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?

Answer:1 ¹/₉ hoursStep-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 + 2C3 = 5C + 5M1 = L + C + MDouble the third equation and subtract the first equation from it:2 = 2L + 2C + 2M1 = 2L + 2C1 = 2MM = 1/2Plugging into the second and third equations, we get:C = 1/10L = 2/5Therefore, 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 = 10/9t = 1 ¹/₉ hoursIt takes them 1 ¹/₉ hours, or 1 hour 6 minutes 40 seconds.