Example Question - quarterly compounding
Here are examples of questions we've helped users solve.
Solving Compound Interest with Quarterly Compounding
The question refers to compound interest being calculated on a savings account. In compound interest, the interest is calculated periodically and added to the principal for the next period of interest calculation. The formula provided in the image is an expression for the total amount in the account at the end of \( n \) years when interest is compounded quarterly: \[ \text{Total amount} = 20,000 \left(1 + \frac{r}{1000}\right)^{4n} \] Here, \( r \) represents the annual interest rate (expressed as a percent) divided by the number of compounding periods in a year (which is 4 for quarterly compounding). Since the interest is compounded quarterly, the annual rate \( r\% \) is divided by 4 to get the rate per quarter and compounded for \( 4n \) times over \( n \) years. The question asks us to find the values of \( r \) and \( n \). However, with the given information, it is not possible to uniquely determine both variables, as we have one equation and two unknowns. Additional information, such as the final amount in the savings account or the length of the investment period, is required to solve for the individual values of \( r \) and \( n \). If further information is provided, please share it, and I can assist you in solving for the variables.
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 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 Accumulated Value of Ordinary Annuity with Quarterly Compounding
The image shows a financial math problem that reads: "What is the accumulated value of deposits of $112,000 made at the end of every six months for three years if interest is at 8.48% compounded quarterly?" We are given the following details: - Regular deposits of $112,000 are made at the end of every six months (semiannually), which constitutes an ordinary annuity. - The total period is three years. - The nominal interest rate is 8.48% per annum, compounded quarterly. To calculate the accumulated value, we need to use the future value formula for an ordinary annuity, adjusting appropriately for the semiannual deposits and quarterly compounding. First, we calculate the effective interest rate per six months, since interest is compounded quarterly. The nominal annual rate is 8.48%, so the quarterly rate is 8.48%/4 = 2.12% per quarter. To get the effective semiannual rate, we use the formula for compound interest for two quarters (six months): \[ (1 + i)^n \] where \( i \) is the quarterly interest rate and \( n \) is the number of quarters in six months. The effective semiannual rate \( i_{\text{semi}} \) is \[ i_{\text{semi}} = (1 + 0.0212)^2 - 1 \] \[ i_{\text{semi}} = (1 + 0.0212)\times(1 + 0.0212) - 1 \] \[ i_{\text{semi}} = 1.04308544 - 1 \] \[ i_{\text{semi}} = 0.04308544 \] \[ i_{\text{semi}} \approx 0.0431 \text{ (or 4.31%)} \] Now that we have the effective semiannual rate, we can calculate the future value of the annuity over three years. For six deposits, the future value formula is: \[ FV = P \times \left( \frac{(1 + i)^n - 1}{i} \right) \] where: \( P \) = periodic payment (in this case, $112,000), \( i \) = effective interest rate per period (4.31% per six months), \( n \) = total number of periods (six periods for three years). Plugging in our values: \[ FV = \$112,000 \times \left( \frac{(1 + 0.0431)^6 - 1}{0.0431} \right) \] Next, we calculate each part in turn: \[ (1 + 0.0431)^6 = (1.0431)^6 \approx 1.2846348545 \] \[ (1.0431)^6 - 1 \approx 0.2846348545 \] \[ \frac{(1.0431)^6 - 1}{0.0431} \approx 6.6014822329 \] \[ FV = \$112,000 \times 6.6014822329 \] \[ FV \approx \$739,365.9304828 \] Thus, the accumulated value of the deposits is approximately $739,365.93.
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