Question

BJ's goal is to have $50,000 saved at the end of Year 5. At the end of Year 2, they can add $7,500 to their savings but they want to deposit the remainder they need to reach their goal today, Year 0, as a lump sum deposit. If they can earn 4.5 percent, how much must they deposit today

Answer 1

$33534.73

Explanation:

Let the lump sum be P.

The interest, I, on a rate, R%, per annum after T years is given by

`I = PRT/100`

The amount, A, is

`A = P + I = P(1 + (RT)/(100))`

After 2 years at 4.5% interest rate, the amount is

`A = P(1+(4.5*2)/(100)=1.09P`

$7500 is added after 2 years. The principal for the beginning of the third year is then

1.09P + 7500

The amount after the next 3 years is

`A = (1.09P + 7500)\left(1+(4.5*3)/(100)\right)=(1.09P + 7500)*1.135`

This is the amount expected to be saved.

`50000=(1.09P + 7500)*1.135`

Solving for P, we have

`1.09P + 7500 = 44052.86`

`1.09P = 36552.86`

`P = 33534.23`