Question

The population of Hyderabad was 68,09,000 in the year 2011. If it increases at the rate of 4.7% per annum. What will be the population at the end of the year 2015.

Answer 1

Hint: As the population is increasing yearly at the rate of 4.7% of what it was last year, this is the case of compound interest. Initial population is given. Find the time period during which the population is increasing from the question and apply the formula \[P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\] to find the final population. Here in the place of ‘p’ we will consider the initial population. ‘r’ and ‘n’ will be the rate and time period respectively.

Complete step-by-step answer:
As we know that compound interest is the type of interest in which the rate of interest is applied on every new amount that we get after a complete duration (here it is one year).
And after the total time the new amount will be the real value.
As given in the question that,
Population of Hyderabad was 68,09,000 in the year 2011 and we had to find the population at the end of year 2015.
So, to find the time duration for which the rate of interest is applied we had to subtract the initial and final years.
So, number of years will be equal to, n = 2015 – 2011 = 4 years
And P will be the initial population i.e. population of Hyderabad in 2011 which is 68,09,000.
And the population is increasing at a rate of 44.7% per year.
So, now putting all the values in the formula of compound interest to find the population after 4 years i.e. at the end of 2015.
So, population = \[68,09,000{\left( {1 + \dfrac{{4.7}}{{100}}} \right)^4}\]
Now solving the above equation to find the population at the end of 2015.
Population = \[68,09,000\left( {\dfrac{{104.7}}{{100}} \times \dfrac{{104.7}}{{100}} \times \dfrac{{104.7}}{{100}} \times \dfrac{{104.7}}{{100}}} \right)\]
Population = \[68,09,000 \times \left( {1.2016} \right) = 81,81,694\]
Hence, the population of Hyderabad at the end of 2015 will be equal to 81,81,694


Note:- Whenever we come up with this type of problem then there is also another method to solve this type of question. Like we can use simple interest formula for each year i.e. \[\dfrac{{P \times r \times t}}{{100}}\] where after each year we must had to add the population increased in the previous year to the principal population. Like if the initial population at 2011 is P and we get increased population at the end of 2011 as A by using a simple interest formula then for calculating increase in population at the end of 2012 (next year) we must use P = P + A. And this goes on for the next four years.