'On air live telly maths secrets: the rule of 76' blog discussion

Former_MSE_Penelope

This is the discussion to link on the back of Martin's blog. Please read the blog first, as this discussion follows it.

Read Martin's "On air live telly maths secrets: the rule of 76" Blog.

Please click reply to discuss below.

MrsBartolozzi

That's a useful rule of thumb, thanks Martin, and I see you put together a table of other "rules", but could you maybe add another column in the table with the actual times/growth increase for comparison?
This would show the accuracy of each rule and which is most suitable to use for different % values.

sjeapes

For the extra nerdy the Wikipedia article has the derivation of this rule and the approximations used (not for the maths averse)

I'm not allowed to post with links but just go to Wikipedia and search for Rule of 72

Regshoe

Sadly I did find Martin's blog interesting. The wikipedia page gives a good mathemtatical explanation of why these numbers work:
http://en.wikipedia.org/wiki/Rule_of_72


"The value 72 is a convenient choice of numerator, since it has many small divisors: 1, 2, 3, 4, 6, 8, 9, and 12. It provides a good approximation for annual compounding, and for compounding at typical rates (from 6% to 10%). The approximations are less accurate at higher interest rates.


For continuous compounding, 69 gives accurate results for any rate. This is because ln(2) is about 69.3%; see derivation below. Since daily compounding is close enough to continuous compounding, for most purposes 69, 69.3 or 70 are better than 72 for daily compounding. For lower annual rates than those above, 69.3 would also be more accurate than 72."


If you are trying to do it in your head then it looks like 72 is a pretty convenient number. That said if you are going to use the rule to try to surprise somebody about how fast their credit card debt or sub-prime loan will double then a higher number would be better (Maybe the Sub-Prime Rule of 76?)


I remember doing something similarly geeky when an accountant colleague (I'm also an accountant) mentioned the rule of 78s (http://en.wikipedia.org/wiki/Rule_of_78s). I told him the name didn't seem quite right and shouldn't it be the "Rule of n*(n+1)/2"? Oddly he just fixed me with an exasperated gaze and didn't respond.....

Edit: Beaten to it (twice!)

MrsBartolozzi - you can see the actual values on the table in my first link.

sjeapes - I assume that's the link you mean!

sjeapes

Regshoe wrote: »

sjeapes - I assume that's the link you mean!

Yes it it. Thank you


If you work out the values for the 'third order E-M approximation (the more accurate approximation given in the article you get the following

For the range given (1 to 50%) the approximation of %interest/3 works rather well as a way of helping decide which number to use

So the approximation would become:
(69 + %rate/3) / %rate


Can't get the table to work but the data is

% rate Rule of … Rule of %/3
1 69.65 69.33
3 70.33 70
5 71.02 70.67
10 72.71 72.33
15 74.37 74
20 76.01 75.67
25 77.62 77.33
30 79.2 79
50 85.29 85.67

sjb92

We were told the rule of 70 during A level economics - just as a way of quickly doing sums in our heads during exams. As we were generally using it to look at the impact of mid-1990s inflation, we didn't need to worry about the different rules for higher interest rates - or more likely it was being kept simple for us.

foreversomeday

Lol Martin - I love how you explain this as the simple way and it has still gone straight over my head :rotfl:

Despite being told I should have gone on to do A Level maths, my division skills have always been lacking.

camaj

There's a very cool book called Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks by Arthur Benjamin that is full of these tricks. I haven't read it but he was a guest on The Colbert Report and I was impressed. I've often relied on my own shortcuts when it comes to mental arithmetic. The same book was published as Think Like a Maths Genius: The Art of Calculating in Your Head for some reason
Amazon.co.uk

Penelope_Penguin

Just been musing on this - does it work for savings, too? IE, if I get a 5% interest on my savings, it'll take 76/5 years for my savings to double

Regshoe

Penelope_Penguin wrote: »

Just been musing on this - does it work for savings, too? IE, if I get a 5% interest on my savings, it'll take 76/5 years for my savings to double

Yes it's exactly the same, assuming you don't withdraw anything and the rate remains constant. You'll find that for savings you probably want to use 70 or 72 though, as the rates are likely to be fairly low.

Also, although you could double your money in 14 and a bit years at 5%, it wouldn't be "worth" twice as much due to inflation.