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7. [0/0.6 Points] DETAILS \( _{\text {MY NOTES }} \) AUFEXC4 11.2.027. 1/100 Submissions Used Calculate the future value of \( \$ 6,000 \) earning \( 7 \% \) interest compounded quarterly for 6 years. (Round your answer to

Ask by Todd Little. in the United States
Dec 02,2024

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The future value of $6,000 with 7% quarterly interest over 6 years is about $9,098.66.

Solución

To calculate the future value of $6,000 earning 7% interest compounded quarterly for 6 years, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the future value - \( P \) is the principal amount ($6,000) - \( r \) is the annual interest rate (7% or 0.07) - \( n \) is the number of times interest is compounded per year (quarterly, so 4) - \( t \) is the number of years (6) Substitute the given values into the formula and calculate the future value. Calculate the value by following steps: - step0: Calculate: \(6000\left(1+\frac{0.07}{4}\right)^{4\times 6}\) - step1: Divide the terms: \(6000\left(1+\frac{7}{400}\right)^{4\times 6}\) - step2: Add the numbers: \(6000\left(\frac{407}{400}\right)^{4\times 6}\) - step3: Multiply the numbers: \(6000\left(\frac{407}{400}\right)^{24}\) - step4: Simplify: \(6000\times \frac{407^{24}}{400^{24}}\) - step5: Rewrite the expression: \(400\times 15\times \frac{407^{24}}{400^{24}}\) - step6: Reduce the numbers: \(15\times \frac{407^{24}}{400^{23}}\) - step7: Rewrite the expression: \(5\times 3\times \frac{407^{24}}{400^{23}}\) - step8: Rewrite the expression: \(5\times 3\times \frac{407^{24}}{25^{23}\times 16^{23}}\) - step9: Rewrite the expression: \(5\times 3\times \frac{407^{24}}{5^{46}\times 16^{23}}\) - step10: Reduce the numbers: \(3\times \frac{407^{24}}{5^{45}\times 16^{23}}\) - step11: Multiply the fractions: \(\frac{3\times 407^{24}}{5^{45}\times 16^{23}}\) The future value of $6,000 earning 7% interest compounded quarterly for 6 years is approximately $9,098.66.

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The Deep Dive

To find the future value of an investment compounded quarterly, you can use the formula: \[ FV = P \times \left(1 + \frac{r}{n}\right)^{nt} \] where \( P \) is the principal amount (initial investment), \( r \) is the annual interest rate (decimal), \( n \) is the number of times interest is compounded per year, and \( t \) is the number of years. For your example, with \( P = 6000 \), \( r = 0.07 \), \( n = 4 \), and \( t = 6 \), the calculation gives you a future value of approximately \( \$ 9,098.24 \). Now, let’s talk about how this applies in real life! People often use compounding to grow their savings, like for retirement funds. By investing now and allowing interest to compound over time, you can substantially increase your savings while you focus on other things in life. It's money making money! Lastly, one common mistake is forgetting to convert the interest rate into decimal form or not properly handling the compounding frequency. Always double-check your values! Remember, using a financial calculator or spreadsheet software can also simplify this process and reduce errors.

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