Some of you might have heard of "Rule of 72" which simply states that money invested in an account bearing interest r, will double its principal in 72/r years when compounded annually. If you invest a $1 at 3% interest compounded annually, you will end up with $2 in approximately 72/3=24 years.
So , what is Rule of 114? Well, what if we want to know the time it takes to triple our initial principal given an interest rate, r. Just as you guess it, to triple our initial investment at an interest rate of 3% compounded annually, it will take approximately 114/3= 38 years. Lets see how we obtain the number 114 using the same reasoning for getting 72 in the rule of 72.
The amount of money we end up with, F, is given by this compounding formula, F=P(1+r/100)^n where P =initial principal, r= interest, n=years/period
To triple our initial investment P, we can algebraically represent F=3P ( tripling). By substitution,
Then take the log on both sides,
log 3= n log ( 1+r/100)
n= log 3/(log(1+r/100)
If r= 1, n=110.40962 and n*r=110.4962
If r= 3, n= 37.167009 and n*r=111.50127
if r=5, n=22.517085 and n*r=112.58542
if r=7, n=16.23753 and n*r=113.66271
if r=11, n=10.527138 and n*r=115.79851
if r=13, n=8.988983 and n*r=116.85678
if r=15, n=7.860596 and n*r=117.90894
Sum of ( nr)= 798.81073 Average ( Sum of nr)= 798.81073/7= 114.11
The average mean came out very close to 114 and a very close estimate of tripling one's money at an interest rate, r for n periods. You can also work out the rule for quadrupling or k-tupling for that matter.