Answers to Problems of the Week - June 14 to June 20

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: Each soccer team has the following players: 1 goalie, 4 defenders, 4 midfielders, and 2 forwards. How many players are on 10 soccer teams?

Answer:  110 players

Solution:  Instead of adding each position individually, we can find the number of players on each team first. There are 1 + 4 + 4 + 2 = 11 players on each soccer team. So, on 10 teams, there are 11 × 10 = 110 players!

Upper Elementary:

Question: Kara and Spencer are practicing their golf swings. They bring 18 golf balls to the course. Kara hits all 18 first, then retrieves all but one. Then it’s Spencer’s turn to hit all the remaining golf balls and retrieve them, but he also loses one. If this pattern continues, who will take the 10th turn, and how many golf balls will he or she hit? Find a way to solve the problem without counting each turn.

Answer:  Spencer hits 9 golf balls.

Solution:  Kara has the odd numbered turns and Spencer has the even numbered turns. Since 10 is an even number, Spencer must have the 10th turn. By the 10th turn, only 9 (not 10) golf balls have gotten lost because the balls get lost at the end of the turn.

Middle School:

Question: Hank and Benjamin play on the same baseball team. Last year, they both had a batting average of .240. Benjamin’s batting average increased by the same amount that Hank’s batting average decreased. If Benjamin has a batting average of .270 this season, then what fractional part of Benjamin’s new batting average is Hank’s new batting average?

Answer:  ⁷/₉

Solution:  Both of their batting averages change by 30%. That means that Hank’s new batting average is 70% of his old average, or 0.7 × .240 = .210. Next, we can find out what fractional part Hank’s batting average is of Benjamin’s by dividing .210 by .270. First, we multiply both numbers by 100 to get ²¹/₂₇, then we can reduce the fraction to ⁷/₉.

Algebra and Up:

Question: Kayla and Margot are both running toward a soccer ball. Kayla is 28 meters away and Margot is 33 meters away. If Kayla can run 2¹/₂ meters per second and Margot can run 2³/₄ meters per second, then how much sooner does the player who reaches the ball first get there?

Answer:  Kayla reaches the ball ⁴/₅ of a second faster than Margot.

Solution:  If Kayla runs 2¹/₂ meters per second, then it takes her 28 meters ÷ 2¹/₂ = 11¹/₅ seconds to reach the ball. Margot reaches the ball in 33 ÷ 2³/₄ = 12 seconds. This means that Kayla got there 12 – 11¹/₅ = ⁴/₅ of a second before Margot.

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

     The perimeter of isosceles triangle ABC is 19. Triangle $$ABD  and triangle $$AEC  are formed by joining vertices of triangle $$ABC with the midpoints of their opposite sides.

The perimeter of triangle $$ AEC is $$ 2 units greater than the perimeter of triangle ABD. What is the length of $$BC?   

Answer: 7