# The Rule of 72 and Compound Growth

### Transcript (PDF)

In this video, we’ll learn two tricks for solving compound interest problems in our heads. Trick 1 is the Rule of 72.

A savings account pays 8% interest per year. At the start of 2018, we deposit $1. When will this initial dollar grow to$2?

The Rule of 72 says that if the interest rate is 8% per year, the doubling time is roughly “72 divided by 8” — or 9 years.

So, after 9 years, our initial dollar doubles to $2. After another 9 years, the$2 double to $4. And after yet another 9 years, the$4 double to $8. Etc. In general, the Rule of 72 says that if a quantity is growing at a rate of x% per time period, the doubling time is roughly “72 divided by x” periods. So if germs in a Petri dish are growing at a rate of 12% per minute, they double roughly once every “72 divided by 12”, or 6 minutes. And if the German population is growing at a rate of 2% per year, it doubles roughly once every “72 divided by 2”, or 36 years. The Rule of 72 is a rough approximation to the actual doubling time. If the growth rate is 8%, it’s very close to the actual doubling time. For lower growth rates, like 2%, it overstates the actual doubling time. Conversely, for higher growth rates, like 12%, it understates the actual doubling time. Now, why does the Rule of 72 work? Where does the mysterious number 72 come from? The explanation is pretty simple but does require a bit of math. If you’re interested, see the description below. Trick 2 is a very simple observation: 2 to the power of 10 is roughly 1,000. That’s because 2 multiplied by itself 10 times equals 1,024. Which is pretty close to 1,000. That’s all there is to Trick 2! Trick 2 seems pretty useless. But it’s actually great for mental arithmetic. It tells us that growing a thousand-fold is roughly the same as doubling 10 times. Let’s now put our two tricks to use. “At an interest rate of 8% per year, how long will it take for$1 to grow to $1,000?” Trick 1 — the Rule of 72 — says that the doubling time is roughly 9 years. Trick 2 says that growing a thousand-fold is roughly the same as doubling 10 times. Altogether then, it’ll take roughly 9 × 10, or 90 years. This rough answer is remarkably close to the actual answer. Let’s try another problem. Zimbabwe is wracked by hyperinflation. The price of an egg is initially Z$1, but rises by 12% per day. How long before the price hits Z$1M? Trick 1 — the Rule of 72 — says that the doubling time is roughly “72 divided by 12”, or 6 days. Now, notice that 1M is a thousand squared. And so growing a million-fold is the same as growing a thousand-fold TWICE. Trick 2 says that growing a thousand-fold is roughly the same as doubling 10 times. So, growing a thousand-fold TWICE must be roughly the same as doubling 20 times. Altogether then, it’ll take roughly 6 × 20, or 120 days. This rough answer is, again, remarkably close to the actual answer. Now for the thumbnail problem. A nasty woman borrows$1 from a loanshark, at an interest rate of 2% per day. Roughly how many years before her debt hits $1B? I’ll end here, but you should be able to solve this on your own within a minute. #### Footnotes  Let t be the actual answer. Then 1.08t = 1,000. Taking base-10 logs, we have t log 1.08 = 3. Rearranging, t = 3 / log 1.08 = 89.75652…  Let t be the actual answer. Then 1.12t = 1,000,000. Taking base-10 logs, we have t log 1.12 = 6. Rearranging, t = 6 / log 1.12 = 121.90656… #### Thumbnail Problem At 2% daily interest, how long for$1 to grow to \$1B?

SOLUTION:

• Doubling time ≈ 72 ÷ 2 = 36 days.
• 1B = 1000³. So, ↑1B ≈ Doubling 30 times.
• Answer ≈ 36 × 30 = 1080 days — or roughly 3 years.

(The actual answer is 1046.5… days — or 2.867… years. )

#### Why the Rule of 72 Works

A quantity grows at a rate of $x\%$ per period. Let t be the doubling time. Then $\left(1+x\%\right)^{t} =2.$

Take the natural logarithm (ln) and rearrange: $t\ln\left(1+x\%\right) =\ln2 \implies t =\frac{\ln2}{\ln\left(1+x\%\right)}.$

As you may know (Wikipedia), if $x\%$ is small, then $\ln(1+x\%)\approx x\%$. Hence: $t \approx\frac{\ln2}{x\%}.$

Now, $\ln2=0.6931\dots$. So: $t \approx\frac{0.6931\dots}{x\%} =\frac{69.31\dots}{x}.$

Altogether then, for a small growth rate $x\%$, the doubling time is approximately $69\div x$.

So it should really be the Rule of 69. Why do we change it to the Rule of 72 instead?

The reason is that 72 is a much “nicer” number, in the sense that it has plenty of factors (2, 3, 4, 6, 8, 9, 12, 18, 24, 36), hence greatly facilitating the mental arithmetic.

So boys and girls, now you know. When someone asks you where the mysterious number 72 comes from, you can tell her that 72 is simply a “nice” number that happens to be close to $100\ln2=69.31\dots$.

#### Super Secret Bonus Material!

The Rule of 72 also works “in reverse”. That is, if a quantity is shrinking by $x\%$ per time period, the halving time is roughly $72\div x$ periods. The proof is almost the same as before:

A quantity grows at a rate of $-x\%$ per period. (Equivalently, it shrinks at a rate of $x\%$ per period.) Let t be the halving time. Then $\left(1-x\%\right)^{t} =0.5.$

Take the natural logarithm (ln) and rearrange: $t\ln\left(1-x\%\right) =\ln0.5 \implies t =\frac{\ln0.5}{\ln\left(1-x\%\right)}.$

As you may know (Wikipedia), if $x\%$ is small, then $\ln(1-x\%)\approx-x\%$. Hence: $t \approx\frac{\ln0.5}{-x\%}.$

Moreover, $\ln0.5=-\ln2$. Altogether then: $t \approx\frac{-\ln2}{-x\%} =\frac{\ln2}{x\%} =\frac{0.6931\dots}{x\%} =\frac{69.31\dots}{x}.$

Altogether then, for a small shrinking rate $x\%$, the halving time is approximately $69\div x$.

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