Example Question - compound interest calculation
Here are examples of questions we've helped users solve.
Calculating the Value of TSA Account After 20 Years with Regular Deposits
题目问的是，如果Arnold每个季度末存入200美元到税收优惠年金（TSA）账户中，而且账户的年化回报率是9%（每季度复利），那么在第80次存款之后（即20年后），TSA账户的价值是多少？ 要解这个问题，我们需要使用复利公式计算定期投资的未来价值。由于存款是每季度进行，我们将使用公式来计算一个年金终值，这是一个普通年金，因为存款是在每个期间的末尾进行的。 年金终值的公式是： \[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \] 其中： \( FV \) 是年金的未来价值， \( P \) 是每期付款金额， \( r \) 是每期的利率， \( n \) 是付款期数。 在这个问题中: \( P = \$200 \)，这是Arnold每季度存款的数额； \( r = \frac{0.09}{4} = 0.0225 \)，因为年利率是9%，我们将其除以4来得到每季度的利率； \( n = 80 \)，这是Arnold在20年内的存款次数，因为一年有4个季度，所以20年有80个季度。 代入公式计算： \[ FV = 200 \times \left( \frac{(1 + 0.0225)^{80} - 1}{0.0225} \right) \] 现在我们进行计算。 首先计算 \((1 + 0.0225)^{80}\)，然后根据结果减去1，再除以0.0225，然后乘以200。 当我们完成上述计算后，就可以得到Arnold在20年后TSA账户的价值。如果你需要具体的数值计算结果，请告诉我，我会继续完成这个计算过程。
Compound Interest Calculation with Monthly Compounding
To solve for the amount Jace will have in the account after one year with an annual percentage rate of 3.8% and monthly compounding interest, we use the formula for compound interest: 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 for, in years. Given: P = $2,000 r = 3.8% or 0.038 (as a decimal) n = 12 (since interest is compounded monthly) t = 1 year Plugging these into the formula gives us: A = 2000(1 + 0.038/12)^(12*1) A = 2000(1 + 0.0031667)^(12) A = 2000(1.0031667)^(12) Calculating this out, we get: A ≈ 2000(1.038565) A ≈ 2077.13 So, after rounding to the nearest penny, Jace will have approximately $2077.13 in the account after one year. The question also asks for the annual percentage yield (APY). The APY takes into account the effect of monthly compounding and is calculated using the formula: APY = (1 + r/n)^(n) - 1 Using the same values for r and n: APY = (1 + 0.038/12)^(12) - 1 APY = (1.0031667)^(12) - 1 APY ≈ 0.038818 or 3.8818% After converting this to a percentage and rounding to the nearest hundredth of a percent, the APY is approximately 3.88%.
Compound Interest Calculation for Principal Amount
Por supuesto, para resolver esta pregunta, usaremos la fórmula del interés compuesto, la cual es: \( A = P \cdot (1 + r/n)^{nt} \) Donde: - \( A \) es el monto futuro o cantidad acumulada después de \( t \) años, incluyendo el interés. - \( P \) es el capital principal o cantidad inicial. - \( r \) es la tasa de interés anual (en decimal). - \( n \) es el número de veces que el interés se capitaliza por año. - \( t \) es el tiempo en años. En este caso, nos están pidiendo encontrar \( P \), el capital principal. Tenemos el monto futuro \( A = \$32,158 \), la tasa de interés anual \( r = 6\% \) o \( 0.06 \) en decimal, y el tiempo es de \( t = 7.5 \) años. Ya que el interés se capitaliza anualmente, \( n = 1 \). Reordenando la fórmula para despejar \( P \), obtenemos: \( P = A / (1 + r/n)^{nt} \) \( P = \$32,158 / (1 + 0.06/1)^{1 \cdot 7.5} \) Calculamos \( (1 + 0.06)^{7.5} \): \( (1.06)^{7.5} \approx 1.5694634 \) Y ahora dividimos: \( P = \$32,158 / 1.5694634 \) \( P \approx \$20,477.95 \) Por lo tanto, el capital inicial invertido fue aproximadamente de \$20,477.95.
Compound Interest Calculation
Para resolver el problema, debemos usar la fórmula del monto para una inversión con interés compuesto, que es: \[ A = P \cdot (1 + r)^n \] Donde: - \( A \) es el monto final que se tendrá después de \( n \) períodos. - \( P \) es el capital inicial o principal, que en este caso son $10,000. - \( r \) es la tasa de interés por período, que es del 10% anual o 0.10 en su forma decimal. - \( n \) es el número de períodos que el dinero estará invertido, que son 10 años en este caso. Sustituyendo los valores dados en el problema en la fórmula, obtenemos: \[ A = 10,000 \cdot (1 + 0.10)^{10} \] \[ A = 10,000 \cdot (1.10)^{10} \] Calculamos \( (1.10)^{10} \) primero: \[ (1.10)^{10} = 2.5937424601 \] (aproximadamente) Luego multiplicamos este resultado por $10,000: \[ A = 10,000 \cdot 2.5937424601 \] \[ A \approx 25,937.42 \] Por lo tanto, el monto final que se generará después de 10 años de una inversión inicial de $10,000 con una tasa de interés compuesto anual del 10% es aproximadamente $25,937.42.
