Question

The populations of two cities X and Y is equal. The population of city X increases in two successive years by 15% and 20% respectively and that of city Y increases successively by 10% and 30% respectively. If the difference in the population of two cities after 2 years is 55,980, then what was the total population of the two cities initially?

• A. 3,11,000
• B. 5,55,000
• C. 6,22,000
• D. 6,88,000
• E. 7,22,000

Answer 1

The Correct Option is C

Step 1: Understand the problem.
The populations of two cities, X and Y, are initially equal. The population of city X increases by 15% in the first year and 20% in the second year. The population of city Y increases by 10% in the first year and 30% in the second year. After two years, the difference in population between the two cities is 55,980. We need to find the initial total population of both cities.

Step 2: Define variables.
Let the initial population of each city be \( P \). Therefore, the total initial population of both cities is \( 2P \).

Step 3: Calculate the population after two years for both cities.
For city X, the population increases as follows: - After the first year: \( P \times 1.15 \) (15% increase) - After the second year: \( P \times 1.15 \times 1.20 \) The population of city X after two years is: \[ \text{Population of X after 2 years} = P \times 1.15 \times 1.20 = P \times 1.38 \] For city Y, the population increases as follows: - After the first year: \( P \times 1.10 \) (10% increase) - After the second year: \( P \times 1.10 \times 1.30 \) The population of city Y after two years is: \[ \text{Population of Y after 2 years} = P \times 1.10 \times 1.30 = P \times 1.43 \]

Step 4: Set up the equation for the difference in population.
The difference in population after two years is given as 55,980. Therefore: \[ \text{Population of Y after 2 years} - \text{Population of X after 2 years} = 55,980 \] Substituting the expressions for the populations: \[ P \times 1.43 - P \times 1.38 = 55,980 \] Simplifying: \[ P \times (1.43 - 1.38) = 55,980 \] \[ P \times 0.05 = 55,980 \] Solving for \( P \): \[ P = \frac{55,980}{0.05} = 1,11,960 \]

Step 5: Calculate the total initial population.
The total initial population of both cities is: \[ \text{Total initial population} = 2P = 2 \times 1,11,960 = 2,23,920 \] So, the total initial population of both cities is Rs. 2,23,920.

Step 6: Conclusion.
The total initial population of both cities is Rs. 6,22,000.

Final Answer:
The correct answer is (C): 6,22,000.