Question

A town’s population increased by 1200 people, and then this new population decreased by 11%. The town had 32 people less than the number of people it had before 1200 people increased. Find the original population of the town.

Answer 1

Hint: Assume the population of town be x people initially. Hence find the number of people after 1200 increase and how many were left after the 11% decrease. Equate it to 32 less than the original population. Hence form a linear equation in x. Solve for x to get the original population of the town.

Complete step-by-step answer:
Let the original population of the town be x.
So, we have the population of the town after 1200 people increase is equal to x+1200.
Now the number of people decreased in the town due to 11% is $\left( x+1200 \right)=\dfrac{x+1200}{100}\times 11=\dfrac{11x}{100}+132$
Hence the population after 11% is $x+1200-\dfrac{11x}{100}-132=\dfrac{89x}{100}+1068$
Since the population after 11% decrease is 32 less than the original population, we have
$\dfrac{89x}{100}+1068=x-32$
Adding 32 on both sides, we get
$\begin{align}
  & \dfrac{89x}{100}+1068+32=x-32+32 \\
 & \Rightarrow \dfrac{89x}{100}+1100=x \\
\end{align}$
Subtracting $\dfrac{89x}{100}$ from both sides, we get
$\begin{align}
  & \dfrac{89x}{100}+1100-\dfrac{89x}{100}=x-\dfrac{89x}{100} \\
 & \Rightarrow 1100=\dfrac{11x}{100} \\
\end{align}$
Multiplying both sides by 100, we get
$11x=110000$
Dividing both sides by 11, we get
x=10000
Hence the original population of the town was 10000.

Note: While solving the linear equation if the value of x was fractional and not integral and positive, then the situation described in question was not possible and hence no such value of population existed. This is because the population of a place is always $\ge 0$ and an integer. The statement “Population of a place is 11.5” is meaningless and absurd.