Problem K. 593. (September 2018)

K. 593. A disobedient class were trying to boycott the PE lesson. Therefore, when given the task of throwing a ball along the running track as far as they can, they threw the balls towards the fence instead, in the hope of getting the balls across the fence and halting the lesson while the balls are collected. 1/6 of the students threw 5 balls each, half the class threw 4 balls per student, one student threw 6 balls and all the rest of them threw 2 balls each. They managed to throw 75% of the balls over the fence. Since it took a lot of of time to get all these 66 balls back, the students were punished. They were made to run 3 laps each on the 200-metre running track. How many kilometres did they run altogether?

(6 pont)

Deadline expired on October 10, 2018.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. Az osztály tanulóinak számát jelölje \(\displaystyle x\). Ekkor a dobott labdák száma \(\displaystyle \frac x6\cdot5+\frac x2\cdot 4+1\cdot6+\left(x-\frac x6-\frac x2-1\right)\cdot2=\frac x6\cdot(5+12+12-2-6)+4=\frac72x+4\). A kidobott labdák ennek \(\displaystyle 3/4\) részét adják, azaz \(\displaystyle \frac34\cdot\left(\frac72x+4\right)=66\), amiből \(\displaystyle \frac{21}{8}x=66-3=63=3\cdot21\), és így \(\displaystyle x=3\cdot8=24\). Mivel a tanulók fejenként \(\displaystyle 600\) métert futottak, ezért összesen \(\displaystyle 24\cdot600=14400\) m, vagyis \(\displaystyle 14,4\) km lett a megtett táv.

Statistics:

244 students sent a solution.
6 points:	185 students.
5 points:	12 students.
4 points:	5 students.
3 points:	2 students.
2 points:	2 students.
1 point:	1 student.
0 point:	1 student.
Unfair, not evaluated:	4 solutionss.
Not shown because of missing birth date or parental permission:	32 solutions.

Problems in Mathematics of KöMaL, September 2018