The Rule of 70

One of our new projects at MRUniversity will be to produce a video Dictionary of Economics. The first video in that series, The Rule of 70 is below.

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Excellent video, you made me grab a piece of paper to find out why "70". So, it's just ln(2)=0.693...... and using interest rate in % units.

The best educational material is not the one that answers a question but the one that makes you look for an answer. Bravo.

It's more memorable as 72. "The rule of 70" sounds like a description of the oligarchy in a city in Ancient Greece. Somehow 72 avoids that fate, and also has the advantage of being divisible easily by more integers.

That's true, but as Axa says, shouldn't it be more accurate as the Rule of 69? More sexy and connotes the Bryan Adams song, as well as Woodstock.

69 is surely too sexy, and will set adolescents giggling in an unseemly way.

Or a more elitist version of America's The 700 Club.

I also hear "rule" and "72" as somewhat rhyming.

I learned it as 72 because it neatly divides by 2, 3, 4, 6, 8, and 9. Since 7 and 10 are easy to compute and round to what the rule of 70 would give anyway...

69 is the worst in this regard... 3 and 23...

Yeah, I use 72 to get a first quick estimate. If I want a little more accuracy I use 69, but usually if I need more accuracy then at that point I'll switch to a calculator or spreadsheet.

The math "looks like" y * exp(0.05*t) = 2*y; solve for t ??? I don't think so. The math looks like exp(RT) = 2 or RT = ln(2) or T = ln(2)/R where T is time and R is interest rate. Is it really that much more difficult to whip out your phone and divide the log of 2 by R than to whip out your phone and do it right? Is the rule of 70 about as useful as a slide rule in your pocket? (The nice thing about T = ln(2)/R is that it easily generalizes to T = ln(M)/R where M is the multiplier...but perhaps since this requires understanding high school math, accountants and most economists aren't comfortable with know, social science graduates.

sorry I meant: divide 70 by R compared to divide ln(2) by R. I argue that the actual difference is minimal so why not do it right? Must be because they don't remember or never did understand the underlying math.

70/x is a mental calculation. Neither a smartphone or a piece of paper are needed..........just trained neurons, that's the beauty of the simple things.

Correct, no calculator or phone is as fast as a human who can do simple arithmetic in their head -- and the rule of 70 (or 72) permits us to do compound interest calculations in our heads, many times faster than the calculations can be done with a calculator. Fast enough to use in conversation, if one's brain can multi-task, i.e. do the arithmetic while also listening or even talking.

If you want to be exact it's ln(2)/ln(1+R).

I'm surprised that shortcuts like this are still used.

In that long-ago time before electronic calculators, such shortcuts were handy but today with ubiquitous calculating devices it seems easy enough to calculate doubling time if you know the interest rate and how often it is compounded. Even if you don't want to deal with logarithms and exponents, there's always Excel's formulas.

Interesting. I learned this trick as a child and have used it many, many times sense. (Then again, economics/finance gives many cause to use it). But I think tricks like this are part of learning basic numeracy. So many our students have so little numeracy that the calculator can spit out a ridiculous answer and the students will follow it blindly. So many students go through high school just smashing buttons, that teachers are often trying to get the calculators out of the students hands and encouraging them to *think* about the problem. I hope tricks like this continue to be thought.


Yes, aside from speed the other advantage of these mental calculations is exactly what you describe: the ability to have an estimate of what the answer is, independent of the calculations, programming, software, and all the bugs and typos involved with calculating the answer with high accuracy.

When the computer spits out the results of its calculations and the answer is "19", how do we know if that answer is correct? If we have the time and resources we can exhaustively test the algorithm and the coding and the data entry. Faster and easier is knowing that if that answer is supposed to be say the trillions of dollars of US debt then we're in the right ballpark; if it's the annualized percent rate of return on an investment that doubled in valued between 2010 and 2017 then we instantly know something is very wrong with the calculations.

Mathematicians are notorious for often being rather poor at doing arithmetic, because what they're doing is totally different from arithmetic. But people who work with quantitative data should ideally be able to do calculations and estimations in their head, quickly. Because it so greatly speeds up the detection of errors when I can look at a result and within 3 seconds say "that can't be right". People who are dependent on calculators and spreadsheets to give them the answer can't do that.

If you're teaching on a whiteboard and you want to demonstrate to students that a seemingly small change going from a growth rate of 2% per year to 3% per year over a normal lifespan of, say, 72 (not 70!) years means the difference between GDP growing by 8 times instead of 4 times the last thing you want is for students to be looking at their phones to divide logs of 2 by R.

Rule of 72 is better than Rule of 70 [over a larger span of interest rates]

Rule of K is better than Rule of 72, for the same reason:

I learned it as rule of 72, but that was in a higher interest rate environment.

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