Question

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

Answer 1

Hint: Assume that the original population of the town is \[x\] . Obtain the new population of the town for the increase of 1200. In the \[{{2}^{nd}}\] case, the new population is decreased by its 11%. Now, obtain the population after a decrement of 11%. It is given that after the decrement of 11%, the town now had 32 fewer people than it did before the 1200 increase, i.e., \[\left( x+32 \right)=\left( x+1200 \right)\left( 1-\dfrac{11}{100} \right)\] . Now, solve it further and calculate the value of \[x\] .

Complete step by step answer:
According to the question, we are given that A town’s population increased by 1200 people, and then this new population is decreased by 11%. The town now had 32 fewer people than it did before the 1200 increase. We are asked to find the original population of the town.
First of all, let us assume that the original population of the town is \[x\] ………………………(1)
Here, we have two cases.
 In \[{{1}^{st}}\] case, the population is increased by 1200 ………………………………..(2)
Now, from equation (1) and equation (2), we have
The new population of the town = \[\left( x+1200 \right)\] ………………………………..(3)
In \[{{2}^{nd}}\] case, the population is decreased by 11% ……………………………………(4)
Now, from equation (1) and equation (4), we get
The new population after decrementing 11% = \[\left( x+1200 \right)\] - 11% of \[\left( x+1200 \right)\] = \[\left( x+1200 \right)\left( 1-\dfrac{11}{100} \right)\] …………………………………(5)
It is also mentioned in the question that after the decrement of 11%, the town now had 32 fewer people than it did before the 1200 increase …………………………………(6)
Now, from equation (1), equation (5), and equation (6), we get
\[\begin{align}
  & \Rightarrow \left( x+32 \right)=\left( x+1200 \right)\left( 1-\dfrac{11}{100} \right) \\
 & \Rightarrow \left( x+32 \right)=\left( x+1200 \right)\times \dfrac{89}{100} \\
 & \Rightarrow 100\left( x+32 \right)=89\left( x+1200 \right) \\
 & \Rightarrow 100x-89x=89\times 1200-100\times 32 \\
 & \Rightarrow 11x=100\left( 1068-32 \right) \\
 & \Rightarrow 11x=100\times 1034 \\
 & \Rightarrow x=\dfrac{103400}{11} \\
\end{align}\]
\[\Rightarrow x=9400\] ……………………………..(9)
Therefore, the original population of the town is 9400.


Note:
For this type of question, one might do a silly mistake while forming the equation using the information that after a decrementing of 11%, the town now had 32 fewer people than it did before the 1200 increase. One might form the equation as \[x=\left( x+1200 \right)\left( 1-\dfrac{11}{100} \right)+32\] . This is wrong. Since the population original population is 32 less than the population after the decrement of 11% so, 32 must be added to the original population to make it equal to the population after the decrement of 11%. Therefore, the correct equation is \[\left( x+32 \right)=\left( x+1200 \right)\left( 1-\dfrac{11}{100} \right)\] .