Problem K. 572. (January 2018)

K. 572. Tom Sawyer and Huckleberry Finn were painting the fence together. It would take Tom 3 hours to paint the whole fence alone, and it would take Huck 4 hours to do it alone. However, when they work together, their working speed decreases by 20% since they are doing pranks on each other continually. The two of them started working at noon, but after a while Huck was getting bored, so he decided to go fishing instead. Tom spent 10 minutes trying to persuade him to continue (during that time, neither of them did any painting at all), without success. So he threw a dead rat at Huck, and finished the job alone. He was done at 2:34. When did Huckleberry Finn stop painting?

(6 pont)

Deadline expired on February 12, 2018.

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

Megoldás. Tom összesen \(\displaystyle 2\) óra \(\displaystyle 24\) percen, azaz \(\displaystyle \frac{12}{5}\) órán át festett. A közösen munkával töltött idő legyen \(\displaystyle x\) óra. Ekkor, mivel Tom \(\displaystyle 1\) óra alatt a munka harmadát, Huck pedig a negyedét végezné el, az alábbi összefüggés írható fel: \(\displaystyle \left(\frac x3+\frac x4\right)\cdot0,8+\left(\frac{12}{5}-x\right)\cdot\frac13=1\). Rendezve: \(\displaystyle \frac{7}{12}\cdot\frac45\cdot x+\frac{12}{15}-\frac13x=1\), majd \(\displaystyle 28x+48-20x=60\), vagyis \(\displaystyle 8x=12\), ahonnan \(\displaystyle x=\frac32\) adódik. Tehát Huckleberry Finn \(\displaystyle 13\) óra \(\displaystyle 30\) perckor hagyta abba a festést.

Statistics:

99 students sent a solution.
6 points:	74 students.
5 points:	8 students.
4 points:	2 students.
3 points:	8 students.
2 points:	4 students.
1 point:	3 students.

Problems in Mathematics of KöMaL, January 2018