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A small business owner contributes $1,000 at the end of each quarter to a retirement account that earns 10% compounded quarterly.

(a) After how many quarterly payments, n, will this annuity account be worth at least $50,000? (Round your answer UP to the nearest quarter.) 17 quarters
(b) Now assume the owner has $50,000 in the same account continuing to earn interest. In addition, the business owner starts making contributions of $3,000 at the end of each quarter into another account with this same interest rate. What will the total value of both accounts be after 15 more years?

Respuesta :

Answer:

Account balance after 15 years is $627,964

Explanation:

Detailed step-wise solution is given in the images attached.

Ver imagen hamzafarooqi188
Ver imagen hamzafarooqi188
TomShelby

Answer:

a ) 32.84

b) future value after 15 ears and additional contribution of 3,000 dolalr each

627.964,26‬

Explanation:

Given the formula for an annuity we have to solve for time (n)

[tex]C \times \frac{(1+r)^{time} -1 }{rate} = PV\\[/tex]

C  $1,000.00

time n

rate 0.025

PV $50,000.0000

[tex]1000 \times \frac{(1+0.025)^{n} -1}{0.025} = 50000\\[/tex]

[tex](1+0.025)^{n}= 1 + \frac{50000\times0.025}{1000}[/tex]

[tex](1+0.025)^{n}= 2.25[/tex]

We now solve using logarithmic properties:

[tex]-n= \frac{log 2.25}{log(1.025)[/tex]

n =  32.84  

b)

Value of the 50,000 dollars

[tex]Principal \: (1+ r)^{time} = Amount[/tex]

Principal 50,000.00

time 60.00 (15 years x 4 quarter per year)

rate 0.02500

[tex]50000 \: (1+ 0.025)^{60} = Amount[/tex]

Amount 219,989.49

Value of the additional contributions:

[tex]C \times \frac{(1+r)^{time} -1}{rate} = FV\\[/tex]

C 3,000.00

time 60

rate 0.025

[tex]3000 \times \frac{(1+0.025)^{60} -1 }{0.025} = FV\\[/tex]

FV $407,974.7699

Total:

219,989.49   +  407,974.7699  = 627.964,26‬

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