At a dinner party, every two guests share one bowl of rice between them, every three guests share one bowl of dal between them, and every four guests share one bowl of meat between them. If the total number of dishes served (rice bowls + dal bowls + meat bowls) is 65, how many guests attended the party?

Introduction:
This is a word problem involving ratios and inverse relationships. Instead of one dish per guest, dishes are shared among groups of guests, and you must deduce the number of guests from the total number of shared dishes. Such problems are common in aptitude exams to test algebraic modeling skills.

Given Data / Assumptions:

• Total number of guests = n (unknown).
• Every 2 guests share 1 bowl of rice → number of rice bowls = n/2.
• Every 3 guests share 1 bowl of dal → number of dal bowls = n/3.
• Every 4 guests share 1 bowl of meat → number of meat bowls = n/4.
• Total number of dishes (bowls) = 65.
• We assume n is a positive integer and that divisions like n/2, n/3, n/4 give integer counts of bowls, meaning n is a common multiple where needed.

Concept / Approach:
The total dishes are the sum of the rice, dal and meat bowls. We translate the sharing statements into an equation in n and then solve for n. This is effectively solving a linear equation with fractional coefficients.

Step-by-Step Solution:
Step 1: Express dish counts in terms of n.Rice bowls = n/2.Dal bowls = n/3.Meat bowls = n/4.Total dishes = n/2 + n/3 + n/4 = 65.Step 2: Use a common denominator (12).n/2 = 6n/12, n/3 = 4n/12, n/4 = 3n/12.So total dishes = (6n + 4n + 3n) / 12 = 13n / 12.Step 3: Set equal to 65 and solve.13n / 12 = 65.Multiply both sides by 12: 13n = 65 * 12 = 780.n = 780 / 13 = 60.

Verification / Alternative check:
With 60 guests: rice bowls = 60/2 = 30; dal bowls = 60/3 = 20; meat bowls = 60/4 = 15. Total dishes = 30 + 20 + 15 = 65, matches the given condition exactly.

Why Other Options Are Wrong:
74, 82, 58 and 48 do not satisfy the equation n/2 + n/3 + n/4 = 65 when substituted. For example, with n = 48, dishes would be 24 + 16 + 12 = 52, not 65.

Common Pitfalls:
Some students add the denominators incorrectly or forget to find a common denominator before combining fractions. Others misinterpret the wording and think each pair gets its own dish without using fractions at all. Proper algebraic translation is the key step.

Final Answer:
There were 60 guests at the party.