Example Question - future value calculation
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Calculating Future Value and APY of a Savings Account
The question in the image is asking to calculate the future value of a savings account with an annual percentage rate of 3.8% with interest compounded monthly after one year, given an initial deposit of $2,000. Additionally, it asks to calculate the annual percentage yield (APY) for the savings account. To solve the first part of the question, we use the future value formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] 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 (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: - \( P = \$2,000 \) - \( r = 3.8\% = 0.038 \) (as a decimal) - \( n = 12 \) (since the interest is compounded monthly) - \( t = 1 \) year \[ A = 2000 \left(1 + \frac{0.038}{12}\right)^{12 \times 1} \] \[ A = 2000 \left(1 + 0.0031667\right)^{12} \] \[ A = 2000 \left(1.0031667\right)^{12} \] \[ A ≈ 2000 \times 1.03884 \] \[ A ≈ 2077.68 \] Jace will have approximately $2077.68 in the account after one year. For the second part of the question, we calculate the annual percentage yield (APY) which takes into account the effect of compound interest. APY can be calculated using the formula: \[ APY = \left(1 + \frac{r}{n}\right)^n - 1 \] Using the same values for \( r \) and \( n \), we have: \[ APY = \left(1 + \frac{0.038}{12}\right)^{12} - 1 \] \[ APY = \left(1.0031667\right)^{12} - 1 \] \[ APY ≈ 1.03884 - 1 \] \[ APY ≈ 0.03884 \] \[ APY ≈ 3.88\% \] The annual percentage yield (APY) for the savings account is approximately 3.88%.
Calculating Future Value and Annual Percentage Yield of Savings Account
The image contains a question which says: "A bank features a savings account that has an annual percentage rate of 3.8% with interest compounded monthly. Jace deposits $2,000 into the account. a) How much money will Jace have in the account in 1 year? Round answer to the nearest penny. b) What is the annual percentage yield (APY) for the savings account? Round to the nearest hundredth of a percent." Let's solve part a) first. We are given that the annual interest rate is 3.8% and it is compounded monthly. To find the total amount in the account after one year, we will use the formula for compound interest: A = P(1 + r/n)^(nt) Where: A = the future value of the investment/loan, including interest P = the principal investment amount ($2,000 in this case) r = the annual interest rate (decimal) (3.8% or 0.038) n = the number of times that interest is compounded per year (12 for monthly) t = the time the money is invested or borrowed for, in years (1 year in this case) Plugging in the values: A = 2000(1 + 0.038/12)^(12*1) A = 2000(1 + 0.0031667)^(12) A = 2000(1.0031667)^(12) Now use a calculator to evaluate the expression: A ≈ 2000(1.0031667)^12 A ≈ 2000 * 1.038726 A ≈ 2077.45 So, after one year, Jace will have approximately $2077.45 in the account, rounding to the nearest penny. For part b), the annual percentage yield (APY) takes into account the effect of compound interest over the year. It is calculated by the formula: APY = (1 + r/n)^(n) - 1 Again, substitute the values: APY = (1 + 0.038/12)^(12) - 1 APY = (1 + 0.0031667)^(12) - 1 Now evaluate the expression using the calculator: APY ≈ (1.0031667)^12 - 1 APY ≈ 1.038726 - 1 APY ≈ 0.038726 To express this as a percentage, multiply by 100: APY ≈ 0.038726 * 100 APY ≈ 3.8726% Rounding to the nearest hundredth of a percent, we get an APY of approximately 3.87%.
Calculating Future Value of an Account with Compound Interest
The image shows a mathematical formula for the future value of an account and asks for the value of the account after 8 years. The formula provided is a standard compound interest formula: \[ A(t) = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A(t) \) is the amount of money accumulated after time \( t \), including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (in decimal form). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time the money is invested for, in years. Given in the problem: - \( P = 5280 \) (the principal amount) - \( r = 0.042 \) (the annual interest rate, in decimal form) - \( n = 12 \) (since interest is compounded monthly, there are 12 compounding periods per year) - \( t = 8 \) (the time period in question, namely 8 years) Plugging these values into the formula, we get: \[ A(8) = 5280 \left(1 + \frac{0.042}{12}\right)^{12 \times 8} \] Calculating each part step by step: First, divide the annual interest rate by the number of compounding periods: \[ \frac{r}{n} = \frac{0.042}{12} = 0.0035 \] Then add 1 to this result: \[ 1 + \frac{r}{n} = 1 + 0.0035 = 1.0035 \] Calculate the exponent, which is the number of compounding periods times the number of years: \[ nt = 12 \times 8 = 96 \] Now raise the previous result to this exponent: \[ \left(1 + \frac{r}{n}\right)^{nt} = 1.0035^{96} \] And finally, multiply this by the principal amount: \[ A(8) = 5280 \times 1.0035^{96} \] Using a calculator, you can compute: \[ 1.0035^{96} \approx 1.368616 \] Multiplying this by the principal amount, we get: \[ A(8) = 5280 \times 1.368616 \approx 7226.933 \] So the value of the account after 8 years is approximately $7226.93.
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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