1550.35 – Sandwiches

Al and Bob are just sitting down to lunch when they are joined by their friend Carol. Al has brought five sandwiches to eat, Bob has brought three, but Carol has not brought any. She does have four dollars, however. The three of them divide the food evenly, and then Carol gives Al and Bob her four dollars to pay for her share. How should Al and Bob share the money?

Solution

Al and Bob together had 8 sandwiches. That means each person ate 83\dfrac{8}{3}, or 2232 \dfrac{2}{3} sandwiches. In order for Al to get from 5 to 2232 \dfrac{2}{3} sandwiches, he would have given 2132 \dfrac{1}{3}, or 73\dfrac{7}{3}, of his own to Carol. Bob would have given 13\dfrac{1}{3} to get from 3 to 2232 \dfrac{2}{3}.

Check: 213+13=2232 \dfrac{1}{3} + \dfrac{1}{3} = 2 \dfrac{2}{3} or 83\dfrac{8}{3}.

This means that Al contributed 7 times as much to Carol’s lunch as Bob did. To be compensated appropriately, Al would get 7 times as much of the money as Bob, or 78\dfrac{7}{8} of Carol’s money. 78\dfrac{7}{8} of $4.00 is $3.50. Bob, meanwhile, gets 18\dfrac{1}{8} of the $4.00, which is $0.50.