SOLUTION: Ten thousand dollars is deposited in an account that pays 6% annual interest. How long does it take to double the initial deposit if the interest is compounded annually? monthly? d

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Question 1130906: Ten thousand dollars is deposited in an account that pays 6% annual interest. How long does it take to double the initial deposit if the interest is compounded annually? monthly? daily?
Found 2 solutions by solver91311, greenestamps:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


Compound Interest Formula:



Where is the future value, is the present value, is the annual interest rate as a percent, is the number of compounding periods per year, and is the number of years in the term of the investment.

If the value of the investment doubles, then , so solve:





and



For

Hint: Start by taking the logarithm of both sides and then using


John

My calculator said it, I believe it, that settles it

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The doubling time is independent of the initial amount.

At 6% annual interest for t years, the growth factor over the initial amount is

%281%2B%28.06%2Fn%29%29%5E%28n%2At%29 where n is the number of compounding periods per year.

To find the doubling time, we want the growth factor to be equal to 2. So

(a) for annual compounding, the number of periods per year is 1:

%281%2B%28.06%2F1%29%29%5E%28%281%2At%29%29+=+2

(b) for monthly compounding, the number of periods per year is 12:

%281%2B%28.06%2F12%29%29%5E%28%2812%2At%29%29+=+2

(c) for daily compounding, it is customary to use 360 as the number of periods per year:

%281%2B%28.06%2F360%29%29%5E%28%28360%2At%29%29+=+2

The unknown (number of years, t) is an exponent in each of these equations; to solve each equation you need to use logarithms, or a graphing calculator. Here is a solution for monthly compounding using logarithms.

%281%2B%28.06%2F12%29%29%5E%28%2812%2At%29%29+=+2

%281.005%29%5E%28%2812%2At%29%29+=+2
12t%2Alog%281.005%29+=+log%282%29
t+=+log%282%29%2F%2812%2Alog%281.005%29%29 = 11.58 years, to 2 decimal places

You can work the cases for annual and daily compounding. The daily compounding should take less time than the monthly to double, and the annual compounding should take longer then the monthly to double.