Problem C. 1823. (October 2024)
C. 1823. Smart Pali once went to the market to sell \(\displaystyle 30\) apples. He planned to ask for one kreuzer for every three apples, i.e. he expected a total income of \(\displaystyle 10\) kreuzers. At the market, he met another person who was also selling apples. The other seller also had 30 apples for sale, but he gave two apples for one kreuzer, so he was expecting a total income of \(\displaystyle 15\) kreuzers. Smart Pali got tired of the hustle and bustle of the market and handed over his \(\displaystyle 30\) apples to the other person, instructing him to sell them so that five apples would cost two kreuzers. Pali said he would come back later for his share of the income.
\(\displaystyle a)\) If this person sold all 60 apples following Pali's instructions and kept the income he originally expected, how many kreuzers were left for Smart Pali?
\(\displaystyle b)\) For how many kreuzers should the 60 apples have been sold in order for both sellers to receive their originally planned income?
Based on a short story by Kálmán Mikszáth
(5 pont)
Deadline expired on November 11, 2024.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. \(\displaystyle a)\) Ha a \(\displaystyle 60\) almát az ember úgy adja el, ahogy Okos Pali kigondolta, vagyis \(\displaystyle 5\) almáért kér \(\displaystyle 2\) krajcárt, akkor ebből összesen \(\displaystyle 24\) krajcár bevétel származik, hiszen \(\displaystyle 60=12\cdot 5\) és \(\displaystyle 12\cdot 2=24\). Ebből levonva a saját tervezett \(\displaystyle 15\) krajcár bevételét, Okos Palinak \(\displaystyle 10\) helyett csak \(\displaystyle 9\) krajcár marad.
\(\displaystyle b)\) A tervezett összes bevétel \(\displaystyle 25\) krajcár. Ha egy alma ára \(\displaystyle x\), akkor \(\displaystyle 60\cdot x=25\), azaz \(\displaystyle \displaystyle{x=\frac{25}{60}=\frac{5}{12}}\).
A \(\displaystyle 60\) alma darábját tehát \(\displaystyle \displaystyle{\frac{5}{12}}\) krajcárért árulva az összes bevétel pontosan \(\displaystyle 25\) krajcár lett volna, és így mindketten megkapták volna a tervezett bevételüket.
Megjegyzés. Okos Pali almáinak egységára \(\displaystyle \displaystyle{\frac{10}{30}=\frac{1}{3}}\) volt, a másik piaci árus almáinak egységára pedig \(\displaystyle \displaystyle{\frac{15}{30}=\frac{1}{2}}\). Könnyen látható, hogy az eredetileg tervezett összes bevételt akkor kapták volna meg, ha az új egységár a két eredeti egységár számtani közepe lett volna, hiszen \(\displaystyle \displaystyle{\frac{\frac{1}{2}+\frac{1}{3}}{2}=\frac{5}{12}}\).
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322 students sent a solution. 5 points: 230 students. 4 points: 23 students. 3 points: 17 students. 2 points: 7 students. 1 point: 1 student. Not shown because of missing birth date or parental permission: 31 solutions.
Problems in Mathematics of KöMaL, October 2024