Question

The population of town A is \[96000\] more than town B. If $62000$ people move from town B to town A, the ratio between the population of town A and that of town B will be $11:1$. Find the original population of the two towns.
A) ${\text{A}} = 1,40,000{\text{ and B = 90,000}}$
B) ${\text{A = 1,50,000 and B = 52,000}}$
C) ${\text{A = 1,80,000 and B = 84,000}}$
D) ${\text{A}} = 1,90,000{\text{ and B = 60,000}}$

Answer 1

Hint: The relation between initial populations of towns A and B are given. Using this we can equate the ratio after some people moved from B to A. Simplifying the equation we get the original populations of town A and town B.

Complete step-by-step answer:
Given that the population of town A is \[96000\] more than town B.
Let the original population of town B be $x$.
So we have the original population of town A will be $x + 96000$.
It is given that if $62000$ people move from town B to town A, the ratio between the population of town A and that of town B will be $11:1$.
Therefore after the movement of people from B to A,
$ \Rightarrow \dfrac{{{\text{Population of A }}}}{{{\text{Population of B}}}} = \dfrac{{11}}{1}$
Population of A after the movement is $(x + 96000) + 62000$
Population of A after the movement is $x - 62000$
$ \Rightarrow \dfrac{{{\text{x + 96000 + 62000 }}}}{{{\text{x - 62000}}}} = \dfrac{{11}}{1}$
$ \Rightarrow \dfrac{{{\text{x + 158000 }}}}{{{\text{x - 62000}}}} = \dfrac{{11}}{1}$
Cross multiplying we get,
$x + 158000 = 11(x - 62000)$
$ \Rightarrow x + 158000 = 11x - 682000$
Simplifying we get,
$158000 + 682000 = 11x - x$
$ \Rightarrow 10x = 840000$
Dividing both sides by .$10$ we get,
$ \Rightarrow x = 84000$
$ \Rightarrow x + 96000 = 84000 + 96000 = 180000$
So, the original population of town A is $1,80,000$ and that of town B is $84,000$.
$\therefore $ The answer is option C.

Note: By the definition of ratios, if two quantities have ratio $a:b$, we can write those quantities as $ak,bk$ for some value $k$. Like that we can consider the new population of the towns as $11k and 1k$. Thus, we get two equations in two variables $(x,k)$ and solving this we can find the answer.