Michelle needs to save $7,000 for school for the next two years. She found a bank that offers a 9% interest rate compounded annually. What does she need to deposit at the beginning of the year to have enough money for school? Refer to the Present Value Table.
To determine the amount Michelle needs to deposit at the beginning of each year, we can use the formula for the future value of an ordinary annuity:
FV = PMT × [(1 + r)^n - 1] / r
Where:
FV = Future Value (amount needed for school) = $7,000
PMT = Yearly deposit
r = Interest rate per period = 9%
n = Number of periods (in this case, two years)
To solve for PMT (the yearly deposit), we rearrange the formula as follows:
PMT = FV × r / [(1 + r)^n - 1]
Now, we can substitute the given values into the formula:
PMT = $7,000 × 0.09 / [(1 + 0.09)^2 - 1]
Using a calculator, we simplify the equation:
PMT ≈ $7,000 × 0.09 / [(1.09)^2 - 1]
PMT ≈ $7,000 × 0.09 / [1.1881 - 1]
PMT ≈ $7,000 × 0.09 / 0.1881
PMT ≈ $7,000 × 0.4789
PMT ≈ $3,351.30
Therefore, Michelle needs to deposit approximately $3,351.30 at the beginning of each year to have enough money for school.
To find out how much Michelle needs to deposit at the beginning of the year to have enough money for school, we can use the concept of present value.
The present value (PV) is the current value of a future sum of money, taking into account the interest rate and time. In this case, Michelle needs to save $7,000 for a period of two years with an interest rate of 9% compounded annually.
To calculate the present value, we can use the Present Value Table. The table provides us with a factor that we can multiply by the future value to find the present value.
First, let's find the present value factor for an interest rate of 9% compounded annually over two years. Look for the row representing 2 years and column representing 9% in the table.
The factor for 2 years and 9% interest rate is 0.8219. This means that multiplying the future value by this factor will give us the present value.
Next, multiply the future value ($7,000) by the present value factor (0.8219) to find the present value:
Present Value = Future Value x Present Value Factor
Present Value = $7,000 x 0.8219
Present Value = $5,753.3 (rounded to the nearest dollar)
Therefore, Michelle needs to deposit approximately $5,753.3 at the beginning of the year to have enough money for school.