You invest $4500 into an account that earns 6.5% interest compounded monthly. How long will it take for the account to double in its value?
solve for m months in
(1+.065/12)^m = 2
To determine how long it will take for the account to double, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Ending amount (in this case, double the initial investment)
P = Principal amount (initial investment)
r = Annual interest rate (in decimal form)
n = Number of times the interest is compounded per year
t = Number of years
In this case, the initial investment (P) is $4500, and we want to find out how long it will take for the account to double, so the ending amount (A) will be $9000.
The annual interest rate is 6.5%, which needs to be converted to a decimal, so r = 0.065.
Since the interest is compounded monthly, the compounding period (n) is 12.
Substituting all the values into the formula, we have:
9000 = 4500(1 + 0.065/12)^(12t)
Now, we need to solve for t, the number of years.
To simplify the equation further, divide both sides by 4500:
2 = (1 + 0.065/12)^(12t)
Now, take the logarithm on both sides to solve for t:
log(2) = log[(1 + 0.065/12)^(12t)]
Using logarithm properties, we can bring the exponent down:
log(2) = (12t) * log(1 + 0.065/12)
Finally, we solve for t by dividing both sides by 12 * log(1 + 0.065/12):
t = [log(2)] / [12 * log(1 + 0.065/12)]
Using a scientific calculator or an online calculator, we can calculate the value of t.
After finding the value of t, it will represent the number of years it will take for the account to double.