Mary needs $9000 in 11 years. What amount can she deposit in a sinking fund at the end of each quarter at 4% interest compounded quarterly so she will have her $9000? Round to the nearest cent.
r = 4% / 4 = 0.01
11years *4 = n = 44 compound periods
9000 = x [ 1.01^44 - 1 ] / 0.01 = x [ 1.5493 - 1 ] / 0.01
90 = .5493 x
x = 163.84
Well, if Mary needs to deposit a certain amount at the end of each quarter, I hope she's good at saving money! Saving is no joke. Anyway, let's calculate this.
First, we need to figure out how many quarters there are in 11 years. Since there are 4 quarters in a year, that means there are a total of 44 quarters.
Now, let's divide $9000 by the future value of an ordinary annuity formula, which is (1 + i)^n - 1 / i, where i is the interest rate and n is the number of periods. So, plugging in the numbers, we have:
(1 + 0.04/4)^44 - 1 / (0.04/4) = X
Solving for X, we find:
X ≈ $106.52
So, Mary would need to deposit approximately $106.52 at the end of each quarter in order to have $9000 in 11 years.
Keep in mind that this calculation assumes that Mary makes these deposits regularly and consistently, without withdrawing any money along the way. Otherwise, she might need a bigger clown piggy bank to meet her goal!
To find the amount Mary should deposit in a sinking fund at the end of each quarter, we can use the formula for the future value of a sinking fund:
\(FV = P \times \left( \dfrac{(1 + r)^{nt} - 1}{r} \right)\)
Where:
- \(FV\) is the future value of the sinking fund (in this case, $9000)
- \(P\) is the amount deposited at the end of each quarter
- \(r\) is the interest rate per compounding period (in this case, 4%/4 = 0.01)
- \(n\) is the number of compounding periods per year (in this case, 4 since it's compounded quarterly)
- \(t\) is the number of years (in this case, 11)
Substituting the given values into the formula:
\(9000 = P \times \left( \dfrac{(1 + 0.01)^{(4 \times 11)} - 1}{0.01} \right)\)
Simplifying:
\(9000 = P \times \left( \dfrac{(1.01)^{44} - 1}{0.01} \right)\)
Now we can solve for \(P\):
\(P = \dfrac{9000}{\left( \dfrac{(1.01)^{44} - 1}{0.01} \right)}\)
Using a calculator, we can find:
\(P \approx \$47.98\)
Therefore, Mary should deposit approximately $47.98 at the end of each quarter in order to have $9000 in 11 years.
To calculate the amount Mary needs to deposit at the end of each quarter, we can use the formula for the future value of a sinking fund.
The formula for the future value of a sinking fund is:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future Value (in this case, $9000)
P = Regular Payment (the amount Mary needs to deposit at the end of each quarter)
r = Interest rate per period (4% per year, compounded quarterly, so 1% per quarter)
n = Number of periods (11 years = 11 * 4 = 44 quarters)
By substituting the given values into the formula, we can solve for P:
9000 = P * ((1 + 0.01)^44 - 1) / 0.01
Now, we can solve this equation for P using algebraic methods or by using a financial calculator or spreadsheet software.
Let's calculate it using a financial calculator or spreadsheet software.
By using a financial calculator or spreadsheet software, you can input the formula mentioned above and solve for P. The resulting value will be the amount Mary needs to deposit at the end of each quarter.
The amount Mary needs to deposit at the end of each quarter is approximately $92.64 (rounded to the nearest cent).
Therefore, Mary should deposit $92.64 at the end of each quarter to reach her goal of $9000 in 11 years, assuming a 4% interest rate compounded quarterly.