To find out how much Dan needs to invest today as a lump sum to reach $32,000 in six years with an 8% interest rate, we can use the formula for calculating the future value of a lump sum investment:
\[ FV = PV \times (1 + r)^n \]
Where:
- \( FV \) is the future value (which is $32,000 in this case)
- \( PV \) is the present value (the amount Dan needs to invest)
- \( r \) is the interest rate per period (8% or 0.08)
- \( n \) is the number of periods (6 years)
Substituting the given values into the formula:
\[ 32,000 = PV \times (1 + 0.08)^6 \]
Now, solve for \( PV \):
\[ PV = \frac{32,000}{(1 + 0.08)^6} \]
\[ PV = \frac{32,000}{(1.08)^6} \]
\[ PV = \frac{32,000}{1.586874} \]
\[ PV ≈ 20,167.59 \]
So, Dan needs to invest approximately $20,167.59 today as a lump sum to reach $32,000 in six years at an 8% interest rate.
