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Question 1096173: You want to have $6,000 saved up for a new car in 4 years. How much should you deposit each quarter into an account paying 8% compounded quarterly?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
We will use this formula
FV = P*( (1+i)^n - 1 )/i
where generally,
FV = future value of annuity
i = interest rate per period
n = number of periods
Specifically we can say
FV = target amount of money we want four years into the future
i = interest rate per quarter
n = number of quarters
In this case, we are given
FV = 6000
i = (interest rate in decimal form)/(compounding frequency) = 0.08/4 = 0.02
n = (compounding frequency)*(number of years) = 4*4 = 16
Plug FV = 6000, i = 0.02, and n = 16 into the formula. Then solve for P
FV = P*( (1+i)^n - 1 )/i
6000 = P*( (1+0.02)^16 - 1 )/0.02
6000 = P*( (1.02)^16 - 1 )/0.02
6000 = P*(1.372786 - 1 )/0.02
6000 = P*(0.372786/0.02)
6000 = P*18.6393
6000 = 18.6393*P
18.6393*P = 6000
18.6393*P/18.6393 = 6000/18.6393
P = 321.900501
P = 321.91
So the answer is $321.91
I rounded up to the nearest penny so we could clear the hurdle. Notice how if P = 321.90, then we have FV equal to...
FV = P*( (1+i)^n - 1 )/i
FV = 321.90*( (1+0.02)^16 - 1 )/0.02
FV = 321.90*( (1.02)^16 - 1 )/0.02
FV = 321.90*(1.372786 - 1 )/0.02
FV = 321.90*(0.372786/0.02)
FV = 321.90*18.6393
FV = 5999.99067
FV = 5999.99
Showing that we come up 1 cent short of our goal
On the other hand, if we have P = 321.91, then FV is...
FV = P*( (1+i)^n - 1 )/i
FV = 321.91*( (1+0.02)^16 - 1 )/0.02
FV = 321.91*( (1.02)^16 - 1 )/0.02
FV = 321.91*(1.372786 - 1 )/0.02
FV = 321.91*(0.372786/0.02)
FV = 321.91*18.6393
FV = 6000.177063
FV = 6000.18
we haven't landed on the exact value of $6000 but overshot it (which is better than coming up short). So this confirms we have the correct answer of $321.91
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