Problem K. 670. (November 2020)

K. 670. Grandma bought two candles: the red candle was 2 cm longer than the blue one. On All Saints' Day she lit the red candle at 5:30 p.m. then she lit the blue candle at 7 p.m. and let them burn all the way down. The two candles were equal in length at 9:30 p.m. The blue candle burned out at 11 p.m and the red one finished at 11:30 p.m. What was the initial length of the red candle?

(6 pont)

Deadline expired on December 10, 2020.

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

Megoldás. A piros gyertya \(\displaystyle 6\) óra alatt, a kék gyertya \(\displaystyle 4\) óra alatt égett le. A piros gyertyából \(\displaystyle 4\) óra alatt \(\displaystyle 2\) cm-rel több égett le, mint a kékből \(\displaystyle 2\) óra \(\displaystyle 30\) perc alatt. Legyen a piros gyertya eredeti hossza \(\displaystyle x\) cm, ekkor a kék gyertya hossza \(\displaystyle x–2\) cm. A piros gyertyából \(\displaystyle 1\) óra alatt \(\displaystyle x/6\), \(\displaystyle 4\) óra alatt \(\displaystyle 4x/6\) cm-nyi ég el. A kék gyertyából egy óra alatt \(\displaystyle \frac{x-2}{4}\), \(\displaystyle 2,5\) óra alatt \(\displaystyle \frac{2,5(x-2)}{4}\) cm ég el. A megadott összefüggés alapján \(\displaystyle \frac{2,5(x-2)}{4}+2=\frac{4x}{6}\). Innen rendezéssel kapjuk, hogy \(\displaystyle x= 18\). Tehát a piros gyertya eredetileg \(\displaystyle 18\) cm hosszú volt.

Statistics:

128 students sent a solution.
6 points:	82 students.
5 points:	8 students.
4 points:	4 students.
3 points:	7 students.
2 points:	12 students.
1 point:	8 students.
0 point:	2 students.
Unfair, not evaluated:	4 solutionss.
Not shown because of missing birth date or parental permission:	1 solutions.

Problems in Mathematics of KöMaL, November 2020