Question

The population of a town increases \[5%\] annually. If it is \[15,435\] now, what was it \[2\] years ago?
(A) \[14,000\]
(B) \[13,473\]
(C) \[12,345\]
(D) \[10,145\]

Answer 1

Hint: In this question, we have to find the population which was two years ago. The population after two years is given to us and also the increment in the population annually and with the help of this, we have to find the population two years ago. First, we will find the population which was one year ago and with the help of this, we will find the population which was two years ago.

Complete step-by-step solution:
The population increases annually every year by some percentage so the resulting population will be the sum of the actual population of the town and the increase in the percentage of the actual population.
The concept of population is very important in mathematics as it comes in various competitive examinations. Annually means that the percentage is increasing every year. Suppose it is given that the population increases by \[5\%\] in the town so it means that the \[5\%\] of the total population increases in the town.
In the above question, it is given that the population of a town increases \[5\%\] annually and after two years it becomes \[15,435\]. Here in this question, we have to find the population two years ago.
Let the population after two years be ‘P’.
Then, \[P=15,435\]
Let the population that is one year ago be,
\[\text{Population one year ago} ={{p}_{1}} \]
And since the population of a town increases \[5\%\] annually, therefore, the population after one year will be as follows.
\[\begin{align}
  & {{p}_{1}}+\dfrac{5{{p}_{1}}}{100}=p \\
 & \Rightarrow \dfrac{105{{p}_{1}}}{100}=15435 \\
\end{align}\]
After solving the above equation we will get the population which was one year ago.
\[\begin{align}
  & {{p}_{1}}=\dfrac{15435\times 100}{105} \\
 & {{p}_{1}}=14700 \\
\end{align}\]
So the population one year ago will be \[{{p}_{1}}=14700\].
Now the original population will be \[{{p}_{1}}\] and here we will find the population two years ago and we also know that the population increases \[5\%\] annually. So the population two years ago will be given as shown below.
Let the population two years ago be ‘\[{{p}_{2}}\]’.
\[\begin{align}
  & {{p}_{2}}+\dfrac{5{{p}_{2}}}{100}={{p}_{1}} \\
 & \Rightarrow \dfrac{105{{p}_{2}}}{100}=14700 \\
 & \Rightarrow {{p}_{2}}=\dfrac{14700\times 100}{105} \\
\end{align}\]
After solving this we get the population which was two years ago.
So the population two years ago was, \[{{p}_{2}}=14000\]
So the correct answer will be option(A) \[14,000\].

Note: The population of India is increasing day by day and hence it leads to various problems like poverty, increment in birth rate, child labor, lack of resources, etc. The families in which there are more than five members result in poverty and hence our country is not a developed country.