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Question 1092767: If $6,000 is placed in an account with an annual interest rate of 5%, how long will it take the amount to triple if the interest is compounded annually?
How long would it take $3,200 to grow to $8,300 if the annual rate is 4.7% and interest in compounded monthly?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the formula to use if 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.
your first problem yields the following adaptation of that formula.
3 * 6000 = 6000 * (1+.05)^n
the time periods are in years, therefore r = 5% / 100 = 5/100 = .05.
in this formula, you always use the decimal equivalent of the percent which is the percent divided by 100.
divide both sides of this formula by 6000 to get 3 = (1.05)^n
take the log of both sides of this equation to get log(3) = log((1.05)^n).
the property of logs states that log(x^a) = a*log(x).
your equation becomes log(3) = n * log(1.05)
divide both sides of this equation by log(1.05) to get log(3) / log(1.05) = n
solve for n to get n = 22.51708531.
the initial amount of 6000 will triple in 22.51708531 years.
the formula becomes:
18000 = 6000 * (1.05) ^ 22.51708531.
this results in 18000 = 18000, confirming the solution is correct.
your second problem is a little different because the time periods are not in years, but the same formula applies.
you divide your annual interest rate by the number of compounding periods per year and you multiply your number of years by the number of compounding periods per year.
f = p * (1+r)^n
f = 8300
p = 3200
r = 4.7 / 100 / 12 = .0039166667
n = what you want to find.
the annual rate percent is 4.7%
you need to divide that by 100 to get the annual rate of .047 per year.
you then need to divide that by 12 to get the monthly rate of .0039166667
the formula of f = p * (1+r) ^ n becomes 8300 = 3200 * (1 + .0039166667) ^ n
since you are compounding monthly, then n will be in months.
divide both sides of this equation by 3200 to get 8300 / 3200 = (1 +.0039166667) ^ n.
take the log of both sides of the equation to get log(8300/3200) = log(1+.0039166667)^n).
properties of logs states that log(x^a) = a*log(x).
your equation becomes log(8300/3200) = n * log(1+.0039166667).
divide both sides of this equation by log(1+.0039166667) to get log(8300/3200) / log(1+.0039166667) = n
solve for n to get n = 243.822124.
confirm by replacing n in the original equation by 243.822124 and you should get 8300 = 8300.
the equation becomes 8300 = 3200 * (1 + .0039166667) ^ 243.822124 which results in 8300 = 8300 which confirms the solution is correct.
the solution is that 3200 will become 8300 in 243.822124 months.
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