Answers to Problems of the Week - June 7 to June 12

Rewards for Problems of the week:

5 stars for the correct answer at your level; 10 stars for the next level; 20 stars each for 2+ levels up.

Lower Elementary: 

Question: A clown makes 10 balloon giraffes, 16 balloon poodles, 5 balloon monkeys, and a balloon octopus. How many balloon animals does the clown make?

Answer:  32

Solution:  Let’s add the animals together, one species at a time. There are 10 giraffes and 16 poodles, and 10 + 16 = 26. To add the 5 monkeys, we can count 4 up to 30 and 1 more up to 31. One more animal—the octopus—makes 32!

Upper Elementary: 

Question: A spinning prize wheel has four different varieties of prizes. 20% of the prizes are items from a prize case. 30% of the prizes are tickets to local events and museums. 40% of the prizes are cash. The other 2 prizes are all-expense-paid trips. How many different prizes are on the wheel?

Answer:  20

Solution:  First we need to know what percentage of the prizes are all-expense-paid trips. The rest of the prizes add up to 20% + 30% + 40% = 90%, so the other 10% of the prizes are all-expense-paid trips. If 2 is 10% of the total, and 10% × 10 = 100%, then the total number of prize options is 2 × 10 = 20.

Middle School: 

Question: A dunk tank is ⁴/₅ full. After Tom gets dunked, ¹/₄ of the water splashes out. After that, there are 450 gallons of water left. How many gallons of water does the dunk tank hold when it’s full?

Answer:  750 gallons

Solution:  If ¹/₄ of the water in a ⁴/₅–full tank leaks out, then the tank is now ³/₅ full. If 450 gallons makes up ³/₅ of the tank’s capacity, then ¹/₅ of it must be 450 ÷ 3 = 150 gallons. 150 gallons, 5 times makes ⁵/₅ of the tank’s capacity, and 150 × 5 = 750 gallons.

Algebra and Up:

Question: The paper cone of a snowcone is 5 inches deep and 3 inches wide at its opening. If the cone is filled to the top with snow and then a perfect hemisphere of snow is placed on top, what is the volume of the snow in cubic inches?

Answer:  6π inches³

Solution:  The volume of the hemisphere of snow is equal to half of ⁴/₃πr³, or ²/₃πr³ (we can do this because half of ⁴/₃ is ²/₃). The volume of the cone is πr² × ^h/₃. We know that h = 5 and r = 1.5, so altogether, the volume of the snowcone is:

(²/₃ × π × 1.5³) + (π × 1.5² × ⁵/₃)

Altogether, the above equals 2.25π + 3.75π = 6π, or approximately 18.85.

Challenge problem to take home. 10 stars for the correct answer. 

+\red{R}+\yellow{E}?

In the circle puzzle below, solid arrows denote multiplication while dashed arrows denote addition. Both marked circles represent the same number, and all the other circles contain distinct integers.

What is the value of the marked circles?

Answer: - 4