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,
$\Rightarrow {\displaystyle \frac{\text{Population of A }}{\text{Population of B}}}={\displaystyle \frac{11}{1}}$
Population of A after the movement is $(x+96000)+62000$
Population of A after the movement is $x-62000$
$\Rightarrow {\displaystyle \frac{\text{x + 96000 + 62000 }}{\text{x - 62000}}}={\displaystyle \frac{11}{1}}$
$\Rightarrow {\displaystyle \frac{\text{x + 158000 }}{\text{x - 62000}}}={\displaystyle \frac{11}{1}}$
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 $11kand1k$. Thus, we get two equations in two variables $(x,k)$ and solving this we can find the answer.