Problem K. 571. (January 2018)

K. 571. The headmaster of a school issued a decree that the legs of students' trousers must not be shorter than one fifth of their height. In the investigation of Sam's trousers length, the ethical committee concluded that the legs of his trousers were shorter than allowed, by exactly \(\displaystyle \frac{2}{7}\) of the allowed minimum length. In addition, they also established that a 3-cm increase of the length of his trousers legs would still make it 20% shorter than allowed. How tall is Sam?

(6 pont)

Deadline expired on February 12, 2018.

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

Megoldás. Legyen Samu magassága \(\displaystyle x\) cm. Ekkor a megengedett nadrágszár-hossz \(\displaystyle \frac x5\), Samu nadrágjának hossza tehát ennek \(\displaystyle \frac57\) része, azaz \(\displaystyle \frac x7\). Ha ehhez hozzáadunk 3 cm-t, akkor \(\displaystyle \frac x5\) \(\displaystyle 80\)%-át, azaz \(\displaystyle \frac{4x}{25}\)-öt kapunk. Tehát az alábbi összefüggés érvényes: \(\displaystyle \frac x7+3=\frac{4x}{25}\). Rendezve: \(\displaystyle 25x+3\cdot7\cdot25=4x\cdot7\), amiből \(\displaystyle 525=3x\), vagyis \(\displaystyle x = 175\) cm. Tehát Samu \(\displaystyle 175\) cm magas.

Statistics:

146 students sent a solution.
6 points:	103 students.
5 points:	17 students.
4 points:	10 students.
3 points:	5 students.
2 points:	3 students.
1 point:	2 students.
0 point:	4 students.
Unfair, not evaluated:	2 solutionss.

Problems in Mathematics of KöMaL, January 2018