Up Rules Of Population Rule Of 70 and Rule Of 72 Compared The Scales Of e

Rules Of Population

Introduction

Much more obscure than the Rule Of 70 are the Rule Of 110 (for trebling and thirding) and the Rule Of 140 (for quadrupling and quartering). Other rules exist for other factors, but these two will be sufficient to confirm some natural patterns in nature.

Rule Of 110

Taking a growth rate of r, the accurate value X is calculated as:

X = Ln 3 / Ln (1 + (r / 100))

Taking a growth rate of r, the accurate value Y is calculated as:

Y = Ln 3 / Ln (1 - (r / 100))

Curiously enough this difference seems to equate to the Natural Logarithm of 3 (Ln 3 = 1.098612), which is where we get the Rule Of 110 in the first place.  This confirms a similar finding for the Rule Of 70.

Ln 3 = X + Y

This can be written:

X = ABS( Y) + Ln 3

 Trebling Period Thirding Period X + Y Growth Rate Rule Of 110 Accurate Value (X) Rule Of 110 Accurate Value (Y) 1 110 110.409624 110 -109.3110026 1.098621 2 55 55.47810764 55 -54.37945872 1.098649 3 36.66666667 37.16700967 36.66666667 -36.06831495 1.098695 4 27.5 28.01102276 27.5 -26.91226388 1.098759 5 22 22.51708531 22 -21.41824388 1.098841 6 18.33333333 18.85417668 18.33333333 -17.75523427 1.098942 7 15.71428571 16.23757367 15.71428571 -15.13851178 1.099062 8 13.75 14.27491459 13.75 -13.17571468 1.0992 9 12.22222222 12.74822067 12.22222222 -11.6488641 1.099357 10 11 11.52670461 11 -10.42717266 1.099532

Rule Of 140

Taking a growth rate of r, the accurate value X is calculated as:

X = Ln 4 / Ln (1 + (r / 100))

Taking a growth rate of r, the accurate value Y is calculated as:

Y = Ln 4 / Ln (1 - (r / 100))

Curiously enough this difference seems to equate to the Natural Logarithm of 4 (Ln 4 = 1.386294), which is where we get the Rule Of 140 in the first place.  Again, a natural pattern is confirmed.

Ln 4 = X + Y

This can be written:

X = ABS( Y) + Ln 4

 Growth Rate Quadrupling Period Quartering Period X + Y Rule Of 140 Accurate Value (X) Rule Of 140 Accurate Value (Y) 1 140 139.3214338 140 -137.9351279 1.386306 2 70 70.00557756 70 -68.61923698 1.386341 3 46.66666667 46.8995445 46.66666667 -45.51314613 1.386398 4 35 35.34597537 35 -33.95949604 1.386479 5 28 28.41339817 28 -27.02681467 1.386583 6 23.33333333 23.79132209 23.33333333 -22.40461117 1.386711 7 20 20.4895367 20 -19.10267502 1.386862 8 17.5 18.01293668 17.5 -16.62590083 1.387036 9 15.55555556 16.08646345 15.55555556 -14.69922992 1.387234 10 14 14.54508179 14 -13.15762696 1.387455

Conclusion

Although not as popular as the Rule Of 70, it was worth examining the Rule Of 110 and the Rule Of 140 to confirm the suspicion that the Natural Logarithm of a factor (in this case 3 and 4) appears to be roughly the difference between the time it takes a population to grow by that factor ( multiply by 3 or multiply by 4) and the time it takes to shrink by that factor (divide by 3 or divide by 4).

This brief investigation also confirm that shrinkage time is always the smaller of the two (the time to multiply and the time to divide).