Question

The present population of a village is 130,000. If the population increases each year by \[4\% \] of what it had been at the beginning of each year, then the population of the village after four years will be \[x\]. Find the sum of the digits of \[x\].

Answer 1

Hint: First we will assume that \[{P_0}\] is the present population and \[P\] is the population after 4 years. Then we will use the formula to calculate the present population, \[P = {P_0} \times {\left( {1 + \dfrac{r}{{100}}} \right)^n}\], where \[n\] is the time in years and \[r\] is the rate of increment and then substitute the above values of \[{P_0}\], \[r\] and \[n\].

Complete step by step answer:

We are given that the present population of a village is 130,000. If the population increases each year by \[4\% \] of what it had been at the beginning of each year, then the population of the village after four years will be \[x\].

Let us assume that\[{P_0}\] is the present population and \[P\] is the population after 4 years.
Now we will use the formula to calculate the present population, \[P = {P_0} \times {\left( {1 + \dfrac{r}{{100}}} \right)^n}\], where \[n\] is the time in years and \[r\] is the rate of increment.

Finding the value of \[{P_0}\], \[r\], and \[n\] for the above formula, we get

\[{P_0} = 130,000\]
\[r = 4\]
\[n = 4\]

Substituting the above values of \[{P_0}\], \[r\] and \[n\] in the above formula, we get
\[
   \Rightarrow P = 130,000 \times {\left( {\dfrac{{100 + 4}}{{100}}} \right)^4} \\
   \Rightarrow P = 130,000 \times {\left( {\dfrac{{104}}{{100}}} \right)^4} \\
   \Rightarrow P = 130,000 \times {\left( {\dfrac{{26}}{{25}}} \right)^2} \\
   \Rightarrow P = 152082 \\
 \]

Thus, the population at the end of 4 years is 152082.

Finding the sum of the digits of the number 152082, we get
\[ \Rightarrow 1 + 5 + 2 + 0 + 8 + 2 = 18\]

This implies that 18 is the required answer.

Note: In this question, we assume the value of population after 4 years ago with any of the variable and then use it to find the original value. Also, we will use the formula for calculating the present population, \[{\text{Present population = }}{P_0} \times {\left( {1 + \dfrac{r}{{100}}} \right)^n}\], where \[{P_0}\] is the present population, \[P\] is the population after 4 years., \[n\] is the time in years and \[r\] is the rate of increment. Students need to find the sum of the digit of the value, as this is the possible mistake.