Compound Interest Calculation Function with Variables
To write a function for the amount that will be in the account after \( t \) years, we can use the 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: - \( P = \$5,280 \) - the annual interest rate \( r = 4.2\% = 0.042 \) (as a decimal) - interest is compounded monthly, so \( n = 12 \) Substitute these values into the compound interest formula to define the function for \( A \): \[ A(t) = 5280 \left(1 + \frac{0.042}{12}\right)^{12t} \] This function can be used to calculate the amount in the account after any number of years \( t \).
Compound Interest Calculation Result
The question in the image asks to calculate the amount of money that Adam will have in the bank account after 24 years with an initial deposit of £3,660 and an annual compound interest rate of 3.25%. To solve this, we use the formula for compound interest, which is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: \( A \) = the amount of money accumulated after n years, including interest. \( P \) = the principal amount (the initial amount of money). \( r \) = the annual interest rate (decimal). \( n \) = the number of times that interest is compounded per year. \( t \) = the time the money is invested for, in years. In this case: \( P = £3,660 \) \( r = 3.25\% = 0.0325 \) (in decimal form) \( n = 1 \) (interest is compounded once per year) \( t = 24 \) years Substituting these values into the formula gives us: \( A = £3,660 \left(1 + \frac{0.0325}{1}\right)^{1 \times 24} \) \( A = £3,660 \left(1 + 0.0325\right)^{24} \) \( A = £3,660 \times 1.0325^{24} \) Now we need to calculate \( 1.0325^{24} \) and multiply it by £3,660 to find the final amount. \( 1.0325^{24} \approx 2.0398873 \) (rounded to 7 decimal places for precision) Now, we will multiply this by the principal amount: \( A \approx £3,660 \times 2.0398873 \) \( A \approx £7,465.97 \) (rounded to two decimal places) After 24 years, Adam will have approximately £7,465.97 in the account.
Continuous Compounding Calculation for Future Value
The question is asking to calculate the future value of a deposit that is compounded continuously. The formula for calculating the future value, A, when compounded continuously is given by: A = P * e^(rt) Where: P is the principal amount (initial deposit), r is the annual interest rate (expressed as a decimal), t is the time the money is invested for in years, and e is the base of the natural logarithm (approximately equal to 2.71828). Based on the information provided in the image: P = $5500 r = 3.6% or 0.036 (as a decimal) t = 8 years We substitute these values into the formula to calculate the future value, A: A = 5500 * e^(0.036 * 8) Next, we compute the exponent: 0.036 * 8 = 0.288 Now, calculate e raised to this exponent: e^0.288 = 2.71828^0.288 ≈ 1.33404 (using a calculator) Finally, we multiply this by the principal amount: A ≈ 5500 * 1.33404 ≈ 7337.22 So the future value after 8 years is approximately $7337.22. If you are asked for the amount of interest earned, you would subtract the principal from the future value: Interest earned = A - P Interest earned ≈ 7337.22 - 5500 ≈ 1837.22 Therefore, the amount of interest earned is approximately $1837.22. Remember to verify these calculations using a calculator for precision.
Comparing Present Value of $8000 in 10 Years with $6433 Now
The question asks you to find the present value of $8000 in 10 years given an interest rate of 2.2% compounded quarterly. To decide which is larger, $6433 now or $8000 in 10 years, we need to calculate the present value of the $8000 using the present value formula for compound interest. The present value formula is given by: \[ PV = \frac{FV}{(1 + \frac{r}{n})^{n \times t}} \] where: PV = present value FV = future value ($8000) r = annual interest rate (2.2% or 0.022) n = number of times the interest is compounded per year (quarterly, so 4 times a year) t = number of years (10) Now let's calculate PV: \[ PV = \frac{8000}{(1 + \frac{0.022}{4})^{4 \times 10}} \] \[ PV = \frac{8000}{(1 + 0.0055)^{40}} \] \[ PV = \frac{8000}{(1.0055)^{40}} \] Using a calculator: \[ PV \approx \frac{8000}{(1.0055)^{40}} \approx \frac{8000}{2.48832} \approx 3215.77 \] Therefore, the present value of $8000 in 10 years at an interest rate of 2.2% compounded quarterly is approximately $3215.77. Now, comparing the present value of $8000 in 10 years ($3215.77) with $6433 now, it is clear that $6433 now is larger. Please note: The final answer has been rounded to the nearest cent as requested.
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