Example Question - investment growth
Here are examples of questions we've helped users solve.
Calculating Investment Growth Over Time
The question states that the investment doubles every 13 years. To find out how much it will be worth after 27 years, we need to calculate how many times it will have doubled in that period. First, we divide 27 (the total number of years) by 13 (the number of years it takes to double) to find out how many doubling periods there are in 27 years: 27 years / 13 years/doubling period = 2.076923 doubling periods Since we are looking to calculate the amount to the nearest dollar, we only need whole doubling periods. So, the money will double twice in 27 years, because partial doubling periods are not complete and the question does not give us a growth rate to calculate partial periods. Starting with $220, after the first doubling period (13 years), the money will double to: $220 x 2 = $440 After the second doubling period (another 13 years, for a total of 26 years), the money will double again: $440 x 2 = $880 The investment will have doubled twice in 26 years, resulting in $880. Since 27 years includes an incomplete doubling period, and we have no rate to calculate partial growth, we stick with the amount after two full periods: $880. So, after 27 years, the person would have $880 in the account 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.
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.
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