Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem C. 970. (January 2009)

C. 970. Rose, Ivy and Violet decided to solve all problems in their book of mathematics problems. Rose solves a problems, Ivy solves b problems and Violet solves c problems a day. (Each problem is only solved by one of the girls.) If Rose solved 11 times as many problems a day, Ivy solved 7 times as many and Violet solved 9 times as many, they would finish the job in 5 days. If Rose solved 4 times as many, Ivy solved 2 times as many and Violet solved 3 times as many problems a day, they would take 16 days. How many days does it take them to get done with the solutions?

(5 pont)

Deadline expired on February 16, 2009.

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

Megoldás. Felírhatjuk a következő egyenletet:

5(11a+7b+9c)=16(4a+2b+3c),

amiből rendezés után kapjuk, hogy:

 (1) b=3a+c.

Ha n nap alatt készülnek el a megoldásokkal, akkor még azt is tudjuk, hogy:

5(11a+7b+9c)=n(a+b+c),

amibe behelyettesítve az 1)-ben b-re kapott kifejezést, kapjuk, hogy:

160a+80c=4na+2nc.

Végül mindkét oldalt osztva (4a+2c)-vel:

n=40.

Ez valóban lehetséges, ha pl. a=c=1 és b=4, és a feladatok száma 240.

### Statistics:

 229 students sent a solution. 5 points: 184 students. 4 points: 10 students. 3 points: 9 students. 2 points: 9 students. 0 point: 6 students. Unfair, not evaluated: 11 solutions.

Problems in Mathematics of KöMaL, January 2009