Introduction
We, all live in a society by forming communities. Many people reside in such communities/societies. The number of people keeps on changing. There occurs many events such as birth and in-migration that increase the population. While in other times, the death of a person or out-migration causes the population to decrease. The marriage of a person can either increase or decrease the population depending upon whether the person stays or leaves the place.
Key Terminologies:
Population: Population is the total number of people residing in a particular place over a particular period of time. Population is a variable factor that changes its size over time. In some countries, the population might be growing. While, in other countries, the population might be constant or decreasing, over time.
Factors affecting the size of the population: The size of the population of a place depends on a number of factors, such as birth, death, and migration.
Population Growth: It is the increase in the number of individuals residing in a particular place over a particular period of time.
Formulae for Population Growth
Let us consider that we have a certain initial population P at the start of a certain year. Let the population be growing at a constant rate of R% interest per annum. The population is compounded in a regular interval of T years. In the following derivation section, we shall write R/100 as R%. Then, we have,
For First Year
Initial Population = Rs. P
Simple Population Growth = PTR/100 = PTR%
Final Population = P + PTR% = P (1 + TR%) ---- (1)
For Second Year
Initial Population = Rs. P (1 + TR%)
Simple Population Growth = {P (1 + TR%) } T R%
Final Population = P ( 1 + TR%) + { P (1 + TR%) } TR%
= {P (1 + TR%)}(1 + TR%)
= P (1 + TR%)2 ---- (2)
For Third Year
Initial Population = Rs. P (1 + TR%)2
Simple Population Growth = {P (1 + TR%)2 } T R%
Final Population = P ( 1 + TR%)2 + { P (1 + TR%)2 } TR%
= {P (1 + TR%)2}(1 + TR%)
= P (1 + TR%)3 ---- (3)
What we can analyze from the result of cases (1), (2), and (3) are the following:
- The size of population P each year is dependent upon the initial size of population P.
- The size of population P each year is dependent upon the nth power of the quantity (1 + TR%) where n is the number of years.
Thus, we can generalize a formula for the Final Population at a fixed rate of growth and compound yearly as:
$$\text{Pt = P}\left [1 \ + \ \dfrac{\text{R}}{100} \right ]^\text{T}$$
Similarly, we can deduce the formula for Population Growth at a fixed rate of growth and compounded yearly as:
$$\text{Pg = P}\left [1 \ + \ \dfrac{\text{R}}{100} \right ]^\text{T} \ - \ \text{P}$$
$$\text{Pg = P}\left [ \left ( 1 \ + \ \dfrac{\text{R}}{100} \right )^\text{T} \ - \ 1 \right ]$$
where Pt represents the final population after T years and Pg represents the Population growth in the duration of T years.
Other Important Formulae
Let, in each case, the final population after T years be Pt and the increased population after T years be Pg.
1. When the population grows annually but the rate is different in different years:
$$\text{Pt = P}\left (1 \ + \ \dfrac{\text{R}_1}{100} \right )\left (1 \ + \ \dfrac{\text{R}_2}{100} \right )\left (1 \ + \ \dfrac{\text{R}_3}{100} \right ) \ .... \ \left (1 \ + \ \dfrac{\text{R}_n}{100} \right )$$
$$\text{Pg = P}\left [ \left (1 \ + \ \dfrac{\text{R}_1}{100} \right )\left (1 \ + \ \dfrac{\text{R}_2}{100} \right )\left (1 \ + \ \dfrac{\text{R}_3}{100} \right ) \ .... \ \left (1 \ + \ \dfrac{\text{R}_n}{100} \right ) \ - \ 1 \right ]$$
where R1, R2, R3, ... , Rn are the rate of population growth at Years T1, T2, T3, ...., Tn.
2. When the population grows annually but the time is given in 'T' years and 'M' months:
$$\text{Pt = P}\left [1 \ + \ \dfrac{\text{R}}{100} \right ]^\text{T}\left [ 1 + \dfrac{\text{MR}}{1200} \right ]$$
$$\text{Pg = P}\left [ \left ( 1 \ + \ \dfrac{\text{R}}{100} \right )^\text{T}\left ( 1 + \dfrac{\text{MR}}{1200} \right ) \ - \ 1 \right ]$$