Example Question - compound interest formula
Here are examples of questions we've helped users solve.
Solving Compound Interest Problem with Incomplete Information
The image shows a compound interest problem where Aladdin invests $20,000 in a savings account that pays compound interest at the rate of r% per year. The interest is compounded quarterly, and the formula shown is for the total amount of the investment at the end of n years. The formula is: Total amount = 20,000 * (1 + r/1000)^(4n) However, to solve for the values of r and n, we would need additional information such as the final total amount or the time period of the investment. Without additional information, there are infinite possibilities for r and n that would satisfy this equation. If there is more data provided, please share it, and I can help you solve for r and n accordingly. Otherwise, it's not possible to determine the values of r and n with the information given.
Solving compound interest problem with missing total amount
Based on the image, I see a mathematical problem involving the formula for compound interest. However, the total amount that the investment grew to is not provided in the question as visible in the image, and we need this to find the values of 'r' (the interest rate) and 'n' (the number of years). The formula given is: Total amount = 20,000(1 + r/1000)^(4n) The '4' in the exponent signifies that interest is compounded quarterly (4 times per year). If you have the total amount that the investment grew to, you can then replace 'Total amount' in the formula with that number. Once you have the total amount, we can take the following steps to solve for 'r' and 'n': 1. Divide both sides by 20,000 to isolate the compound interest factor. 2. Take the fourth root (since the interest is compounded quarterly) of both sides to get rid of the power of '4n'. 3. Solve for '1 + r/1000' by taking the 20th root (since we are left with '(1 + r/1000)^20' after step 2). 4. Once you have the value for '1 + r/1000', you can then subtract 1 and multiply by 1000 to get the value of 'r'. 5. To find 'n', you would then use the log operation with the value found for '1 + r/1000' and the total amount (divided by the initial 20,000). Please provide me with the total amount or any other missing information so that I can guide you through the solving process in detail.
Solving Compound Interest with Quarterly Compounding
The formula in the image is a compound interest formula where interest is compounded quarterly. The standard compound interest formula 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 (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time the money is invested for, in years. But in the formula provided in the image, it is shown as: \[ \text{Total amount} = 20,000 \left(1 + \frac{1}{1000}\right)^{20} \] Firstly, we need to relate this to the standard formula to find \( r \) and \( n \). Given that the interest is compounded quarterly, \( n \) should be 4 (because there are four quarters in a year). From the standard formula, we can adjust \( \frac{r}{n} \) to look like \( \frac{1}{1000} \), which implies: \[ \frac{r}{n} = \frac{1}{1000} \] \[ \frac{r}{4} = \frac{1}{1000} \] \[ r = \frac{1}{1000} \times 4 \] \[ r = \frac{4}{1000} \] So the annual interest rate \( r \) as a decimal is: \[ r = 0.004 \] To convert this to a percentage, we multiply by 100: \[ r \times 100 = 0.004 \times 100 = 0.4\% \] Now, we need to identify \( t \) from the formula provided in the image: The \( 20 \) in the exponent should represent \( nt \), which implies: \[ nt = 20 \] Since we already determined \( n = 4 \), we can solve for \( t \): \[ 4t = 20 \] \[ t = \frac{20}{4} \] \[ t = 5 \] Therefore, the annual interest rate \( r \) is 0.4% per year and the time \( t \) is 5 years.
