Problem K. 522. (November 2016)

K. 522. Originally, a theatre sold tickets for 1000 forints (HUF, Hungarian currency) a piece for a particular performance. There are 300 seats altogether. Since the full-priced tickets did not sell so well, the theatre issued a coupon. With the coupon, tickets were available for 25% less (so all further customers bought their tickets in this way). The income achieved from the tickets sold for the reduced price was found to be half as much as the income from those with full price. In spite of all these marketing efforts, there were still 50 tickets remaining. For what price should they be sold immediately before the performance so that the total income should reach 80% of the maximum achievable income originally planned?

(6 pont)

Deadline expired on December 12, 2016.

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

Megoldás. Összesen 250 db teljes árú, illetve kuponos jegyet adtak el. Legyen \(\displaystyle x\) db a 750 Ft-os jegyek száma, és \(\displaystyle 250–x\) az 1000 Ft-os jegyek száma. A kuponos jegyek eladásából feleannyi bevétel származott, mint a teljes árú jegyekéből, ezért \(\displaystyle (250–x) \cdot 1000 = 2 \cdot 750 \cdot x\). Innen \(\displaystyle x = 100\). Az eladott 150 db 1000 Ft-os és 100 db 750 Ft-os jegyből összesen \(\displaystyle 225\,000\) Ft bevételük származott, a cél a \(\displaystyle 300\,000\) Ft 80%-a, azaz \(\displaystyle 240\,000\) Ft-os bevétel elérése. A fennmaradó 50 jegyből így még \(\displaystyle 15\,000\) Ft-ot kell beszedni, így egy jegy ára 300 Ft lesz közvetlenül az előadás előtt.

Statistics:

120 students sent a solution.
6 points:	107 students.
5 points:	3 students.
4 points:	2 students.
3 points:	3 students.
2 points:	3 students.
0 point:	1 student.
Unfair, not evaluated:	1 solutions.

Problems in Mathematics of KöMaL, November 2016