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.