How long, to the nearest year, will it take an investment to triple if it is continuously compounded at 18% per year?
I used A=Pe^rt to get the wrong answer 20 years. Don't understand what is wrong
Thanks
3P = Pe^.18t
3 = e^.18t
ln3 = .18t
1.0986 = .18t
t = 6.10
This makes sense, considering the rule of 72, which states that at r%, the amount doubles about every 72/r years.
So, we expect this to double every 4 years.
You made a common mistake while using the formula A = Pe^rt. This formula is used for calculating the future value of an investment when it is compounded continuously. However, in this case, you are trying to find the time it takes for an investment to triple, given the annual growth rate of 18%.
To solve this problem, you need to use the formula for compound interest. The formula for compound interest is given by:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal or initial investment
r = the annual interest rate (in decimal form)
n = the number of times interest is compounded per year
t = the time period in years
In your case, the investment is continuously compounded, so we can consider n as infinite. Therefore, the formula becomes:
A = Pe^(rt)
Now, let's solve the problem using the correct formula:
We want the investment to triple, so the future value (A) will be 3 times the initial investment (P). The interest rate (r) is 18%, which is equivalent to 0.18. We need to find the time (t) in years.
3P = Pe^(0.18t)
Cancelling out the 'P' on both sides:
3 = e^(0.18t)
To solve for 't', we need to take the natural logarithm (ln) on both sides:
ln(3) = ln(e^(0.18t))
Using the logarithmic property ln(a^b) = b*ln(a):
ln(3) = 0.18t * ln(e)
Now, ln(e) is equal to 1, so the equation simplifies to:
ln(3) = 0.18t
Now, solve for 't' by dividing both sides by 0.18:
t = ln(3) / 0.18
Using a calculator, ln(3) is approximately equal to 1.10. Therefore:
t ≈ 1.10 / 0.18 ≈ 6.11
Rounding to the nearest year, it will take approximately 6 years for the investment to triple if it is continuously compounded at a rate of 18% per year.