SOLUTION: Find the time required for an investment of 5000 dollars to grow to 8300 dollars at an interest rate of 7.5 percent per year, compounded quarterly.

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Question 1149069: Find the time required for an investment of 5000 dollars to grow to 8300 dollars at an interest rate of 7.5 percent per year, compounded quarterly.
Found 3 solutions by Theo, MathTherapy, ikleyn:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
formula is f = p * (1 + r) ^ n

p = 5000
f = 8300
r = 7.5 / 100 / 4 = .01875
n = what you want to find.

formula becomes 8300 = 5000 * (1 + .01875) ^ n
divide both sides by 5000 to get 8300/5000 = (1 + .01875) ^ n
take the log of both sides to get log(8300/5000) = log((1 + .01875) ^ n)
this becomes log(8300/5000) = n * log(1 + .01875)
divide both sides of log(1 + .01875) to get log(8300/5000) / (log(1.01875) = n
solve for n to get n = 27.28289636.

5000 becomes 8300 in 27.28289636 quarters.
divide that by 4 to get 6.820724091 years.

note that a quarterly compound rate of .01875 becomes an effective yearly rate of 1.01875 ^ 4 - 1 = .077135866.

bring that out 6.820724091 years and you get f = 5000 * (1.077135866) ^ 6.820724091 = 8300.

to find the quarterly rate, you divide the annual rate by 4.
to find the effective annual rate, you add 1 to the quarterly rate and raise that to the fourth power and then subtract 1.
the rate is the percent divided by 100.
put it all together and you get:
nominal annual rate = 7.5%
divide by 100 to get a nominal annual .075.
effective annual rate = (1 + .075/4) ^ 4 - 1 = .0771358658

you can use the quarterly rate to find the number of quarters, or you can use the effective annual rate to find the number of years.
the number of quarters divided by 4 is the number of years.
both ways will get the same answer if you did it right.





Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
Find the time required for an investment of 5000 dollars to grow to 8300 dollars at an interest rate of 7.5 percent per year, compounded quarterly.
Use the FV formula: 

t or time required is: highlight_green%28matrix%281%2C2%2C+6.820724091%2C+years%29%29

Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
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            After the presented analytical solution/solutions,  the correct answer should be formulated in terms
            of integer number of years OR integer number of compounding periods. 


In this context, the correct answer should go and should be read in this form:


    (1)  7 years is enough;  6 years is not enough.


    (2)  7*4 = 28 compounding quarter periods is enough;  27 compounding quarter periods are not enough.