Vijay can do a piece of work in 4 hours. Ajay can do it in 28 hours. With the assistance of Amit, they completed the work in 3 hours. In how many hours can Amit alone do it?

Given:

Vijay can do the work in 4 hours.

Ajay can do the work in 28 hours.

Vijay, Ajay, and Amit together can complete the work in 3 hours.

Calculation:

Total Work (LCM of 4, 28, and 3):

Prime factorization of 4 = 2²

Prime factorization of 28 = 2² × 7

Prime factorization of 3 = 3

LCM(4, 28, 3) = 2² × 3 × 7 = 4 × 3 × 7 = 12 × 7 = 84 units (Total Work)

Efficiency of Vijay (E_V) = Total Work / Time taken by Vijay = 84 / 4 = 21 units/hour

Efficiency of Ajay (E_A) = Total Work / Time taken by Ajay = 84 / 28 = 3 units/hour

Efficiency of (Vijay + Ajay + Amit) (E_V+A+Am) = Total Work / Time taken by (Vijay + Ajay + Amit) = 84 / 3 = 28 units/hour

Efficiency of Amit (E_Am):

We know that E_V+A+Am = E_V + E_A + E_Am

28 = 21 + 3 + E_Am

28 = 24 + E_Am

E_Am = 28 - 24 = 4 units/hour

Time taken by Amit = Total Work / E_Am = 84 / 4 = 21 hours

∴ Amit alone can do the work in 21 hours.