Dishes Puzzle

Algebraic Solution:

      Let x be the number of guests => the total number of rice dishes is x/2, since x guests divided by 2 guests per rice dish equals x/2 rice dishes.  Using similar reasoning with the other two food types, there must be (x/2) + (x/3) + (x/4) dishes in all.  But, we are given that there are 65 total dishes.  Therefore,
      (x/2) + (x/3) + (x/4) = 65
=> (6x + 4x + 3x)/12 = 65
=> 13x = 780
=> x = 60
Thus, there must be 60 guests in total, consuming 60/2 = 30, 60/3 = 20, and 60/4 = 15 dishes of rice, broth, and meat, respectively.

    
Non-Algebraic Solution:

      Notice that the number of guests must be divisible by 2, 3, and 4 => the number of guests must be divisible by lcm(2,3,4) = 12*.  Each group of 12 guests consume (12/2) + (12/3) + (12/4) = 6 + 4 + 3 = 13 dishes.  [Picture a table with 13 dishes on it and 12 chairs around it]  Since we are given that there are 65 dishes in all, there must be (65/13) = 5 groups(tables) of 12 guests, for a total of 60 guests.

*computation of the least common multiple (lcm) of 2, 3, and 4: both 2 and 3 are prime, 4 = 2x2 => lcm(2,3,4) = 3x2x2 = 12.


Cultural Context:
      In certain cultures, food sharing is socially acceptable.  Some students may be confused by the fact that people are sharing food at all (why doesn't each guest get their own dish?) - those familiar with pay-by-the-plate events.  Others may ask why all of guests don't line up to eat from three large dishes (buffet-style).  If you explain that the question is merely hypothetical (doesn't have to take place in China), then learners should have an easier time grasping the style of eating outlined (culture-free).