The dishes problem

"How many guests are there?" said the official.
"I don't know.", said the cook, "but every 2 used a dish of rice, every 3 used a dish of broth, and every 4 used a dish of meat between them". There were 65 dishes in all. How many guests were there?

Taken from A puzzle from 4th century CE, China from the Sunzi Suan Jing 孙子算经

My first thought when reading this is that the number of guests must be a multiple of 12, since that is the least common multiple of 2, 3 and 4. 

My next thought was to write out that every guest had half a bowl of rice, one third of a bowl of broth and one quarter of a bowl of meat. Changing the wording helped me understand the problem better and find a formula to solve it.

I ended up with the equation:

guest + guest + guest = 65

  2           3            4

I simplified that and found that there were 60 guests total.