Today, the two most populous countries in the World are PR China and India. The table below summarizes the essential data for our further analysis.
 Population  Population Growth Rate  Part of Total Population  Rel. Growth Quotient  World  6,790,062,216  1.133%  100.00%  1.00000  PR China  1,338,612,968  0.655%  19.714%  0.99527  India  1,156,897,766  1.407%  17.038%  1.00271 

The first two columns are actual data and the last two are calculated. The "Part of the total population" entry is simply calculated as the ratio of the population of that entry to the entry in the World population expressed in percent with 5 significant figures. We see that today China and India make up almost 37% of the total World Population.
The last entry needs some explanation. It has to do with the way we calculate compounding quantities. The population growth rate is the ratio of the increase of the population in the given year to the present population expressed in percent. Let's say the population in year 1 is
P_{1} and in the year after is
P_{2}. Then, the increase in population during that year is Δ
P =
P_{2} 
P_{1} and the growth rate is given as:
gr = ΔP/P_{1} * 100% = (P_{2}  P_{1})/P_{1} * 100%.
We can solve this equation for
P_{2} to get:
P_{2} = P_{1} * (1 + gr/100%).
The factor in the parenthesis occurs very often and I call it the
Growth Quotient q:
q = 1 + gr/100%.
For example, India has a population growth rate of 1.407%. This means it has a population growth quotient 1.01407. If the growth rate is negative, for example 0.601%, then the growth quotient is less than 1, in this case 1.00000  0.00601 = 0.99399. Because the growth rates were reported with three decimal places, the growth quotient has 5 decimal places, as reflected in the fourth column of the above table.
Now, the ratio of the population in the third year to the population in the second year is the same as the one in the second year is to the population in the first, i.e. it is equal to the growth quotient
q (assuming stays the same, hence cetris paribus):
So, we have:
P_{3} = P_{2} * q = P_{1} * q^{2}
P_{4} = P_{3} * q = P_{1} * q^{3}
...
P_{n} = P_{1} * q^{n  1}
So, what is the relative growth quotient? Well, if we want to answer the question how the ratio between two populations changes over the years, then we use the compounding formula for both of them. Then the ratio becomes:
r_{n} = P'_{n}/P_{n} = (P'_{1} * q'^{n  1})/(P_{1} * q^{n  1})
r_{n} = r_{1} * (q'/q)^{n  1}, r_{1} = P'_{1}/P_{1}
We see that the ratio of the growth quotients is essential in determining how the ratio of two compounding populations changes over time. So, the
Relative Growth Quotient q_{rel} is equal to:
q_{rel} = q/q_{TOT}
and it determines how the relative part of a certain population changes with respect to the whole. If the relative growth quotient is smaller than 1, then the relative part of that country's population in the World will decrease, like for China, for example, while in the other case it would increase, like for China.
We may ask ourselves, ceteris paribus, after how many years (
x =
n  1) will India's Population become equal to that of PR China. Using the compound growth formula, we get the equation:
1,156,897,766 * 1.01407^{x} = 1,338,612,968 * 1.00655^{x}.
With some algebraic manipulation, we can transform this equation in the form:
(1.01407/1.00655)^{x} = 1,338,612,968/1,156,897,766
This is the canonical form of an exponential equation, where the unknown is in the exponential. Taking the logarithm of both sides and using the properties of logarithms, we get:
x*log(1.01407/1.00655) = log(1,338,612,968/1,156,897,766)
x*[log(1.01407)  log(1.00655)] = log(1,338,612,968)  log(1,156,897,766)
x*(0.00607  0.00284) = 9.12666  9.06329
0.00323*x = 0.06337
x = 19.6
So, we expect their populations be equal in 20 years.
Another, more interesting question is to answer whether the relative part of these two nations combined increases or decreases. To answer that, we write the formula for the relative part of their population is a function of years passed:
r(x) = 0.19714*0.99527^{x} + 0.17038*1.00271^{x}
Because the relative part of PR China decreases, while the relative part of India increases, it is not straightforward to see if this will increase or decrease. Calculus provides the answer. Taking the derivative with respect to
x (and using (
q^{x})' =
q^{x}*ln
q), we get:
r'(x) = 0.19714*ln(0.99527)*0.99527^{x} + 0.17038*ln(1.00271)*1.00271^{x}
r'(x) = 9.3469*10^{4}*0.99527^{x} + 4.6111*10^{4}*1.00271^{x}
r'(0) = 9.3469*10^{4} + 4.6111*10^{4} = 4.7358*10^{4} < 0
This means that the relative part of these two nations will actually decrease. The trend will continue as long as
r'(
x) < 0. This is as long as:
9.3469*10^{4}*0.99527^{x} + 4.6111*10^{4}*1.00271^{x} < 0
4.6111*10^{4}*1.00271^{x} < 9.3469*10^{4}*0.99527^{x}
(1.00271/0.99527)^{x} < 9.3469/4.6111
x*[log(1.00271)  log(0.99527)] < log(9.3469)  log(4.6111)
x*(0.00118  (0.00206)) < 0.9707  0.6638
0.00324*x < 0.3069
x < 94.7
In the next 95 years!