Math Olympiad
posted by Usher .
Lara starts at the bottom of a long staircase. She climbs exactly 2/3 of the stairs. Then, she goes back down exactly 1/2 of the way to the bottom. From that spot, she climbs exactly 2/3 of the way to the top. Finally from there, she climbs 6 stairs to reach the top. How many stairs are in the staircase?

Assume that there are x steps in all. Lara goes up (2x/3) steps and returns half the number of steps taken to 2x/6 or to (x/3) up from start.
After this there are (2x/3) steps still to go of which she once again takes (2/3)(2x/3)=(4x/9) steps to reach a spot which 6 steps away from the top.
Hence, x/3 + 4x/9 + 6 = x or x=27 steps.