Question : The population of town B is 300% more than that of town A. For the next two years, the population of A increases by $x$% per year, and that of B decreases by the same percentage per year. After 2 years, if the population of A and B become equal, then the value of $x$ is ________.
Option 1: $30 \frac{2}{3}$
Option 2: $33 \frac{1}{3}$
Option 3: $40$
Option 4: $25$
Correct Answer: $33 \frac{1}{3}$
Solution :
Given :
The population of town B is 300% more than that of town A.
Let the population of town A be 100.
Then, the population of town B = 400
According to the question,
$100 × \frac{100 + x}{100}×\frac{100 + x}{100} = 400 × \frac{100 -x}{100} ×\frac{100 -x}{100}$
$\Rightarrow (100\ +\ x)^2\ =\ 4(100\ -\ x)^2$
$\Rightarrow \frac{100\ +\ x}{100\ -\ x}\ =\ \frac{2}{1}$
$\Rightarrow (100+x) =(200-2x)$
$\therefore x = \frac{100}{3}\% = 33\tfrac{1}{3}\%$
Hence, the correct answer is $33\tfrac{1}{3}\%$.
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