Anne Thorne deposits $100 at the end of each month into her savings account which pays 6% interest compounded monthly. How much will be in her account at the end of 2 ½ years?
i = .06/12 = .005
n =2.5(12) = 30
amount = 100( 1.005^30 - 1 )/.005
= $3228.00
To calculate the total amount in Anne Thorne's account at the end of 2 ½ years, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the total amount after time t
P = principal amount (initial deposit)
r = annual interest rate (6% in this case, which is written as 0.06)
n = number of times interest is compounded per year (monthly in this case, so n = 12)
t = time in years (2 ½ years, which is written as 2.5 years)
First, let's calculate the total number of deposits made over 2 ½ years. Since Anne makes monthly deposits, we multiply the number of years by 12:
Number of Deposits = 2.5 years * 12 months/year = 30 deposits
Next, let's calculate the total principal amount deposited:
Total Principal Amount = $100/deposit * 30 deposits = $3,000
Now, we can substitute the values into the compound interest formula:
A = $3,000 * (1 + 0.06/12)^(12 * 2.5)
Simplifying the formula:
A = $3,000 * (1 + 0.005)^(30)
Now, we solve for A:
A = $3,000 * (1.005)^(30)
Using a calculator or spreadsheet, we find that (1.005)^30 ≈ 1.1802.
So the final calculation is:
A ≈ $3,000 * 1.1802
Calculating this value gives us approximately:
A ≈ $3,540.60
Therefore, there will be approximately $3,540.60 in Anne Thorne's account at the end of 2 ½ years.
To calculate the amount in Anne Thorne's account at the end of 2 ½ years, we need to calculate the future value of monthly deposits and compounded interest.
Step 1: Calculate the number of months in 2 ½ years.
2 ½ years = 2.5 years
Months = 2.5 years * 12 months/year
Months = 30 months
Step 2: Calculate the future value of monthly deposits using the formula for the future value of an ordinary annuity:
Future Value = P * ((1 + r)^n - 1) / r
Where:
P = Monthly deposit amount = $100
r = Monthly interest rate = 6% / 12 months = 0.005
n = Number of months = 30 months
Future Value = $100 * ((1 + 0.005)^30 - 1) / 0.005
Future Value ≈ $3,572.71 (rounded to the nearest cent)
Therefore, at the end of 2 ½ years, there will be approximately $3,572.71 in Anne Thorne's savings account.