# Proportional Reasoning, Ratios, and Rates

Bob, Jerry, Tom, and Larry ate a cake. They took turns, eating it just one person at a time. Each of them was eating exactly for the period of time that it would take the other three to eat half of the cake.

How much faster would they be done eating this cake if they were eating all at once instead of taking turns (assuming each of them would eat at the same pace)?

Clarifications:

(1) They ate the whole cake.

(2) They could switch more than three times, but the total amount of time that each of them was eating was equal to the amount of time that it would take the other three to eat half of the cake. So by rearranging their turns if necessary, we can assume that each of them ate their portion of the cake in one sitting. E.g. first Bob ate for the time that it would take Jerry, Tom, and Larry to eat half of the cake. Then Jerry ate for the time that it would take Bob, Tom, and Larry to eat half of the cake. Then it was Tom's turn. Then Larry's turn. It happened so that exactly at the end of Larry's turn, the whole cake was gone.

(3) The problem does not say that all four rates are equal. Only that each person always eats at the same rate, no matter whether they are alone at the table or with their friends.

After four turns the ones who weren't munching on the main cake ate $4*\frac 12 = 2$ cakes worth. And those who ate on the main cake ate $1$ cake. So in the time it took for them to eat the main cake, taking turns, they would have eaten $2+1=3$ cakes.