Larry and Curly can build a bookcase together in $2$ hours. Curly and Moe can build a bookcase together in $1 / frac{2}{3}$ hours. Larry, Curly, and Moe can build a bookcase together in $1$ hour. How many hours would it take for Larry and Moe to build a bookcase together?

1 ¹/₉ hours

Step-by-step explanation:

Let's say L is Larry's speed, C is Curly's speed, and M is Moe's speed.

1 = 2 (L + C)

1 = 1 ⅔ (C + M)

1 = 1 (L + C + M)

Solve the system of equations. First, simplify the equations:

1 = 2L + 2C

3 = 5C + 5M

1 = L + C + M

Double the third equation and subtract the first equation from it:

2 = 2L + 2C + 2M

1 = 2L + 2C

1 = 2M

M = 1/2

Plugging into the second and third equations, we get:

C = 1/10

L = 2/5

Therefore, the time it takes Larry and Moe together is:

1 = t (L + M)

t = 1 / (L + M)

t = 1 / (2/5 + 1/2)

t = 1 / (4/10 + 5/10)

t = 1 / (9/10)

t = 10/9

t = 1 ¹/₉ hours

It takes them 1 ¹/₉ hours, or 1 hour 6 minutes 40 seconds.