female population of city a amd b is 820 if the total population of city a and b is in the ratio 3:4 respectively and their difference is 400 then what is the population of both the cities

Let's solve this problem step by step.

### Step 1: Understanding the given information
- The ratio of the populations of City A to City B is 3:4.
- The difference in the populations of City A and City B is 400.
- The total female population of both cities combined is 820.

### Step 2: Set up equations
Let the population of City A be \( 3x \) and the population of City B be \( 4x \).

Given:
1. \( 4x - 3x = 400 \) (The difference between the populations)
2. \( 4x + 3x = 820 \) (The total population based on the female population)

### Step 3: Solve for \( x \)
1. From the difference equation:
   \[
   x = 400
   \]
   
2. Now substitute \( x = 400 \) into the population expressions:
   - Population of City A: \( 3x = 3(400) = 1200 \)
   - Population of City B: \( 4x = 4(400) = 1600 \)

### Conclusion:
- The population of City A is 1200.
- The population of City B is 1600.

Would you like a detailed explanation of any part? Here are some related questions to explore:

1. What if the difference in populations was 500 instead of 400?
2. How would the solution change if the ratio of the populations was 2:3 instead of 3:4?
3. Can you calculate the individual female populations in each city if given the female-to-male ratio?
4. What happens if the total population of the cities increases by 10%?
5. How would the approach differ if you only knew the total population of both cities combined?

**Tip:** When working with ratios, setting up equations that express the given conditions helps in solving for unknowns systematically.

Explore how to solve a ratio and proportion problem involving city populations. Understand the relationship between populations, ratios, and differences. Suitable for Grades 7-9.

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