SOLUTION: A deposit of $2400 in a savings account reaches a balance of $3174.03 after 7 years. The interest on the account is compounded monthly. What is the annual interest rate of the acco

Question 1038748: A deposit of $2400 in a savings account reaches a balance of $3174.03 after 7 years. The interest on the account is compounded monthly. What is the annual interest rate of the account?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
future value = 3174.03.
present amount = 2400.
number of years = 7
interest rate is compounded monthly.
formula to use is f = p * (1+r)^n
f is the future value
p is the present amount
r is the interest rate per time period.
n is the number of time periods.

since the interest rate is compounded monthly, then the interest rate per time period is the annual interest rate divided by 12 and the number of time periods is the number of years * 12.

in the formula, r represents the monthly interest rate and n is calculated to be 7 years * 12 = 84 months.

f is given as 3174.03 and p is given as 2400.

the formula becomes 3174.03 = 2400 * (1+r)^84

divide both sides of this equation by 2400 to get 3174.03 / 2400 = (1+r)^84.

take the log of both sides of this equation to get log(3174.03/2400) = log((1+r)^84).

since log((1+r)^84) = 84 * log(1+r), this formula becomes log(3174.03/2400) = 84 * log(1+r).

divide both sides of this equation by 84 to get log(3174.03/2400)/84 = log(1+r)

solve for log(1+r) to get log(1+r) = .0014452355.

this is true if and only if 10^.0014452355 = 1+r.

simplify to get 1.003333321 = 1+r.

subtract 1 from both sides of this equation to get .003333321 = r

this is the monthly interest rate.

multiply it by 12 to get the annual interest rate.

you get annual interest rate = .03999998522.

this is very close to .04, so i would guess that the annual interest rate is .04 and that the difference in the answer will be due to rounding of the result.

assuming that the annual interest rate is .04, you would confirm the solution is correct by doing the following.

formula is f = p * (1+r)^n
p is 2400
r is .04 per year divided by 12 = .0033333333.... per month.
n is 7 * 12 = 84 months.

formula becomes f = 2400 * (1.003333333...)^84 which results in f = 3174.033273.

round that off to 2 decimal digits and it becomes 3174.03.

this agrees with the problem statement so the solution is good.

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