Paul and Archie take 6 days to assemble a toy plane. Archie and Howard take 5 days to build the same plane while Paul and Howard take only 3 days. How many complete days would the three of them take to build the same toy plane?
Let the plane be 1 P.

Fraction of the plane assembled by Paul and Archie per day
= ¹₆ P

Fraction of the plane assembled by Howard and Archie per day
= ¹₅ P

Fraction of the plane assembled by Paul and Howard per day
= ¹₃ P

Combined rate
= (¹₆ + ¹₅ + ¹₃) ÷ 2
= (⁵₃₀ + ⁶₃₀ + ¹⁰₃₀) ÷ 2
= ²¹₃₀  ÷ 2
= ²¹₃₀  x ¹₂
= ⁷₂₀ P 

Number of days taken by the three of them to build the same toy plane
= 1 ÷ ⁷₂₀
= 1 x ²⁰₇
= 2.86

Since the number of days needed is 2.86 days, therefore the total number of complete days needed to build the toy plane is 3.

Answer(s): 3 days