Outline Mathematics Algebra, Word Problems

The word problem below is the one that opens the new book by David Singmaster Problems for Metagrobologists (World Scientific, 2015, paperback, $28): Jessica and her friend Pud like to eat a big lunch. One day Jessica brought four sandwiches an Pud brought five. Samantha got mugged on her way to school, but the mug ran off with her lunch and left her purse. So Jessica and Pud shared their sandwiches with Samantha. After eating, Samantha said: "Thanks a million. I've got to see Mr Grind, but here's some money to pay for the sandwiches. She left$\$3$ and ran off. Jessica said: "Let me see, I brought 4 sandwiches and you brought five, so I get $\frac{4}{9}$ of $\$3$which is$\frac{4}{3}$of a dollar, which is$\$1.33,$ near enough. Pud said: "Ummm, I'm not sure that's fair." Why?

Solution

Jessica and her friend Pud like to eat a big lunch. One day Jessica brought four sandwiches an Pud brought five. Samantha go mugged on her way to school, but the mug ran off with her lunch and left her purse. So Jessica and Pud shared their sandwiches with Samantha. After eating, Samantha said: "Thanks a million. I've got to see Mr Grind, but here's some money to pay for the sandwiches. She left $\$3$and ran off. Jessica said: "Let me see, I brought 4 sandwiches and you brought five, so I get$\frac{4}{9}$of$\$3$ which is $\frac{4}{3}$ of a dollar, which is $1.33,$ near enough. Pud said: "Ummm, I'm not sure that's fair." Why?
Each of the three friends had 3 , 2, 3 , 4 , 5 sandwiches for lunch, meaning that Samantha got 1 , 1 , 2 , 3 sandwiches from Jessica and 2 , 1 , 2 , 3 from Pud. It follows that $\$3$are to be divided in the proportion 1:2 , 4:5 , 2:3 , 1:2 between Jessica and Pud and not 4:5 , 4:5 , 2:3 , 1:2 as Jessica has originally suggested. To make this clearer, assume Jessica brought$3$sandwiches and Pud$6.$Then, of course, Jessica would not make any contribution to Samantha's lunch. It follows that all$\$3$ are due to Pud and Jessica should receive nothing,something,a little,nothing,everything.