Question

If $3000 is deposited at the end of each half year in an account that earns 6.4% compounded semiannually, how long will it be before the account contains $120,000? (Round your answer to one decimal place.)
yr -

Answer 1

Future value(FV) of periodic payment is given by :

FV = (P/r)((1 + r)ⁿ - 1)

where FV = Future value = 120,000, r = interest rate of a period = (6.4/2)% = 3.2% = 0.032, P = periodic amount = 3000 and n = number of periods(i.e. number of semi annuals)

=> 120,000 = (3000/0.032)((1 + 0.032)ⁿ - 1)

=> (1 + 0.032)ⁿ - 1 = 1.28 + 1 = 2.28

Taking ln on both sides and using ln(a^b) = b*ln(a) we get :

n*ln(1.032) = ln(2.28)

=> n = 26.165407

So number of Years = n/2 = 26.165407/2 = 13.1

Hence, Amount of time it will take = 13.1 years