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8-111.

THE RULE OF 70

The Rule of 70 is a useful rule of thumb for determining the doubling time under steady (exponential) growth. To get the doubling time, the rule tells us to divide 70 by the growth rate, expressed as a percentage. Carry out the following steps to see why the Rule of 70 works as it does.

1. Use $A = Pe^{rt}$ with $A = 2P$ to find the exact doubling time $t$ in terms of the growth rate $r$ without using your calculator. In other words express $t$ in terms of $r$.

$2P = Pe^{rt}$

$t=\frac{\text{ln}2}{r}$

2. If $r$ is the rate of growth (i.e. $0.07$), and $R$ is the percentage rate of growth (i.e. $7\%$), express $r$ in terms of $R$.

$r=\frac{R}{100}$

3. Using your calculator to evaluate $\ln 2$, show that doubling time with growth rate is $t ≈ \frac { 70 } { k }$.

$t=\frac{\text{ln}2}{r} \ \ \ t=\frac{.6931}{\frac{R}{100}} \ \ \text{ Then simplify.}$

4. $100$ $\ln 2$ is closer to $69$ than $70$. Why do we use $70$?

We use $70$ because it makes the mental arithmetic easier.