Final answer:
The Rule of 70 is a shortcut to estimate the time it takes for an investment or quantity to double at a constant compounding growth rate, by dividing the number 70 by the percentage growth rate. The formula is rooted in logarithmic calculations, and it is most accurate for growth rates below 10%. It is similar to the Rule of 72, which uses the number 72 as the divisor.
Step-by-step explanation:
Understanding the Rule of 70
The Rule of 70 is a simple way to estimate the doubling time for a quantity growing at a constant compounding rate. According to the rule, you divide the number 70 by the percentage growth rate (denoted as 'r') to get an approximation of how many years it will take for the initial quantity to double. For example, if an investment has a growth rate of 6% per year, we calculate the doubling time by dividing 70 by 6, which gives approximately 11.67 years.
For rates of growth that are relatively small (specifically rates smaller than 10%), this approximation becomes quite accurate. The formula underlying this rule comes from the use of natural logarithms to solve for the doubling time (t2), which is represented by the equation t2 = ln(2) / ln(1 + p), where 'p' is the annual percentage rate expressed as a decimal.
It is important to note that the Rule of 70 is closely related to the Rule of 72, another heuristic used to estimate doubling times using compounded interest, with the primary difference being the divisor used in the calculation (70 for the Rule of 70, and 72 for the Rule of 72). Both are useful for quickly estimating growth over time. If we consider an investment growing at a rate of 5% per year, for instance, the Rule of 70 would estimate that the doubling time is 70/5, or 14 years.