Question 1149069
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.