Jessica wants to accumulate $14,000 by the end of 5 yr in a special bank account, which she had opened for this purpose. To achieve this goal, Jessica plans to deposit a fixed sum of money into the account at the end of the month over the 5-yr period. If the bank pays interest at the rate of 7% per year compounded monthly, how much does she have to deposit each month into her account?
i = .07/12 = .00583333
n = 12(5) = 60
let her payment be P
P( (1.00583333)^60 - 1)/.005833333 = 14000
solve for P
Suppose 12 people arrive at a bank at the same time. In how many ways can they line up to wait for the next available teller?
To find out how much Jessica needs to deposit each month into her account, we can use the future value formula for monthly compounded interest.
The formula to calculate the future value of a series of deposits is:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future value (the target amount of $14,000)
P = Monthly deposit
r = Monthly interest rate (7% / 12, converted to decimal)
n = Number of periods (5 years * 12 months per year)
Let's now plug in the values into the formula and solve for P:
14,000 = P * ((1 + (0.07 / 12))^(5 * 12) - 1) / (0.07 / 12)
To simplify the calculation, we can first evaluate the expression inside the parentheses:
14,000 = P * ((1 + 0.005833)^60 - 1) / 0.005833
Now, let's calculate the value inside the parentheses:
(1 + 0.005833)^60 = 2.208376
Substituting this back into the equation:
14,000 = P * (2.208376 - 1) / 0.005833
Now, let's simplify further:
14,000 = P * 1.208376 / 0.005833
14,000 * 0.005833 = P * 1.208376
81.662 = P * 1.208376
Finally, solving for P:
P = 81.662 / 1.208376 ≈ $67.58 (rounded to the nearest cent)
Therefore, Jessica needs to deposit approximately $67.58 each month into her account to accumulate $14,000 by the end of 5 years.
To calculate the fixed monthly deposit that Jessica needs to make, we can use the formula for the future value of an ordinary annuity:
FV = P * ( (1+r)^n - 1 ) / r
Where:
FV = Future Value = $14,000 (the amount Jessica wants to accumulate)
P = Monthly deposit amount (what we need to calculate)
r = Monthly interest rate = annual interest rate / 12 = 7% / 12 = 0.07 / 12
n = Total number of compounding periods = 5 years * 12 months = 60 months
Now, let's substitute the values into the formula and solve for P:
$14,000 = P * ( (1 + 0.07/12)^60 - 1 ) / (0.07/12)
To calculate this equation, we can simplify it step by step:
1. Calculate the bracketed portion first:
(1 + 0.07/12)^60 - 1 = (1.005833 - 1)
2. Simplify the equation:
$14,000 = P * (0.005833) / (0.07/12)
3. Rearrange the equation to solve for P:
P = $14,000 / (0.005833) * (0.07/12)
Calculating this, we find the monthly deposit amount (P) that Jessica needs to make is approximately $198.03.