Problem K. 696. (September 2021)

K. 696. In the left front pocket of my jeans I have twice as much money as in the right front pocket, and one third as much as in the right back pocket. I moved 30 forints (HUF, Hungarian currency) from the right front pocket to the left front pocket, and also moved 180 forints from the right back pocket to the left front pocket. Now I have 3 times as much money in the left front pocket as the amount remaining in the right front pocket. How much money did I have initially in each pocket?

(5 pont)

Deadline expired on October 11, 2021.

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

Megoldás. Legyen a jobb első zsebemben \(\displaystyle x\) Ft, ekkor a bal elsőben \(\displaystyle 2x\) Ft, a jobb hátsóban pedig \(\displaystyle 6x\) Ft. Összesen \(\displaystyle 9x\) Ft-om van a három zsebemben. A pénzek áthelyezése után a bal első zsebemben \(\displaystyle 2x+210\) Ft-om lesz, a jobb elsőben \(\displaystyle x-30\) Ft, míg a jobb hátsóban \(\displaystyle 6x-180\) Ft.

Tudjuk, hogy \(\displaystyle 3(x-30) = 2x+210\), ahonnan \(\displaystyle x = 300\).

Tehát kezdetben a bal első zsebemben 600 Ft, a jobb első zsebemben 300 Ft és a jobb hátsó zsebemben 1800 Ft volt.

Miután áttettem a pénzeket a jobb első zsebemben 810 Ft, a bal elsőben 270 Ft, a jobb hátsóban pedig 1620 Ft lett, így valóban teljesül a feltételek: \(\displaystyle 3\cdot 270 = 810\).

Statistics:

171 students sent a solution.
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3 points:	3 students.
2 points:	2 students.
1 point:	2 students.
0 point:	1 student.
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Problems in Mathematics of KöMaL, September 2021