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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.

2 Answers 2

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Let's imagine that they ate not just the one cake but several extra cakes all at the same time.

As bob eats his turn, larry, tom and jerry eat out of their cakes and eat the amount of half a cake. Bob finishes and Larry comes to eat of the original cake and Bob continues on larry's cake. Larry eats his turn and the other three eat of their cakes and eat the quantity of half a cake. And so on.

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.

So they would have finished the cake three times as quickly if they all ate together.

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As far as I can see, the 4-variable equation relating consumption rates to the fact that the cake is completely eaten after all have had a go, has no real solution. So whatever the merits of the first answer, the whole thing is based on a false premise.