Counting guests from dish ratios: At a party there is 1 bowl of rice for every 2 guests, 1 bowl of juice for every 3 guests, and 1 bowl of meat for every 4 guests. If there are 65 bowls in total, how many guests attended?

Introduction / Context:
The counts of bowls are given per guest ratios. For G guests, the total bowls equal G/2 + G/3 + G/4. Setting this total to 65 yields a simple linear equation in G after taking a common denominator.

Given Data / Assumptions:

• Bowl counts: rice = G/2, juice = G/3, meat = G/4.
• Total bowls = 65.
• Guests are a whole number; ratios are exact.

Concept / Approach:
Solve G/2 + G/3 + G/4 = 65. Use the common denominator 12 to combine terms and isolate G.

Step-by-Step Solution:

Verification / Alternative check:
Compute bowls for G = 60: rice = 30, juice = 20, meat = 15; total = 65. Checks out exactly.

Why Other Options Are Wrong:

• 65: Would require bowl/guest ratios summing to 1, which they do not.
• 24 or 48: Produce totals far from 65 given the bowl ratios.

Common Pitfalls:
Adding numerators without a common denominator or forgetting that each term depends on the same G.

Final Answer: