In a city, 35% of the population consists of migrants, 20% of whom are from rural areas. Of the local population, 48% are female, while for rural and urban migrants the female percentages are 30% and 40% respectively. If the total population of the city is 7,28,400, what is the total female population of the city?

Introduction / Context:
This question is a multi-step percentage and population distribution problem. It involves splitting the total population into local residents and migrants, then further dividing migrants into rural and urban groups, and finally applying different female population percentages to each group. Such layered percentage questions are frequently seen in data interpretation and quantitative aptitude sections, and they test your ability to manage multiple proportions cleanly.

Given Data / Assumptions:

• Total population of the city = 7,28,400.
• 35% of the total population are migrants.
• 20% of the migrants are from rural areas; the remaining 80% of migrants are from urban areas.
• 48% of the local (non-migrant) population are female.
• 30% of rural migrants are female.
• 40% of urban migrants are female.
• We must find the total number of females in the whole city.

Concept / Approach:
The strategy is to calculate the sizes of each subpopulation step by step and then apply the respective female percentages. First, we compute the number of migrants and locals using the 35% figure. Then we break migrants into rural and urban migrants using 20% and 80%. Next, we calculate the number of females in each of the three groups: locals, rural migrants, and urban migrants, using their given percentages. Finally, we sum the female counts across all groups to get the total female population. Careful ordering of these steps helps avoid confusion and arithmetic errors.

Step-by-Step Solution:
Step 1: Total population = 7,28,400.Step 2: Migrant population = 35% of 7,28,400 = 0.35 * 7,28,400 = 2,54,940.Step 3: Local population = total population - migrant population = 7,28,400 - 2,54,940 = 4,73,460.Step 4: Rural migrants = 20% of total migrants = 0.20 * 2,54,940 = 50,988.Step 5: Urban migrants = 80% of total migrants = 0.80 * 2,54,940 = 2,03,952.Step 6: Female locals = 48% of local population = 0.48 * 4,73,460 = 2,27,260.8.Step 7: Female rural migrants = 30% of rural migrants = 0.30 * 50,988 = 15,296.4.Step 8: Female urban migrants = 40% of urban migrants = 0.40 * 2,03,952 = 81,580.8.Step 9: Total female population = 2,27,260.8 + 15,296.4 + 81,580.8 = 3,24,138 (rounding to the nearest whole person).

Verification / Alternative check:
As a verification, we can approximate the calculations to check the scale. Migrants are about 2.55 lakh people, locals about 4.73 lakh. If roughly half of locals are female, that is about 2.36 lakh females, which is close to the exact 2.27 lakh. Among migrants, rural migrants are about 51 thousand with 30% females, giving about 15 thousand, and urban migrants are about 2.04 lakh with 40% females, giving about 81 thousand. Adding these approximate values (2.27 lakh, 0.015 lakh, 0.081 lakh) gives around 3.24 lakh, which matches the exact calculation of 3,24,138. This confirms that the arithmetic and logic are consistent.

Why Other Options Are Wrong:
Option 3,49,680 is larger than the total migrant population and would require an unrealistically high female percentage across all groups. Option 5,09,940 is well over half the total population and contradicts the given female percentages. The value 4,12,356 similarly does not arise from any valid combination of the given percentages. Option "None of these" is not correct because we have found a specific value that fits all conditions, namely 3,24,138. Therefore, only 3,24,138 accurately represents the total female population as per the given data.

Common Pitfalls:
Learners sometimes apply the female percentages directly to the total population instead of to each subgroup, which ignores the differences between locals, rural migrants, and urban migrants. Another common error is to treat 20% of the city population as rural migrants rather than 20% of the migrant population. Misplacing decimal points when dealing with large numbers is also frequent. To avoid these problems, always keep track of which percentage applies to which group and perform calculations step by step with clear labels.

Final Answer:
The total female population of the city is 3,24,138.