Problem K. 271. (December 2010)

K. 271. Alex visited his relatives. He took the train and then changed for a bus. The train and the bus travelled at an average speed of 80 km/h and 45 km/h, respectively. The journey by train took the same time as the journey by bus, but Alex covered a 140 km longer distance by train than by bus. Find the total distance travelled by Alex in kilometres.

(6 pont)

Deadline expired on January 10, 2011.

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

Megoldás. Ha Sanyi \(\displaystyle t\) ideig utazott vonattal és ugyanennyi, azaz \(\displaystyle t\) ideig busszal, akkor közben \(\displaystyle s+140\) km-t tett meg vonattal és \(\displaystyle s\) km-t busszal. Az átlagsebességek \(\displaystyle \frac{s+140}{t}=80\) és \(\displaystyle \frac st=45\). Az egyenletrendszert megoldva a busz átlagsebességéből \(\displaystyle t=\frac s{45}\), amivel a vonat átlagsebessége \(\displaystyle 80=\frac{45(s+140)}{s}\). Ezt rendezve \(\displaystyle s=180\) km. Sanyi 180 km-t utazott busszal és 320 km-t vonattal, ezért összesen 500 km-t utazott.

Statistics:

271 students sent a solution.
6 points:	213 students.
5 points:	24 students.
4 points:	10 students.
3 points:	8 students.
2 points:	2 students.
1 point:	1 student.
0 point:	1 student.
Unfair, not evaluated:	12 solutionss.

Problems in Mathematics of KöMaL, December 2010