Question

The population of a town increases at the rate of $6\% $ every year. The present population of the town is $14,000$. Find the population of the town at the end of $4$ years.

Answer 1

Hint: In the given question, we are given that the present population of the town is $14,000$ and it increases at the rate of $6\% $ every year. Now, we have to calculate the population of the town at the end of $4$ years. The question requires the concepts of percentages and geometric progression to be clear in our mind. Such questions require accuracy in arithmetic.

Complete step-by-step solution:
Let’s consider the given problem.
The present population of the town$ = 14,000$
The percent increment in the population$ = 6\% $
So, in simple words, the population of the town in a year becomes $106\% $ of the population in the previous year.
So, we would have to multiply the population of the present year by $\left( {\dfrac{{106}}{{100}}} \right)$ to get the population of the town in the next year.
Now, we have to calculate the population of the town at the end of $4$ years.
So, we would have to multiply the present population of the town by $\left( {\dfrac{{106}}{{100}}} \right)$, $4$times in order to get the required answer.
So, the population of the town at the end of $4$ years$ = 14,000\left( {\dfrac{{106}}{{100}}} \right)\left( {\dfrac{{106}}{{100}}} \right)\left( {\dfrac{{106}}{{100}}} \right)\left( {\dfrac{{106}}{{100}}} \right)$
$ \Rightarrow 14,000{\left( {\dfrac{{106}}{{100}}} \right)^4}$
$ \Rightarrow 14,000{\left( {1.06} \right)^4}$
So, calculating the value of ${\left( {1.06} \right)^4}$, we get,
$ \Rightarrow 14,000\left( {1.262477} \right)$
Simplifying further, we get the population of the town at the end of $4$ years as,
$ \Rightarrow 17674$

Thus, the population of the town at the end of $4$ years is $17,674$.

Note: Though we have a number of methods to solve this problem, it is better to workout with the above-mentioned method and after solving a number of problems in this model, one can consider the method involving the unitary method. Solving problems by the unitary method needs better understanding in the concept of percentage. While solving such a problem using a unitary method, we consider the given number as $100\% $ of itself and then find out the desired percentage of the number. Work as many problems as possible to crack these types of problems in a limited time period.