Example Question - financial mathematics
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Calculating Annual Interest Rate for Compound Interest Formula
To solve this question, you'll need to use the formula for compound interest, which is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (in decimal form, so 5% would be 0.05). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time the money is invested for in years. From the question: - \( P = $700 \) - \( A = $854.13 \) - \( t = 5 \) years - \( n = 4 \) times per year (compounded every 3 months) Now we have to find \( r \). Plugging in the values we have: \[ 854.13 = 700 \left(1 + \frac{r}{4}\right)^{4 \cdot 5} \] First, divide both sides by 700: \[ \frac{854.13}{700} = \left(1 + \frac{r}{4}\right)^{20} \] \[ 1.21932857 = \left(1 + \frac{r}{4}\right)^{20} \] Now take the 20th root of both sides: \[ \sqrt[20]{1.21932857} = 1 + \frac{r}{4} \] \[ 1.0104239684 = 1 + \frac{r}{4} \] Subtract 1 from both sides: \[ 0.0104239684 = \frac{r}{4} \] Multiply both sides by 4 to solve for \( r \): \[ r = 0.0416958736 \] Converting this to a percentage: \[ r = 4.16958736\% \] And now, rounding to three significant figures: \[ r \approx 4.17\% \] So, the annual interest rate is approximately 4.17% to three significant figures.
Calculating Future Value of an Account with Compound Interest
To solve for the value of the account after 8 years, we need to use the given formula for A(t), where t represents the time in years. The initial amount in the account is $5280, the interest rate is 4.2% (or 0.042 as a decimal), and it's compounded monthly (indicated by 12 in the formula). The formula given is: \[ A(t) = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A(t) \) is the amount of money accumulated after t years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time the money is invested for, in years. Given values: - \( P = \$5280 \) - \( r = 0.042 \) (which is 4.2% expressed as a decimal) - \( n = 12 \) (since interest is compounded monthly) - \( t = 8 \) (since we want to know the value after 8 years) Let's plug these values into the formula: \[ A(8) = 5280 \left(1 + \frac{0.042}{12}\right)^{12 \times 8} \] Now calculate the value inside the parentheses first: \[ 1 + \frac{0.042}{12} = 1 + 0.0035 = 1.0035 \] Then raise this value to the power of \( 12 \times 8 \): \[ (1.0035)^{96} \] (since \( 12 \times 8 = 96 \)) Calculating \( (1.0035)^{96} \) gives us approximately: \[ (1.0035)^{96} \approx 1.3991 \] (rounded to four decimal places) Now multiply this result by the principal amount to find \( A(8) \): \[ A(8) = 5280 \times 1.3991 \approx 7392.61 \] Therefore, the value of the account after 8 years is approximately $7392.61.
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