larry starts at the bottom of a long staircase. he climbs exactly 2/3 of the stairs. then he goes back down exactly 1/2 of the way to the bottom. from that spot he climbs exactly 2/3 of the way to the top. finally from there he climbs 6 stairs to reach the top. how many stairs are in the staircase?

18

9+10=21

To determine the total number of stairs in the staircase, we can follow Larry's movements step by step and keep track of the distance he covers.

Let's assume the total number of stairs in the staircase is "x".

1. Larry climbs exactly 2/3 of the stairs:
This means he climbs (2/3)x stairs.

2. Larry goes back down exactly 1/2 of the way to the bottom:
From the point where he had climbed 2/3 of the stairs, he descends (1/2)((2/3)x) stairs.

3. From that spot, he climbs exactly 2/3 of the way to the top:
From the point where he had descended, he climbs (2/3)((1/2)((2/3)x)) stairs.

4. Finally, from there he climbs 6 stairs to reach the top:
Adding the remaining 6 stairs, the total distance Larry covers is (2/3)((1/2)((2/3)x)) + 6.

Now, we can set up an equation to find the total number of stairs:
(2/3)((1/2)((2/3)x)) + 6 = x

Simplifying the equation step by step:

(2/3)((1/2)((2/3)x)) + 6 = x
(2/3)((1/3)x) + 6 = x
(2/6)x + 6 = x
(x/3) + 6 = x
x + 18 = 3x
18 = 2x
9 = x

Therefore, there are 9 stairs in the staircase.