Example Question - annual interest rate
Here are examples of questions we've helped users solve.
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 Continuous Compounding Interest
To solve this problem, we use the formula for continuous compounding interest: A = P * e^(rt) 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) t = the time the money is invested for, in years e = the mathematical constant approximately equal to 2.71828 Given: P = $1,800 r = 3.7% per annum = 0.037 (as a decimal) t = 10 years We can now substitute the given values into the formula to find the value of the account after 10 years: A = 1800 * e^(0.037 * 10) Calculating the exponent first: 0.037 * 10 = 0.37 Now, raise e to the power of 0.37: e^0.37 is approximately 1.4481 Now multiply this by the principal amount: A = 1800 * 1.4481 ≈ 2606.58 Rounding to the nearest dollar, we get A ≈ $2,607. Therefore, the correct answer, rounded to the nearest dollar, is about $2,607 which corresponds to option C.
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.
Calculating Present Value for Compound Interest Investment
To determine the present value required for an investment to grow to a future amount, we will use the present value formula for compound interest: \( PV = \frac{FV}{{(1 + r/n)}^{(nt)}} \) where: - PV = present value (the amount to be invested now) - FV = future value (the desired accumulated amount) - r = annual interest rate (as a decimal) - n = number of times the interest is compounded per year - t = number of years the money is invested Given information: - FV = $110,000 - r = 2% or 0.02 (as a decimal) - n = 4 (since the interest is compounded quarterly) - t = 4 years Plugging these values into the formula gives us the present value: \( PV = \frac{110,000}{{(1 + 0.02/4)}^{(4*4)}} \) \( PV = \frac{110,000}{{(1 + 0.005)}^{16}} \) \( PV = \frac{110,000}{{1.005}^{16}} \) Now we will calculate \(1.005^{16}\) and then divide 110,000 by this result: \( 1.005^{16} = 1.082856 \) (rounded to six decimal places) So, \( PV = \frac{110,000}{1.082856} \) \( PV ≈ 101,576.65 \) Therefore, the amount to be invested now, or the present value needed, is approximately $101,576.65.
Calculating Simple Interest for Given Principal Amount, Interest Rate, and Time
To solve for the simple interest, the formula that is used is: Simple Interest (SI) = P * r * t Where: P = principal amount (initial amount of money) r = annual interest rate (as a decimal) t = time the money is invested or borrowed for, in years In this problem, you are given: P = $542 r = 0.045% per day t = 3 months First, let's convert the daily interest rate to an annual rate and the time to years. Since the problem assumes 360 days in a year, we can find the annual interest rate by multiplying the daily rate by 360: r_annual = 0.045% * 360 = 16.2% We must express this as a decimal when using it in our calculation, so: r_annual = 16.2 / 100 = 0.162 Now convert the time to years. There are 12 months in a year, so: t_years = 3 months / 12 = 0.25 years Now we can apply these values to the simple interest formula: SI = P * r * t SI = $542 * 0.162 * 0.25 Now calculate the simple interest: SI = $542 * 0.0405 SI = $21.951 Rounding to the nearest cent, the simple interest for 3 months is: $21.95 Therefore, the simple interest on $542 at 0.045% per day for 3 months is $21.95.
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