College savings question

Question

Sam Long anticipates he will need approximately $225,000 in 15 years to cover his 3-year-old daughter’s college bills for a 4-year degree. How much would he have to invest today at an 8% interest rate compounded semiannually?

Answer 1

Compound interest formula A = P ( 1 + ^r/_n ) ^nt

where A=final amount, P = initial principal, r = interest rate

n = number of times interest applied per period, t = number of time periods

Now substitute given values (8% is r = 0.08 and "semi-annually" is n = 2) to get:

225000 = P ( 1 + ^0.08/₂ ) ^(2)(15)

=> 225000 = P ( 1.04 ) ³⁰

=> P = 225000 / ( 1.04 ) ³⁰

=> P = 69371.7

hence Sam needs to invest $69,372 today (to nearest dollar).