Question

You deposit $100 into an account that pays 6% annual interest. How long will it take for the balance to reach 1000 if the interest is compounded with the given frequency? A. Quarterly B. Continuously

Answer 1

6% annually = (1+.06/4)^4 = 1.0613635506
6.13635506 % compounded quarterly
When it is continuously compounded, the formula is
annual = e^r -1
annual = (2.718281828 ^ .06) -1
annual = 1.0618365465 -1
annual rate = .0618365465 or 6.18365465%
To see those formulas:
http://1728.org/rate.htm
Now to do the other calcuations.

Answer 2

We need to know how long it takes
$100 to become $1,000 at 6.13635506 %
The formula is:
Years = [log (total) - log(principal)] / log (1 + rate)
Years = (log 1,000 - log 100) / log (1.0613635506)
Years = (3 -2) / 0.025864169
Years = 1 / 0.025864169
Years = 38.6635271566

Now for the continuously compounded interest:
$100 to become $1,000 at 6.18365465 %
The formula is:
Years = [log (total) - log(principal)] / log (1 + rate)
(except for the rate, this is the same as above)
Years = (3-2) / log (1.0618365465)
Years = 1 / 0.0260576689
Years = 38.3764182375

To see those formulas:
http://1728.org/compint2.htm
http://1728.org/rate.htm