Final answer:
To find the number of years it will take a given amount to multiply by a factor of ten at an effective annual interest rate of 12 percent, use the formula for compound interest. The formula is A = P(1 + r/n)^(n*t), where A is the final amount, P is the initial amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. The calculated answer is approximately 6.91 years.
Step-by-step explanation:
To find the number of years it will take a given amount to multiply by a factor of ten, we can use the formula for compound interest. The formula is:
A = P(1 + r/n)^(n*t)
Where:
- A is the final amount
- P is the initial amount
- r is the annual interest rate (as a decimal)
- n is the number of times interest is compounded per year
- t is the number of years
In this case, we want the final amount A to be 10 times the initial amount P. So we can set up the equation:
10P = P(1 + 0.12/1)^(1*t)
Cancelling out the initial amount P on both sides, we have:
10 = (1.12)^t
To solve for t, we can take the logarithm of both sides:
t = log(10)/log(1.12) ≈ 6.91
So it will approximately take 6.91 years for a given amount to multiply by a factor of ten at an effective annual interest rate of 12 percent.