The population of towns A and B is the ratio of 1 : 4. For the next 2 years, the population of A would increase and that of B would decrease by the same percentage every year. After 2 years, their population became equal. What is the percentage change in the population?

Options

A. A)33.33
B. B)66.66
C. C)`25%`
D. D)Not possible

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Define the populations
Let the population of town A be \( P_A \) and the population of town B be \( P_B \). According to the problem, the ratio of their populations is given as:
\[ P_A : P_B = 1 : 4 \]
This means we can express their populations as:
\[ P_A = x \quad \text{and} \quad P_B = 4x \]
where \( x \) is a common multiplier.

### Step 2: Define the percentage change
Let the percentage change in population for both towns be \( x\% \). Therefore, the population of town A after 2 years will increase, while the population of town B will decrease.

### Step 3: Calculate the new populations
After 2 years, the population of town A will be:
\[ P_A' = P_A \times \left(1 + \frac{x}{100}\right)^2 = x \times \left(1 + \frac{x}{100}\right)^2 \]
And the population of town B will be:
\[ P_B' = P_B \times \left(1 - \frac{x}{100}\right)^2 = 4x \times \left(1 - \frac{x}{100}\right)^2 \]

### Step 4: Set the populations equal
According to the problem, after 2 years, the populations of towns A and B become equal:
\[ x \times \left(1 + \frac{x}{100}\right)^2 = 4x \times \left(1 - \frac{x}{100}\right)^2 \]
We can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ \left(1 + \frac{x}{100}\right)^2 = 4 \times \left(1 - \frac{x}{100}\right)^2 \]

### Step 5: Simplify the equation
Taking the square root of both sides gives:
\[ 1 + \frac{x}{100} = 2 \times \left(1 - \frac{x}{100}\right) \]
Expanding the right side:
\[ 1 + \frac{x}{100} = 2 - \frac{2x}{100} \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ \frac{x}{100} + \frac{2x}{100} = 2 - 1 \]
\[ \frac{3x}{100} = 1 \]
Multiplying both sides by 100:
\[ 3x = 100 \]
Dividing by 3:
\[ x = \frac{100}{3} \approx 33.33 \]

### Step 7: Conclusion
The percentage change in the population is:
\[ \text{Percentage change} = 33.33\% \]