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DOUBLING TIME AND HALF LIFE

 


  In any artificial and natural growth at some rate, the quantity will double at regular intervals of time- doubling time.  In any decline or decay process at a fixed rate, in the quantity will halve itself at constant intervals of time-half life.

     For example: at a particular rate of interest, the amount of money will double in constant period of time.There is a very simple formula to calculate doubling time or half life.

 Doubling time or half life = natural log [2] / rate of decay  or growth.
simply
                       t = 0.6931/r
Note:  Rate should be expressed in decimal numbers.
example: 5% = 5/100 = 0.05
the answer will be in the same unit as r

 considering real life problems:
1. If the inflation rate is 6% per annum, when the prices will double?
   t = 0.6931/0.06 = 11.5 years
Every 11.5 years, the prices will be double.

 2. If a financial institution offers 14% per year as a rate of interest[compound interest] on your money, when it will double.
   t = 0.6931/0.14 = 4.95 years
 In nearly 5 years, it will double or every 5 years the money will double as given in the following table.

 YEARS              MONEY
0                         1000
5                         2000
10                       4000
15                       8000
20                       16000
AND SO ON
one day you will become millionaire

 3. A capacitor discharges at the rate of 15%  per second.  The initial current is 1 ampere or 1000 milliampere.  When the current will nearly vanish?

 half-life = 0.6931/0.15 = 4.6 seconds

TIME                CURRENT                                  
0 second            1000 Milli amperes
4.6                      500
9.2                      250
13.8                    125
18.4                    62.5
23                       31.25
27.6                    15.625
32.2                    7.81
.                          .
.                          .
.                          .

 The current becomes negligible after 30 seconds. but it will take infinity of time to become zero.
This simple and great formula solves many real life problems and amazes.

                                                     ----------------------------------

For the serious readers,  

 we know , formula for growth and decay is
  N = Ni*e^rt
 e = 2.71.... = constant
 r = rate of growth
        +ve for growth
        -ve for decay
Ni = initial quantity
N = quantity after time t 
for doubling
Ni = 2Ni
hence
2Ni = Ni*e^rt
2 = e^rt
Taking log to the base e on both sides
ln 2 = ln e^rt
ln 2 = rt*ln e
ln 2 = rt*1
since ln e to the base e is 1
rt = ln 2
t = ln2/r
t = 0.6931/r

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