Question

Out of the world population of approximately \[6.6\] billion,
\[1.2\] billion people live in the richer countries of Europe, North America, Japan and Oceania and is growing at the rate of \[0.25\% \] per year, while other \[5.4\] billion people live in the less developed countries and growing at a rate of \[1.5\% \]. What will be the world population in billions in \[5\] years if we assume that these rates of increase will stay constant for the next \[5\] years. (round answer to \[3\] significant digits)

Answer 1

Hint: Here we will use the property that if the population of the town or country is
\[x\] at present and it is increasing at a rate of \[r\% \] every year then the population of the town for the next year will be as below:

\[{\text{Population for next year}} = x + {\left( {x \times \dfrac{r}{{100}}} \right)^1}\] And if it continues for \[n\]years then the population of the town after \[n\] years will be equals to as below: \[{\text{Population after n year}} = x + {\left( {x \times \dfrac{r}{{100}}} \right)^n}\]

Step-By-Step answer:
Step 1: Total population of the world is given as \[6.6\] billion from which \[1.2\] billion people are richer and live in the countries of Europe, North America, Japan and Oceania and are growing at the rate of \[0.25\% \] per year.
So, after one year the population of the richer people will be equals to as below:
\[ \Rightarrow {{\text{P}}_{Rich}} = 1.2 + {\left( {1.2 \times \dfrac{{0.25}}{{100}}} \right)^1}\] , where
\[{{\text{P}}_{Rich}}\] represents the population of the richer people.
We can write the above expression as below by taking \[1.2\] common:
\[ \Rightarrow {{\text{P}}_{Rich}} = 1.2{\left( {1 + \dfrac{{0.25}}{{100}}} \right)^1}\]
After completing two years the population of the richer people will be equals to as below:
\[ \Rightarrow {{\text{P}}_{Rich}} = 1.2{\left( {1 + \dfrac{{0.25}}{{100}}} \right)^2}\]
This process continues till five years, so the population after five years will be equals to as below:
\[ \Rightarrow {{\text{P}}_{Rich}} = 1.2{\left( {1 + \dfrac{{0.25}}{{100}}} \right)^5}\] ………………………. (1)
Step 2: Similarly, we will repeat step number (1), for calculating the population for less rich people as shown below: So, after one year the population of the less rich people will be equals to as below:
\[ \Rightarrow {{\text{P}}_{LessRich}} = 5.4 + {\left( {5.4 \times \dfrac{{1.5}}{{100}}} \right)^1}\] , where
\[{{\text{P}}_{LessRich}}\] represents the population of the richer people.
We can write the above expression as below by taking \[5.4\] common:
\[ \Rightarrow {{\text{P}}_{LessRich}} = 5.4{\left( {1 + \dfrac{{1.5}}{{100}}} \right)^1}\]
After completing two years the population of the less rich people will be equals to as below:
\[ \Rightarrow {{\text{P}}_{LessRich}} = 5.4{\left( {1 + \dfrac{{1.5}}{{100}}} \right)^2}\]
This process continues till five years, so the population after five years will be equals to as below:
\[ \Rightarrow {{\text{P}}_{LessRich}} = 5.4{\left( {1 + \dfrac{{1.5}}{{100}}} \right)^5}\] ………………………. (2)
Step 3: The total population of the world will be equals to the addition of the richer people and less rich people as shown below:
\[ \Rightarrow {{\text{P}}_{total}}{\text{ = }}{{\text{P}}_{LessRich}} + {{\text{P}}_{Rich}}\]
By substituting the values from the expression (1) and (2) in the above equation, we get:
\[ \Rightarrow {{\text{P}}_{total}}{\text{ = }}5.4{\left( {1 + \dfrac{{1.5}}{{100}}} \right)^5} + 1.2{\left( {1 + \dfrac{{0.25}}{{100}}} \right)^5}\]
By opening inside the brackets of the above equation we get:
\[ \Rightarrow {{\text{P}}_{total}}{\text{ = }}5.4{\left( {\dfrac{{100 + 1.5}}{{100}}} \right)^5} + 1.2{\left( {\dfrac{{1 + 0.25}}{{100}}} \right)^5}\]
By opening the powers of the brackets and writing it into the multiplication form we get:
\[ \Rightarrow {{\text{P}}_{total}}{\text{ = }}5.4 \times \dfrac{{101.5}}{{100}} \times \dfrac{{101.5}}{{100}} \times \dfrac{{101.5}}{{100}} \times \dfrac{{101.5}}{{100}} \times \dfrac{{101.5}}{{100}} + 1.2 \times \dfrac{{1.25}}{{100}} \times \dfrac{{1.25}}{{100}} \times \dfrac{{1.25}}{{100}} \times \dfrac{{1.25}}{{100}} \times \dfrac{{1.25}}{{100}}\]

By simplifying the above equation, we get the answer as below:
\[ \Rightarrow {{\text{P}}_{total}}{\text{ = 7}}{\text{.03 billion}}\]

Note: Students generally make mistakes while solving the big equations, you should take care while doing the multiplication and addition in that part otherwise that will lead to the incorrect answer. Also, while calculating the population for the next few years you should remember that if the population of any town at present is \[x\] which is increasing at a rate of \[r\% \] for the next few years. Then the population for the second year will be equal to the addition of the previous population and the \[r\% \] of the previous population.