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Question 1109513: How long will it take to triple your money at a nominal interest rate
j1 = 12% if simple interest is allowed for part of a year?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! for simple interest, the formula would be:
f = p * r * n + p
r is the interest rate per time period.
n is the number of time periods.
p is the present value
f is the future value.
if you are interested in years, then your interest rate has to be per year.
if you are interested in months, then your interest rate has to be per month.
you always have to adjust your interest rate to be applicable to the time period you are interested in.
your interest rate was given per year.
if you want to solve for years, then the formula would become:
f = p * .12 * n + p
if you assume p = 1000, and you want to triple your money, then f = 3000 and your formula would become:
3000 = 1000 * .12 * n + 1000
you would then solve this formula for n.
you would get n = (3000 - 1000) / (1000 * .12) = 2000 / 120 = 16.666666667 years.
if you wanted to calculate in months, then you would divide .12 by 12 to get .01 per month.
n would then be in months.
the formula would become 3000 = 1000 * .01 * n + 1000
you would solve for n to get n = (3000 - 1000) / (1000 * .01) = 2000 / 10 = 200 months.
with simple interest, the answer would be the same, whether used interest rate per year or interest rate per month.
200 months divided by 12 results in 16.6666666667 years.
that's simple interest.
compound interest is a different story.
with compound interest, the more compounding periods per year, the greater the future value, or the less time it takes to get the same future value.
the compound interest formula is 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.
if you start with 1000 and want to triple it, and your interest rate is .12 per year, and you want to compound once a year, then the formula becomes:
3000 = 1000 * (1 + .12) ^ n
you would then divide both sides of this equation by 1000 to get:
3 = (1 + .12) ^ n
to solve for n, you would take the log of both sides of this equation to get:
log(3) = log((1 + .12) ^ n)
since log(1 + .12) ^ n) is equal to n * log(1 + .12), the formula would become:
log(3) = n * log(1 + .12)
divide both sides of this equation by log (1 + .12) to get:
log(3) / log(1 + .12) = n
solve for n to get:
n = 9.694035413 years.
if you wanted to find out how long it would take with monthly compounding, then you need to adjust r to be per month and n will be the number of months.
the monthly interest rate is .12 / 12 = .01
the formula becomes:
3000 = 1000 * (1 + .01) ^ n
the process is the same.
you would divide both sides of the equation by 1000 to get:
3 = (1 + .01) ^ n
you would take the log of both sides of the equation to get:
log(3) = log((1 + .01) ^ n)
since log(1 + .01) ^ n) is equal to n * log(1 + .01), the formula would become:
log(3) = n * log(1 + .01)
you would divide both sides of this equation by log(1 + .01) to get:
log(3) / log(1 + .01) = n
you would solve for n to get:
n = 110.409624 months.
divide this by 12 to translate to years to get:
n = 9.200802004 years.
with annual compounding, it would take 9.694035413 years.
with monthly compounding, it would take 110.409624 months, which is equivalent to 9.200802004 years.
it takes less time to triple your money with monthly compounding than with annual compounding.
the formula for annual compounding would become f = 1000 * (1 + .12) ^ 9.694035413 = 3000.
the formula for monthly compounding would become f = 1000 * (1 + .01) ^ 110.409624 = 2999.99999999
it doesn't show up as 3000 because there is a small amount of internal rounding being done that doesn't make it come out right on.
if i use an online calculator that displays more digits, then the answer would become n = 110.40962405
using that figure in my calculator, i then get f = 1000 * (1 + .01) ^ 110.40962405 and the result becomes f = 3000.
regardless, 2999.999999 is very close and rounds to 3000 if you need the answer to be within 2 decimal places or so.
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