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 the Tax-Sheltered Annuity Value with Quarterly Compounding
好的，我们要计算的是在每个季度末向以9%的年化季度复利投资的税收优惠年金 (Tax-Sheltered Annuity, TSA) 中存入$200后20年（即80次存款）的价值。 我们可以用年金终值公式来解这个问题： 年金终值 \( FV \) 的计算公式是: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] 其中： \( P \) 是每期支付金额 \( r \) 是每期利率 \( n \) 是支付期数 在这个问题中： \( P = $200 \) 年利率为9%，因为是按季度复利，所以季度利率 \( r = \frac{9\%}{4} = 2.25\% = 0.0225 \) \( n = 20 \times 4 = 80 \) 季度 带入公式，得出: \[ FV = 200 \times \frac{(1 + 0.0225)^{80} - 1}{0.0225} \] 我们可以先计算出 \( (1 + 0.0225)^{80} \)，然后将其余的部分做计算。 \[ (1 + 0.0225)^{80} = (1.0225)^{80} \] 我会先为你计算这个数值。因为我无法直接执行数学运算，我们可以用步骤来处理这个公式： \[ (1.0225)^{80} ≈ 4.8018 \] (这个数值是四舍五入的结果) 现在代入计算出来的 \( (1 + r)^n \) 值： \[ FV = 200 \times \frac{4.8018 - 1}{0.0225} \] \[ FV = 200 \times \frac{3.8018}{0.0225} \] \[ FV ≈ 200 \times 168.9689 \] \[ FV ≈ 33819.78 \] 所以，终值大约是 $33,819.78。当然，这个计算是基于利率恒定，而且每个季度末都投入$200，且在计算中四舍五入得到的近似结果。
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.
Calculating Compound Interest for a Savings Account
To solve this question, we can use the compound interest formula which is: A = P(1 + r/n)^(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 or borrowed for, in years. In this case, a grandmother deposits $5000 in an account that pays 9.5% compounded monthly, and we want to find the value of the account at the child's twenty-first birthday. Therefore, P = $5000, r = 9.5/100 = 0.095 (as a decimal), n = 12 (since interest is compounded monthly), and t = 21 years. Plugging in the values: A = 5000(1 + 0.095/12)^(12*21) A = 5000(1 + 0.00791667)^(252) A = 5000(1.00791667)^(252) Now we can calculate the value of A. A ≈ 5000(1.00791667)^252 Using a calculator to compute this value: A ≈ 5000 * (1.00791667)^252 A ≈ 5000 * 5.98472378 A ≈ 29923.619 So, the value of the account will be approximately $29,923.62 when rounded to the nearest dollar.
Compound Interest Calculation Table
The image shows a problem related to compound interest. An investment of $5000 is deposited into an account where interest is compounded monthly. We are given the interest rate (4% per annum) and the task is to complete the table by filling in the amounts to which the investment grows at the indicated times. The formula for compound interest 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). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time the money is invested for, in years. In this problem: - \( P = $5000 \) - \( r = 4\% = 0.04 \) per year - \( n = 12 \) (since the interest is compounded monthly) We need to calculate the amount \( A \) for each year from 1 to 6. Let's calculate: For year 1: \[ A = 5000 \left(1 + \frac{0.04}{12}\right)^{12 \times 1} \] For year 2: \[ A = 5000 \left(1 + \frac{0.04}{12}\right)^{12 \times 2} \] And so on up to year 6. I will now calculate the amounts for years 1 to 6: For year 1: \[ A_1 = 5000 \left(1 + \frac{0.04}{12}\right)^{12} \] \[ A_1 = 5000 \left(1 + \frac{0.003333}{12}\right)^{12} \] \[ A_1 = 5000 \left(1 + \frac{0.003333}\right)^{12} \] \[ A_1 = 5000 \times 1.04074 \] \[ A_1 = 5203.7 \] For year 2: \[ A_2 = 5000 \left(1 + \frac{0.04}{12}\right)^{24} \] \[ A_2 = 5000 \left(1 + \frac{0.003333}\right)^{24} \] \[ A_2 = 5000 \times 1.08301 \] \[ A_2 = 5415.05 \] Similarly, you can calculate the amounts for years 3, 4, 5, and 6 using the same formula, substituting the appropriate value for \( t \) each time. Make sure to calculate accurately, using either a calculator or computational tool to determine the exact figures as my approximations may be rough. Please fill in the table with the results you compute for each respective year.
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