain consistent: - Compound Amount: \( \$10400.13 \) - Compound Interest: \( \$500.13 \) So, to recap: - The compound amount is \( \$10400.13 \). - The compound interest is \( \$500.13 \).

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