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Question 1151563: If you deposit money today in an account that pays 4% annual interest, how long will it take to double your money?
Found 4 solutions by MathLover1, Theo, ikleyn, Alan3354:
Answer by MathLover1(20850)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period
n is the number of time periods.
with annual compounding and f = 2 * p and r = .04, the formula becomes:
2 = (1 + .04) ^ n
take the natural log of both sides of this equation to get:
ln(2) = ln(1.04^n)
since ln(1.04^n) is equal to n * ln(1.04), this becomes:
ln(2) = n * ln(1.04)
divide both sides of this equation by ln(1.04) and solve for n to get:
n = ln(2) / ln(1.04) = 17.67298769
confirm by replacing n with that in the original equation to get:
f = (1 + .04) ^ 17.67298769
solve for f to get:
f = 2
this confirms the solution is correct.
the solution, assuming annual compounding, is equal to 17.67298769 years.
if you assume monthly compounding, the formula becomes:
2 = (1 + .04/12) ^ n
n becomes the number of months.
take the natural log of both sides of this equation to get:
ln(2) = ln((1+ .04/12) ^ n)
this becomes ln(2) = n * ln(1 + .04/12)
divide both sides of the equation by ln(1 + .04/12) to get:
ln(2) / ln(1 + .04/12) = n
solve for n to get n = 208.2905355
that's the number of months.
divide that by 12 to get number of years = 17.35754463
with monthly compoounding, the number of years to double the money is slightly less because monthly compounding gives a higher effective interest rate per year than annual compounding.
with annual compounding, the effective annual interest rate is .04.
with monthly compounding, the effective annual interest rate is (1 + .04/12) ^ 12 - 1 = .040741543.
that's the rate,
the percent is 100 times that.
.04 = 4%
.040741543 = 4.0741543%
Answer by ikleyn(52858)  (Show Source):
You can put this solution on YOUR website! .
By default, from the problem description we MUST assume that the account is compound,
that the compound period is one year and that the annual rate is 4%.
The condition DOES NOT LEAVE a room to make other assumptions.
I will not repeat calculations of the two other tutors for this case --- they are correct.
But the conclusion that they both make for the answer to be 17.7 years, is WRONG.
The correct answer is THIS:
17 years period is not enough to double the deposited amount;
the 18 years period is just enough.
/\/\/\/\/\/
Alan tries to correct me, but I do not agree with his correction.
The words "an account pays 4% annual interest" COVER all possible variations, making them non-essentials.
Any other treatment / interpretation of the condition (if possible) is a harassment over the common sense.
Answer by Alan3354(69443)  (Show Source):
You can put this solution on YOUR website! The compounding period must be specified.
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It's probably annually, but you must state that.
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It could be monthly, daily, or continuously.
PS Do NOT trust a banker to give you information.
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