Certainly! To determine how much money needs to be deposited now (the present value) in order to obtain [tex]$4,100 after 15 years with an annual interest rate of 6%, we use the formula for the present value of a future sum of money. This formula is derived from the compound interest formula and can be expressed as:
\[ PV = \frac{FV}{(1 + r)^n} \]
where:
- \( PV \) is the present value (the amount of money to be deposited now),
- \( FV \) is the future value (the amount of money to be received in the future, which is $[/tex]4,100 in this case),
- [tex]\( r \)[/tex] is the annual interest rate expressed as a decimal (6% or 0.06 in this case),
- [tex]\( n \)[/tex] is the number of years the money is invested (15 years in this case).
Let's break down the calculation step-by-step:
1. Start with the future value (FV), which is [tex]$4,100.
2. Use the annual interest rate (r), which is 6% or 0.06.
3. Identify the number of years (n), which is 15 years.
4. Plug these values into the present value formula:
\[ PV = \frac{4100}{(1 + 0.06)^{15}} \]
5. Add the interest rate to 1:
\[ 1 + 0.06 = 1.06 \]
6. Raise 1.06 to the power of 15:
\[ 1.06^{15} \approx 2.39656 \]
7. Divide the future value by this compounded factor:
\[ PV = \frac{4100}{2.39656} \]
8. Perform the division:
\[ PV \approx 1710.79 \]
So, the amount of money that needs to be deposited now to obtain $[/tex]4,100 in 15 years with a 6% annual interest rate, compounded annually, is approximately $1,710.79.
