PrepTest 136 · Section 2 · Question 6

The Red Hats on Bus 7

On the field trip bus, Marcus counts the back row. "Most kids back here have red hats," he reports. "So at least one chess club kid has a red hat." Nadia raises an eyebrow. "How do you know any chess club kids are back there?" Marcus shrugs. Priya taps her juice box. "Here is the key that makes it click. If most of the back-row kids are also in chess club, then the red hats and the chess kids have to bump into each other. More than half plus more than half of the same row is more than the whole row. Somebody is in both groups." Lulu counts on her fingers to check: ten kids, six red hats, six chess players. Twelve is bigger than ten! At least two kids are red-hat chess players. Owen adjusts his own red hat and admits he plays chess.

Marcus: Most kids in the back row of the bus wear red hats. So at least one kid in the chess club wears a red hat.

Which assumption, if added, would make Marcus's conclusion follow for sure?

The Case of the Muddy Boots

A witness says most people who walked through the east gate had muddy boots. The prosecutor concludes at least one band member had muddy boots. The defense lawyer frowns. "Where is the link to the band?" The prosecutor smiles and adds one more fact: "Most people who walked through the east gate were band members." The judge does the math out loud. "More than half had mud. More than half were band members. Both groups are inside the same crowd, so they must overlap. At least one band member had muddy boots. That fact clicks the conclusion into place." Mr. Whitfield, sitting in the gallery, polishes his own clean boots very quickly.