Question

The total number of industries in a particular portion of the country is approximately 1600. If the government has decided to increase the number of industries in the area by 20% every year. Find the approximate number of industries after 2 years

Answer 1

Step-1: Compute the value of ${\mathbf{P}}_{\mathbf{n}}$:

${\mathbf{P}}_{\mathbf{n}}\mathbf{=}\mathit{P}{\mathbf{\left(}\mathbf{1}\mathbf{+}\frac{r}{100}\mathbf{\right)}}^{\mathbf{n}}$
where $P_n$ is the number of industries after $n$years
$P$is the no. of industries presently
$r$ is the increment every year
$n$is the no of years after which the new value is to be found.

Let the Present number of industries in a particular portion of the country be $P$.
So, from the given question $P=1600$;
Now, the government has decided to increase the number of industries in the area by $20\%$ every year.
So, let the number of industries after$2$ years be $P_n$.

Now, we need to use the formula${\mathbf{P}}_{\mathbf{n}}\mathbf{=}\mathit{P}{\mathbf{\left(}\mathbf{1}\mathbf{+}\frac{r}{100}\mathbf{\right)}}^{\mathbf{n}}$ to find the value of $P_n$

Step 2 : Applying the formula and substituting the vales in the given formula:
Hence, the number of industries after $2$ years $(P_n)$ = $P{\left(1+{\displaystyle \frac{r}{100}}\right)}^n$

$\Rightarrow P_n=1600{\left(1+{\displaystyle \frac{20}{100}}\right)}^2$
$\Rightarrow P_n=1600\times {\displaystyle \frac{120}{100}}\times {\displaystyle \frac{120}{100}}$
$\Rightarrow P_n=1600\times {\displaystyle \frac{120}{100}}\times {\displaystyle \frac{120}{100}}$
$\Rightarrow P_n=2304$
Hence, the number of industries after $2$years will be $2304$.
Hence, the correct answer is $“2304”.$