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I removed the section on rules for tripling and quadrupling, due to inaccuracy (why not use ln(3) = 1.10?) and lack of source. If there is a good source for the additional rules, somebody should feel free to add it back with a citation, but it needs to note that this generalization does not extend to larger numbers (since it looks like we're scaling linearly, and that's highly misleading.) Rxtreme (talk) 02:35, 24 April 2015 (UTC)[reply]

Adjustment for higher accuracy: I am proposing a formula for higher accuracy. Please guide me as to how to add results of this formula along side the charts with various methods like Rule of 72, Rule of 70, Rule of 69.3 etc. — Preceding unsigned comment added by Ugaap (talk • contribs) 03:07, 20 May 2022 (UTC)[reply]

The compounding table doesn't say how the interest is compounded. Is it continuous? 205.154.230.3 01:16, 19 October 2006 (UTC)[reply]

This page does not explain how or why the rule of 72 works. I hear about it, but can not find an explanation of it. Why is the number 72 ? thank you. 208.240.243.170 00:21, 19 May 2007 (UTC)[reply]

Rule_of_72#Derivation

Fintor kindly provided a pointer to the formulae which yield the solution 69.3~. This is pretty close to 72. The reason 72 is used as a close-enough rule-of-thumb most of the time is that it has many small divisors: 1, 2, 3, 4, 6, 8, 9, and 12. When the interest rate you're considering is one of those numbers, one can do the math in one's head. Rule of 70 works well for interest rates of 5%, 7% and 10%. 76.254.85.140 (talk) 19:49, 3 May 2008 (UTC)[reply]

The statement in the article, "For continuous compounding, 69 gives accurate results for any rate, This is because ln(2) is about 69.3%" is not correct. For continuously compounding interest, it becomes increasingly inaccurate as the interest rate increases. Paralucent (talk) 14:15, 15 October 2008 (UTC)[reply]

Does anyone have any links to spreadsheets, etc that give more accurate answers? [[Special:Contributions/] (talk) 00:01, 30 June 2009 (UTC)[reply]

I believe that this column would be a good one to add to the table. The existing table says what the individual rules provide, but the following column tells what the "perfect" numerator would be. I'd add it myself, but I don't like to mess with the politics of other pages.

I notice that some of the formulas use the rate expressed as a percentage while others us it as a decimal so it is unclear to the reader whether to plug in '8' or '.08' for any given formula. In the exact formulas, especially when using logs, it is standard to use the decimal form so I didn't want to change these - and I certainly didn't want to change the topics title formula to be "The Rule of 0.72"! Prefacing each formula to indicate whether the rate is a decimal or percent seemed too clumsy, so I left everything as is. — Preceding unsigned comment added by 206.130.179.244 (talk) 22:10, 11 June 2013 (UTC)[reply]

The phrase "see derivation below" doesn't actually refer a derivation. In addition, the link that looks like it might be relevant is dead. The derivation is quite simple... The formula for exponential growth is p*exp(r*t/100) where p is the principal, r is the rate, t is the time period, and exp() is exponentiation to the natural base (2.718...). The 100 in the argument is to adjust for r being a percentage rather than a fraction. The units of r and t must match. That is, if t is in years, then r must be percent growth per year. Now, we want doubling time. In a formula that's p*exp(r*t/100) = 2*p. On both sides divide by p and take the natural logarithm. That gives r*t/100 = ln(2). Solve for t: t = 100*ln(2)/r. Since 100 ln(2) is approximately 69.3, we are done. That's the rule of 69. Contrary to an earlier comment, it is accurate for any continuous interest rate, no matter how large. It only becomes inaccurate when interest payments are discrete.

It also becomes inaccurate as you truncate the actual calculated value. It's actually closer to 69.31471805599453, but who wants to use that many digits? For purposes of estimation, 72, 70, and 69 work well. Choose one that's easily divisible by whatever you're trying to divide by (i.e. either the interest rate or the number of periods). Using those 3 values (69, 70, 72), you can easily divide by every integer from 1 to 10, and 12, 14, 3.45, 3.5, 3.6, 6.9, 7.2, (also note 11 x 6.5 is pretty close, as is 13 x 5.5). Remember, folks, it's just an estimate. If you want the actual value, use your calculator or a spreadsheet. — Preceding unsigned comment added by 98.28.156.29 (talk) 11:15, 30 August 2011 (UTC)[reply]

″it is accurate for any continuous interest rate, no matter how large″ - What is a continuous interest rate, is it calculated by the second or the nanosecond?Optymystic (talk) 10:18, 12 January 2016 (UTC)[reply]

″He presents the rule in a discussion regarding the estimation of the doubling time of an investment, but does not derive or explain the rule, and it is thus assumed that the rule predates Pacioli by some time.″ The absence of a derivation or explanation may equally imply that he had no derivation because the rule was only recent discovered, so the final clause should be deleted.Optymystic (talk) 10:12, 12 January 2016 (UTC)[reply]

This page seems to be a very convoluted way of discussing the fact that the solution of the equation is . The in this context seems to be the result of several approximations, one stacked upon the other. The same seems to hold for the approximation . None of this has anything to do with investment; this page discusses simple, universally valid mathematics. Who, when almost everybody has a calculator, needs all these tables and graphs? There must be a simpler way to do explain all of this. In fact, I just did. Maybe there are historical reasons for the presence of all these complications. Maybe it's time to separate mathematics from history? Maybe all of these complications are a result from the fact that one can accumulate interest on a continuous basis, as I assumed. Or one can do so on a daily, weekly, monthly, whatever you like basis. There is no end to these kinds of variations, which financial institutions regularly exploit to their advantage by not being transparent.

The so-called "Rule of 72" is an approximation, used for estimates only. Of course, if you need the exact figure then use a computer or calculator. But there are people who find it useful to be able to make a quick mental assessment of how long it would take to double at a certain compounding rate. Or vice-versa, what rate is required in order to double in a target time. If you are not one of them, don't use it.--Gronk Oz (talk) 22:33, 10 October 2023 (UTC)[reply]