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. Altogether, there are 65 dishes (bowls). How many guests were present at the party?

Introduction / Context:
This problem models a real life sharing situation using simple algebra. At a dinner party, bowls of rice, dal and meat are shared among guests in different group sizes. Using the information about how many guests share each type of dish and the total number of dishes, we have to determine how many guests attended the party. The question tests understanding of ratios, fractions and how to translate words into equations.

Given Data / Assumptions:

- Every two guests share one bowl of rice. - Every three guests share one bowl of dal. - Every four guests share one bowl of meat. - The total number of bowls (rice + dal + meat) is 65. - All sharing is exact, meaning no half bowls or unused portions are counted separately.

Concept / Approach:
Let the number of guests be n. Then, since every two guests share one bowl of rice, the number of rice bowls is n/2. Similarly, the number of dal bowls is n/3, and the number of meat bowls is n/4. The total number of bowls is the sum of these three expressions and is given as 65. We set up the equation n/2 + n/3 + n/4 = 65 and solve for n. Because n represents a number of guests, it must be a positive integer.

Step-by-Step Solution:
Step 1: Let n be the number of guests. Step 2: Number of bowls of rice is n/2 because each bowl serves two guests. Step 3: Number of bowls of dal is n/3 because each bowl serves three guests. Step 4: Number of bowls of meat is n/4 because each bowl serves four guests. Step 5: According to the problem, total dishes are 65, so n/2 + n/3 + n/4 = 65. Step 6: Find a common denominator for 2, 3 and 4, which is 12. Step 7: Rewrite each term with denominator 12: n/2 = 6n/12, n/3 = 4n/12, n/4 = 3n/12. Step 8: Add them: 6n/12 + 4n/12 + 3n/12 = 13n/12. Step 9: So 13n/12 = 65. Step 10: Multiply both sides by 12 to get 13n = 65 * 12. Step 11: Compute 65 * 12 = 780, so 13n = 780. Step 12: Divide by 13 to get n = 780 / 13 = 60.

Verification / Alternative check:
We can verify by computing the number of bowls of each type for 60 guests. Rice bowls = 60/2 = 30. Dal bowls = 60/3 = 20. Meat bowls = 60/4 = 15. Total dishes = 30 + 20 + 15 = 65, matching the problem statement. Thus 60 guests satisfy all the conditions.

Why Other Options Are Wrong:
If we take 74 guests, then 74/2 + 74/3 + 74/4 is not an integer sum equal to 65. Similar checks show that 82 or 58 guests do not satisfy the equation with total dishes equal to 65. Hence those options are not consistent with the sharing rules.

Common Pitfalls:
Some learners incorrectly multiply rather than divide when converting guests to bowls, or they might forget to use a common denominator. Others may misinterpret the statement and assume that there are exactly two, three or four guests in total rather than per bowl. Translating carefully to algebra and using the least common multiple of denominators avoids these issues.

Final Answer:
The number of guests at the dinner party is 60, which corresponds to option A.