1650.13 – Wok Around the Clock

At a large party in a Chinese restaurant (Wok Around the Clock), every three guests shared a bowl of rice, every four guests shared a bowl of soup, and every pair of guests shared a bowl of fish between them.

There were 65 bowls of these three dishes altogether. How many guests were at the party? (There were numerous other dishes, too, but we won't take time to think about them.)

Solution

If $G$ is the number of guests, then there were $\dfrac{G}{3}$ bowls of rice, $\dfrac{G}{4}$ bowls of soup, and $\dfrac{G}{2}$ bowls of fish.

$\begin{align*}\dfrac{G}{3} + \dfrac{G}{4} + \dfrac{G}{2} &= \dfrac{4G}{12} + \dfrac{3G}{12} + \dfrac{6G}{12} \\[1em]&= \dfrac{13G}{12} \\[1em]&= 65 \leadsto G = 60\end{align*}$

(Check: $20 \text{ rice } + 15 \text{ soup } + 30 \text{ fish } = 65 \text{ bowls}$ .)