Attached are Mathnasium's Problems of the Week for the week of January 27! Feel free to have your students solve the problems (solutions included) and ask them how they did it!

**Lower Elementary:**

*Question: *Isaac replaces the six strings on his guitar once every 3 months. How many new guitar strings will he need over the course of 4 years?

*Answer: *96 strings

*Solution: *Since 3 months goes into 12 months 4 times, Isaac replaces 6 + 6 + 6 + 6 = 24 strings in one year. In four years, he replaces 24 + 24 + 24 + 24 = 96 strings. So, Isaac needs 96 new strings.

**Upper Elementary:**

*Question: *Adalyn is making a sign to carry in a parade. She uses a piece of poster board that is 4 feet long and 3 feet wide and folds it in half, hamburger style. What are the area and perimeter of Adalyn’s sign?

*Answer: *The area is 6 square feet and the perimeter is 10 feet.

*Solution: *If Adalyn folds the poster board hamburger style, then the new length of the sign will be half of 4 = 2 feet. Since its dimensions are now 2 feet by 3 feet, that means that the area is 2 × 3 = 6 square feet and the perimeter is 2 + 2 + 3 + 3 = 10 feet.

**Middle School:**

*Question: *Luke has 154 tulips and 105 daffodils. He wants to make the greatest number of bouquets that he can with the same number of tulips and the same number of daffodils in each, and he doesn’t want to have any leftover flowers when he’s done. How many tulips and how many daffodils should be in each bouquet?

*Answer: *22 tulips and 15 daffodils

*Solution: *To solve this problem, we need to find the common factors of 154 and 105: 154 is divisible by 1, 2, 7, 11, 14, 22, 77, and 154, and 105 is divisible by 1, 3, 5, 7, 15, 21, 35, and 105. Their GCF is 7. So, the greatest number of bouquets with the same flowers in each that Luke can make is 7. If we divide the tulips among 7 bouquets, 22 go in each. If we divide the daffodils, 15 go in each.

**Algebra and Up:**

*Question: *Dave, Tom, and Star agree to meet at “the coffee shop by the park.” Unfortunately, there are three coffee shops by the park, and they forget to specify which one. What’s the probability that Dave, Tom, and Star will all go to different coffee shops?

*Answer: *2 out of 9

*Solution: *Dave has a ^{1}/_{1} probability of going to whichever coffee shop he goes to. Tom then has a ^{2}/_{3} probability of going to either of the two coffee shops where Dave didn’t go. If that happens, then there’s a ^{1}/_{3} probability that Star will go to the remaining coffee shop. So, the probability that they all go to different coffee shops is ^{1}/_{1} × ^{2}/_{3} × ^{1}/_{3} = ^{2}/_{9}.