RULE OF 72

RULE OF 72

INTRODUCTION:

THE RULE OF 72 IS A MENTAL TECHNIQUE TO CALCULATE THE NUMBER OF YEARS IN WHICH A SUM OF MONEY  , WHICH BEING COMPOUNDED ON ITSELF , WILL BECOME TWICE THE MONEY INVESTED.

NOTE :

IT WILL ONLY GIVE AN APPROXIMATE RESULT AND NO DEFINITE  VALUE.

THE RULE:

IF A SUM OF MONEY  IS INVESTED IS INVESTED AT A RATE OF r % p.a. THEN THE NUMBER OF YEARS IT WILL TAKE TO DOUBLE ITSELF IS :

n = 72/r

DERIVATION:

WE KNOW,

A=P(1+r)ⁿ

 …………………eq(1)

WHERE  --

A  = FINAL  AMOUNT

P = PRINCIPAL INVESTED

r  = RATE OF INTEREST(EXPRESSED AS A DECIMAL)

n  = TIME TAKEN TO DOUBLE THE PRINCIPAL INVESTED

NOW,  A = 2P

PUTTING THIS VALUE IN eq(1) WE GET:

2P=P(1+r)ⁿ

Or,

2=(1+r)ⁿ

NOW TAKING log (to the base 10) ON BOTH SIDES WE GET:

log 2 = log(1+r)ⁿ

AS, log a^b  = b log a

THEREFORE :

       log 2 = n log(1+r)

AT THIS POINT WE HAVE TO NOTE TWO THINGS:

1.       log 2 = 0.69314718055994530941723212145818

2.       WHEN  r IS VERY SMALL , THEN log(1+r) ≈ r . IN FACT WHEN r IS SMALL ENOUGH THE ERROR PERCENTAGE IS AROUND  5%.

THUS WITH THESE TWO IN HAND THE ABOVE EQUATION CAN BE WRITTEN AS :

0.6931  ≈ n x r

Or,

n = 0.6931/r

NOW  IF r IS EXPRESSED AS A PERCENTAGE ,THEN :

n=0.6931/(r/100)

Or,

n=69.31/r

AS YOU CAN SEE 69.31 IS NOT A GOOD NUMBER TO LOOK AT SO WE CHANGE THE NUMERATOR TO 72 AS:

1.       69.31  ≈ 70 ≈72

2.       72 HAS MANY FACTORS(1,2,3,4,6,8,9,12,18,24,36,72) AS COMPARED TO 70 OR 69.31 SO IT COMES HANDY WHILE MENTAL CALCULATION.

THUS THE ABOVE EQUATION BOILS DOWN TO :

n=72/r

                      WHICH IS THE REQUIRED RESULT.

-LAMBDA