Example Question - interest rate formula
Here are examples of questions we've helped users solve.
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
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 \( t \).
Calculating Simple Interest and Total Repayment Amount
The question involves calculating the simple interest and then determining the total amount of money that the woman has to repay after 4 years. The formula for calculating simple interest is: \( I = P \times r \times t \) where \( I \) is the interest, \( P \) is the principal amount (initial loan amount), \( r \) is the annual interest rate (in decimal form), \( t \) is the time the money is borrowed for, in years. According to the image, the woman borrows $26,000, the interest rate is 3.9%, and the time is 4 years. First, convert the interest rate from a percentage to a decimal by dividing by 100: \( r = 3.9\% = \frac{3.9}{100} = 0.039 \) Now plug the numbers into the formula: \( I = P \times r \times t \) \( I = 26000 \times 0.039 \times 4 \) Now calculate the interest: \( I = 26000 \times 0.039 \times 4 \\ I = 1016 \times 4 \\ I = 4064 \) The interest that will be accrued over 4 years is $4,064. Next, to find the total amount that must be repaid, add the interest to the principal amount: \( Total = P + I \) \( Total = 26000 + 4064 \) \( Total = 30064 \) The woman will have to repay a total of $30,064 after 4 years.
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