SOLUTION: Abe deposits $1500 into a savings account that pays 1.96% per year for t years. How long before the amount doubles if… a. the interest is compounded quarterly (n = 4) using A=

Question 1178981: Abe deposits $1500 into a savings account that pays 1.96% per year for t years. How long
before the amount doubles if…
a. the interest is compounded quarterly (n = 4) using A= P (1+r/n)^nt?
b. the interest is compounded continuously using A= Pe^rt?
c. What is the annual percent yield on this deposit if it is compounding continuously?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the formula you have for discrete compounding is A = P * (1 + r/n) ^ (n*t)
with quarterly compounding, the formula becomes:
P = 1500
r = 1.96% / 100 = .0196
the formula uses the rate, not the percent.
with quarterly compounding, the formula becomes:
f = 1500*(1+.0196/4)^(4*t)
since you want to know when the money will double, f becomes 3000 and the formula becomes:
3000 = 1500*(1+.0196/4)^(4*t)
divide both sides of the formula by 1500 to get:
2 = (1+.0196/4)^(4*t)
take the log of both sides of the equation to get:
log(2) = log((1+.0196/4)^(4*t))
since log((1+.0196/4)^(4*t)) = 4*t*log(1+.0196/4), then the formula becomes:
log(2) = 4*t*log(1+.0196/4)
divide both sides of the equation by log(1+.0196/4) to get:
log(2)/log(1+.0196/4) = 4*t
solve for 4*t to get:
4*t = 141.80489995
solve for t to get:
t = 141.8048995/4 = 35.45122488 years.
that's your solution.
confirm by replacing t with that in the original equation to get:
f = 1500*(1+.0196/4)^(4*35.45122488) = 3000

the continuous compounding formula that you have is A = P*e^(r*t)
A is the future value
P is the present value
e is the scientific constant whose value is 2.718281828.
that's an irrational number that has an infinite number of decimal places.
the display in the calculator shows it rounded to the number of digits that the calculator can display, but the number internally stored in the calculator has many more digits than that, although it does get rounded to the number of digits that the calculator can store internally. i think that's about 18 digits or so.
so, anywhere you see 'e', think 2.718281828 or something close to that, depending on the number of digits that the calculator can display.
with the numbers that you have to work with, the formula becomes:
3000 = 1500*e^(.0196*t)
divide both sides of the formula by 1500 to get:
2 = e^(.0196*t)
you can take the natural log of both sides of the equation or you can take the log separation.
we'll use the natural log.
formula becomes:
ln(2) = ln(e^(.0196*t))
since ln(e^(.0196*t)) = .0196*t*ln(e) and since ln(e) = 1, the formula becomes:
ln(2) = .0196*t
solve for t to get:
t = ln(2)/.0196 = 35.36465207 years.
to confirm, replace t in the original equation to get:
f = 1500*e^(.0196*35.36465207) = 3000.

with quarterly compounding, it takes 35.45122488 years.
with continuous compounding, it takes 35.36465207 years.

RELATED QUESTIONS
James deposits $6000 into a savings account which pays 1% per annum. If the account... (answered by Boreal)

1.A certain bank pays 1% compound interest for their savings account holders. If Nick... (answered by stanbon)

Future value: Sandra wants to deposit 180 dollars each year for her son. If she places... (answered by Boreal)

sandra wants to deposit $80 each year for her son.IF she places her deposits in a savings (answered by 428225)

Alison deposits $500 into a new savings account that earns 5 percent interest compounded... (answered by lynnlo)

An amount of $1,000 is deposited into a bank account that pays 4% interest compounded... (answered by mananth)

Kindly help me to solve this problem: Ken deposits 6,000 into a saving account at t=0. (answered by ankor@dixie-net.com)

Mary and Joe would like to save up $10,000 by the end of three years from now to buy new... (answered by Theo)