Editing Rule of 72 (section)

== Choice of rule ==
The value 72 is a convenient choice of numerator, since it has many small [[divisor]]s: 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, since  [[natural logarithm|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.<ref>Kalid Azad [http://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/ Demystifying the Natural Logarithm (ln)] from BetterExplained</ref> For higher annual rates, 78 is more accurate.

{| class="wikitable sortable" style="text-align:right;"
! Rate
! Actual Years
! Rate &times; Actual Years
! Rule of 72
! Rule of 70
! Rule of 69.3
|-
| 0.25 % || 277.605 || 69.401 || 288.000 || 280.000 || '''277.200''' 
|-
| 0.5 % || 138.976 || 69.488 || 144.000 || 140.000 || '''138.600''' 
|-
| 1 % || 69.661 || 69.661 || 72.000 || '''70.000'''|| 69.300 
|-
| 2 % || 35.003 || 70.006 || 36.000 || '''35.000'''|| 34.650 
|-
| 3 % || 23.450 || 70.349 || 24.000 || '''23.333'''|| 23.100 
|-
| 4 % || 17.673 || 70.692 || 18.000 || '''17.500'''|| 17.325 
|-
| 5 % || 14.207 || 71.033 || '''14.400'''|| 14.000 || 13.860 
|-
| 6 % || 11.896 || 71.374 || '''12.000'''|| 11.667 || 11.550 
|-
| 7 % || 10.245 || 71.713 || '''10.286'''|| 10.000 || 9.900 
|-
| 8 % || 9.006 || 72.052 || '''9.000'''|| 8.750 || 8.663 
|-
| 9 % || 8.043 || 72.389 || '''8.000'''|| 7.778 || 7.700 
|-
| 10 % || 7.273 || 72.725 || '''7.200'''|| 7.000 || 6.930 
|-
| 11 % || 6.642 || 73.061 || '''6.545'''|| 6.364 || 6.300 
|-
| 12 % || 6.116 || 73.395 || '''6.000'''|| 5.833 || 5.775 
|-
| 15 % || 4.959 || 74.392 || '''4.800'''|| 4.667 || 4.620 
|-
| 18 % || 4.188 || 75.381 || '''4.000'''|| 3.889 || 3.850 
|-
| 20 % || 3.802 || 76.036 || '''3.600'''|| 3.500 || 3.465 
|-
| 25 % || 3.106 || 77.657 || '''2.880'''|| 2.800 || 2.772 
|-
| 30 % || 2.642 || 79.258 || '''2.400'''|| 2.333 || 2.310 
|-
| 40 % || 2.060 || 82.402 || '''1.800'''|| 1.750 || 1.733 
|-
| 50 % || 1.710 || 85.476 || '''1.440'''|| 1.400 || 1.386 
|}

Note: The most accurate value on each row is in bold.

[[file:doubling_time_vs_half_life.svg|frame|none|Graphs comparing doubling times and half lives of exponential growths (bold lines) and decay (faint lines), and their 70/''t'' and 72/''t'' approximations. In the [http://upload.wikimedia.org/wikipedia/commons/8/88/Doubling_time_vs_half_life.svg SVG version], hover over a graph to highlight it and its complement.]]