Question

The present population of a town is 35,000. If the population becomes 35,700 next year, the rate of growth is
A.\[2\% \]
B.\[20\% \]
C.\[70\% \]
D.\[7\% \]

Answer 1

Hint: Here we will use the basic formula of the population increment. We will substitute the given values in the formula and simplify it further to get the value of the rate of growth of the population. The rate of growth of population means the increase in percentage in the population during a certain period of time.

Formula used:
Population in \[t\] years is \[{P_t} = {P_0}{\left( {1 + \dfrac{R}{{100}}} \right)^t}\] where, \[{P_0}\] is the initial population and \[R\] is the rate of growth of population

Complete step-by-step answer:
Given present population is 35,000 i.e. \[{P_0} = 35000\].
It is given that if the population becomes 35,700 next year i.e. \[{P_1} = 35700\].
We know that the increase in the population is a compound concept i.e. it follows the compound increasing pattern.
Now we will use the formula \[{P_t} = {P_0}{\left( {1 + \dfrac{R}{{100}}} \right)^t}\] to get the rate of growth by putting the values in it and taking time as 1 year. Therefore, we get
\[ \Rightarrow {P_1} = {P_0}{\left( {1 + \dfrac{R}{{100}}} \right)^1}\]
\[ \Rightarrow 35700 = 35000\left( {1 + \dfrac{R}{{100}}} \right)\]
Now we will simplify the above equation we will get the value of the rate of growth of the population i.e. \[R\].
Dividing both sides by 35000, we get
\[ \Rightarrow \left( {1 + \dfrac{R}{{100}}} \right) = \dfrac{{35700}}{{35000}}\]
Subtracting 1 from both sides, we get
\[ \Rightarrow \dfrac{R}{{100}} = \dfrac{{35700}}{{35000}} - 1\]
Simplifying above equation, we get
\[ \Rightarrow R = \dfrac{{35700}}{{35000}} \times 100 - 100\]
\[ \Rightarrow R = 102 - 100 = 2\% \]
Hence the rate of growth of the population is equal to \[2\% \].
So, option A is the correct option.

Note: Here we should note that the population follows the basic concept of compound interest. While calculating the value of the rate of growth of population from the formula, the growth rate we get will be in terms of the percentage so we don’t have to convert it into the percentage. Whenever the term rate is attached to anything it varies with the time which means it varies i.e. increases or decreases with respect to the time. Time taken in the formula of the population is always or mostly in years.