The ratio of males to females in a city is 7 : 8. Among males, 25% are children and among females, 20% are children. If the number of adult females is 1,56,800, what is the total population of the city?

Introduction / Context:
This question tests combined use of ratios and percentages in a population problem. It requires careful distinction between adults and children among both male and female populations and then reconstruction of the total population from partial information.

Given Data / Assumptions:

• Ratio of males to females = 7 : 8.
• Among males, 25% are children and 75% are adults.
• Among females, 20% are children and 80% are adults.
• Number of adult females = 1,56,800.
• We must find the total population (males plus females, adults plus children).

Concept / Approach:
Let the numbers of males and females be 7x and 8x respectively. Since 80% of females are adults, adult females count is 0.80 * 8x which is given. Solving for x provides actual male and female populations. The total population is then the sum of 7x and 8x. The internal adult and child counts can also be verified using the given percentages.

Step-by-Step Solution:
Step 1: Let number of males = 7x and number of females = 8x. Step 2: Among females, 80% are adults, so adult females = 0.80 * 8x = 6.4x. Step 3: Adult female population is given as 1,56,800, so 6.4x = 1,56,800. Step 4: Solve for x: x = 1,56,800 / 6.4. Step 5: Calculate x = 24,500. Step 6: Total males = 7x = 7 * 24,500 = 1,71,500. Step 7: Total females = 8x = 8 * 24,500 = 1,96,000. Step 8: Total population = 1,71,500 + 1,96,000 = 3,67,500.

Verification / Alternative check:
Check adult females: 80% of 1,96,000 = 0.80 * 1,96,000 = 1,56,800, which matches the given data. Adult males would be 75% of 1,71,500 = 1,28,625 and male children 42,875. Female children are 20% of 1,96,000 = 39,200. Sum of all four categories (adult males, adult females, male children, female children) is 1,28,625 + 1,56,800 + 42,875 + 39,200 = 3,67,500, which agrees with the total population computed earlier.

Why Other Options Are Wrong:
Other totals such as 4,12,480 or 5,44,700 would not yield an adult female population exactly equal to 1,56,800 when the ratio 7 : 8 and the given percentages are applied. They fail to maintain consistency between overall ratio, adult percentages and the specific adult female count.

Common Pitfalls:
A common mistake is to assume that 1,56,800 is total females instead of adult females, which leads to wrong calculations. Another error is miscomputing the adult percentage, for example using 20% instead of 80% for adult females. Always pay close attention to whether a percentage relates to adults or children, and to which subgroup it applies.

Final Answer:
The total population of the city is 3,67,500.