Question : The population of country A decreased by p% and the population of country B
decreased by q% from the year 2020 to 2021. Here 'p' is greater than ‘q’. Let 'x' be the ratio of the population of country A to the population of country B in the given year. What is the percentage decrease in 'x' from 2020 to 2021?
Option 1: $\frac{100(p-q)}{(100-q)}$
Option 2: $\frac{100(p-q)}{(100+P)}$
Option 3: $\frac{100(p-q)}{(100-p)}$
Option 4: $\frac{100(p-q)}{(100+q)}$
Correct Answer: $\frac{100(p-q)}{(100-q)}$
Solution :
Let the initial population of Country A and B be $J$ and $K$ respectively.
Hence, $x = \frac{J}{K}$
Let the ratio of the population of country A to the population of country B for the next year be $x_{1}$.
According to the question,
$(J \times (1 - p\%)) : (K \times (1 - q\%)) = x_{1}$.
⇒ $\frac{J}{K}\times \frac{100−p}{100−q}=x_{1}$.
Now, required % decrease
= $\frac{x-x_{1}}{x}\times 100$%
= $\frac{\frac{J}{K}-\frac{J}{K}\times \frac{100−p}{100−q}}{\frac{J}{K}}×100$%
= $ \frac{p-q}{100-q}\times 100$%
= $ \frac{100(p-q)}{100-q}$%
∴ The percentage decrease in 'x' from 2020 to 2021 is $ \frac{100(p-q)}{100-q}$%.
Hence, the correct answer is $\frac{100(p-q)}{100-q}$.
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