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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

Respuesta :

adelekeademolu

Answer:

$33534.73

Step-by-step explanation:

Let the lump sum be P.

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

[tex]I = PRT/100[/tex]

The amount, A, is

[tex]A = P + I = P(1 + \frac{RT}{100})[/tex]

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

[tex]A = P(1+\dfrac{4.5\times2}{100}=1.09P[/tex]

$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

[tex]A = (1.09P + 7500)\left(1+\dfrac{4.5\times3}{100}\right)=(1.09P + 7500)\times1.135[/tex]

This is the amount expected to be saved.

[tex]50000=(1.09P + 7500)\times1.135[/tex]

Solving for P, we have

[tex]1.09P + 7500 = 44052.86[/tex]

[tex]1.09P = 36552.86[/tex]

[tex]P = 33534.23[/tex]

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