Beauty of Mathematics

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry-- Bertrand Russel.
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Sunday, 3 November 2013

Rule of 72



                                  Double your money!
     
                                       
A student aspiring to appear in various competitive examinations is frequently required to calculate the no. of years in which a sum is doubled under certain rate of Compound Interest when the compounding is done annually. Why only students, even Managers and Investors are often required to calculate the no. of years in which a sum will double itself at a particular rate of interest.

There are generally two ways to calculate it: 

(1)  By Compound Interest Formula, or
(2)  By using scientific calculators or some spreadsheet programmes that have functions to find the accurate doubling time. 

Students are required to calculate it by using a formula while professional can resort to the scientific calculators or some specialized software. The formulae for calculating the CI is given below:

·         A  =  P(1+r/100)n

(Where A is the Amount, P is the Principal or Sum, r is the rate of interest and n is the number of years.)

·         Compound Interest = Amount – Principal.

As the formula for calculation of Compound Interest is not that easy for mental calculation purposes; if an alternative simple formula is provided it can rule out use of both the above options (1) and (2).

Well, indeed there is a simple approximate formula for the above calculation which can be and is used by informed bankers, Investors, Project Engineers, maths teachers, and students alike.
The simple formula is:                         
                                                                       nr =72
This is also known as Rule of 72.

Where n is the no. of years in which a sum will double itself and r is the rate of Compound Interest under which the sum will double itself. The formula is valid when the compounding is done once in a year and the rate of Interest is between 1% to 20%.

(It will be worthwhile to note that in real life, rate of interest is seldom charged beyond this limit of 1% to 20 %.)

Now, the question is how to use this formula. 

Example:  In how many years will a sum of Rs. 5000 double itself at a Compound Interest of?
(a)  8%
(b)  9%
(c)  12%

Solution: Here we can apply the simple formula nr=72 for approximate calculations. In the first case r is given as equal to 8, hence nx8= 72 and n=9.Hence the sum will double itself in 9 years in the first case. Similarly n=8 years in the second case and n=6 years in the third case.
And if we use the traditional Compound Interest Formula our results will be as follows:
(a)  9.006 years
(b)  8.04 years
(c)  12.09 years

Having seen the use of formula, let us now see the logic or derivation of this Rule of 72.

The original formula for Compound Interest is as below:

                           A  =  P(1+r/100)n

Where A is the Amount, P is the Principal or Sum, r is the rate of interest and n is the number of years.

If a sum is doubled then A=2P. Putting the value of A in terms of P, we get
 2P= P (1+r/100)n
2= (1+r/100)n

Taking the natural logarithm of both the sides, we get

loge 2 = n loge (1+r/100); and
n= (loge 2)/ { loge(1+r/100)}
From natural logarithm table loge 2 =0 .693.
n= 0.693/ { loge(1+r/100)}
Expanding the logarithmic series loge(1+r/100), we get
loge(1+r/100)= (r/100) – (1/2)(r/100)2 + (1/3)(r/100)3 -…                                                (For -100 < r ≤100)
Now we can write
n= 0.693/{(r/100) – (1/2)(r/100)2 + (1/3)(r/100)3 -…}

If we take small value of r from 1 to 20, we can conclude from the above that
Approximately n=72/r or nr=72.

There is another way to derive it without using the logarithmic series.

 2P= P (1+r/100)n
2= (1+r/100)n
n= (log 2)/ {log (1+r/100)}

Let us make a table of values with the help of a scientific calculator by putting different values of r in the above formula and then by calculating corresponding values of n and nr…

Rate of Interest ( r)
No. of Years (n)
nr
1
69.66071689 
69.66071689 
3
23.44977225
70.34931675
5
14.20669908
71.03349541
7
10.24476835
71.71337846
9
8.043231727
72.38908554
11
6.641884618
73.0607308
13
5.671417169
73.72842319
15
4.959484455
74.39226682

If we take the average of the nr values, we get 72.04092314, which is quite close to 72, and so our Rule of 72 seems to be a very close estimate for doubling money at an annual interest rate of r% for n number of years.


If we calculate the no of years with actual Compound Interest Formula and by the Rule of 72, the minute difference in both the approaches will be as per the chart given below:

Rate of interest %
Years as calculated by the Rule of 72
Actual Years as calculated by the Compound InterestI Formula
3%
24
23.45
4%
18
17.673
5%
14.4
14.21
6%
12
11.896
7%
10.3
10.24
8%
9
9.006
9%
8
8.04
10%
7.2
7.273


From the above table, it is very clear that if we practically use the Rule of 72; a lot of calculating efforts can be reduced. And that we can calculate the doubling amount period mentally without using the calculator!

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