At a marriage party, 32% of the people are women, 54% are men and the remaining 196 are children. How many men are present at the party?

Introduction / Context:
This question involves percentages and complements of percentages. We are given the percentage of women, the percentage of men, and the absolute number of children, and we need to determine the number of men present at the party. It is a typical problem of converting between percentages and absolute numbers.

Given Data / Assumptions:

• Percentage of women at the party = 32%.
• Percentage of men at the party = 54%.
• The remaining people are children, whose number is 196.
• The total number of people at the party is denoted by N.

Concept / Approach:
We know that:

• Women = 32% of N.
• Men = 54% of N.
• Children = remaining percentage of N.

The remaining percentage is 100% - (32% + 54%) = 14%. We also know that 14% of N equals 196 children. From this, we can find N and then compute 54% of N to get the number of men.

Step-by-Step Solution:
Step 1: Find the percentage of children. Children percentage = 100% - 32% - 54% = 14%. Step 2: Set up the relation between percentage and number of children. 14% of N = 196. Step 3: Solve for N. 14 / 100 * N = 196. N = 196 * 100 / 14. N = 196 * (100 / 14) = 196 * (50 / 7) = 28 * 50 = 1400. Step 4: Compute number of men, which is 54% of N. Men = 54% of 1400 = 54 / 100 * 1400. Men = 0.54 * 1400 = 756.

Verification / Alternative check:
We can also compute the number of women and children and check that totals match 1400. Women = 32% of 1400 = 0.32 * 1400 = 448. Children are given as 196. Sum of men, women and children = 756 + 448 + 196 = 1400, which equals the total number of people N, confirming the correctness of the calculation.

Why Other Options Are Wrong:
Option b (448) is actually the number of women, not men. Option c (332) and option d (324) do not correspond to 54% of 1400 and would give wrong totals when combined with the other groups. Only 756 satisfies the percentage condition and results in consistent totals.

Common Pitfalls:
A common mistake is to think that 196 is 32% or 54% of the total instead of the remainder. Another error is to miscalculate the total by adding percentages incorrectly or failing to convert percentages into fractions before solving. Always ensure that the sum of percentages equals 100% and that the complement percentage is correctly used to link the known count with the total.

Final Answer:
The number of men present at the party is 756.