Final answer:
To calculate the time it takes for a company's sales to double at a 12% annual growth rate, we use the compound interest formula without relying on the rule of 72. Solving for t in the equation 2 = (1 + 0.12)^t results in approximately 6.12 years, which is the time required for the sales to double.
Step-by-step explanation:
To determine how long it will take a sum of money (or sales, in this case) to double at a certain growth rate without using approximations, one can use the Rule of 72. This rule is a simplified way to estimate the number of years required to double the invested money at a given annual fixed rate of interest. However, it's more accurate for smaller growth rates and might yield an approximation for higher rates, such as 12%. Instead, we would use the formula for compound interest: A = P(1 + r/n)(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial sum of money), r is the annual interest rate (decimal), n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.
For sales to double with a growth rate of 12% per year, we would set the equation up as follows: 2P = P(1 + 0.12)t. Simplifying, we would get 2 = (1 + 0.12)t and solving for t gives us t = log(2) / log(1.12), which calculates to approximately 6.12 years. So, it will take a little over 6 years for the company's sales to double at a 12% growth rate per year.