Problems of the Week – December 4 to December 8

Lower Elementary:
Question: Alexis brings 2¹/₂ sandwiches to a picnic. Brett brings twice as many sandwiches as Alexis. How many sandwiches do Alexis and Brett have altogether?
Answer: 7¹/₂ sandwiches
Solution: If Alexis brings 2¹/₂ sandwiches, then Brett brings 2¹/₂ + 2¹/₂ = 5 sandwiches. If we add up Alexis’s sandwiches and Brett’s sandwiches, we get 2¹/₂ + 5 = 7¹/₂ sandwiches.

Upper Elementary:
Question: If Cody has 20 game tokens to spend at the arcade. He spends $2.40 to get 12 more. How much did Cody’s game tokens cost altogether?
Answer: $6.40
Solution: Since Cody spends $2.40 to buy 12 tokens, each token costs $2.40 ÷ 12 = $0.20. Cody has 20 + 12 = 32 tokens altogether, so his game tokens cost $0.20 × 32 = $6.40.

Middle School:
Question: Darla has a number of movies in her collection; ¹/₄ of the movies are comedies, ¹/₃ are documentaries, ¹/₆ are horror movies, and the rest are dramas. The total number of movies in Darla’s collection is between 20 and 30. Exactly how many documentaries does she have?
Answer: 8 documentaries
Solution: Darla can’t own half a movie—that doesn’t make any sense. So, since we know Darla’s movie collection can be divided evenly by 4, 3, and 6, we’re looking for a number between 20 and 30 that is divisible by those three numbers. The only number that fits the bill is 24. So, since ¹/₃ of 24 is 8, Darla must have 8 documentaries.

Algebra and Up:
Question: A syllogism is a logic puzzle that asks you to draw a conclusion based on the logical values or truth values of a series of statements, like these:

1) I’m thinking of a shape that is concave.
2) All triangles are convex.
3) A shape is convex if and only if it is not concave.

What conclusion can you draw about the shape I’m thinking of based on the statements above?
Answer: I am not thinking of a triangle.
Solution: If we start with the third statement and apply it to the first, we can conclude that I am not thinking of a convex shape. If we know that I’m not thinking of a convex shape, then we can conclude that I cannot possibly thinking of a triangle because that would contradict the second statement—all triangles are convex.