March 20, 2017

State competitions for 2017 are in full swing! Let’s see if we can hang with the state competitors by trying a few problems from the 2016 state competition.

A bag contains 4 blue, 5 green and 3 red marbles. How many green marbles must be added to the bag so that 75 percent of the marbles are green?

[Sprint #7]       

There are 4 + 5 + 3 = 12 marbles. If we let x equal the number of green marbles added, then we can set up the equations (5 + x) ÷ (12 + x) = 3/4. Solving for x, we get 20 + 4x = 36 + 3x → x = 16.

 

Alex can run a complete lap around the school track in 1 minute, 28 seconds, and Becky can run a complete lap in 1 minute, 16 seconds. If they begin running at the same time and location, how many complete laps will Alex have run when Becky passes him for the first time?

[Sprint #15]      

Alex runs at a rate of 88 seconds per lap and Becky runs at a rate of 76 seconds per lap. Let x = the length of the lap. After one second, Alex has run 1/88 of a lap and Becky has run 1/76 of a lap. The difference is 1/76 – 1/88 = 3/1672. At a separation of 3/1672 laps per second, it will take 1672/3 seconds to reach one lap difference. In this time, Alex will have run 1672/3 ÷ 88 = 6 1/3 laps or 6 complete laps.

 

The student council at Round Junior High School has eight members who meet at a circular table. If four officers must sit together in any order, how many distinguishable circular seating orders are possible? Two seating orders are distinguishable if one is not a rotation of the other.

[Sprint #22]      

For the four officers, there are 4! = 24 ways for seating. For the four other members, there are also 24 ways for seating. Combining these two, the total number of possible seating orders is 24 × 24 = 576.

 

Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2
contains ONLY PROBLEMS.