A and B together can complete a piece of work in 25 days, B and C together can complete the same piece of work in 36 days, while C and A together can complete it in 30 days. If A, B, C, and D together can complete this piece of work in 18 days, then in how many days can D alone complete this piece of work?

Correct option is C

Given:

A and B together can complete the work in 25 days.

B and C together can complete the work in 36 days.

C and A together can complete the work in 30 days.

A, B, C, and D together can complete the work in 18 days.

Solution:

Work rate of A + B $$= \frac{1}{25}$$​​

Work rate of B + C $$= \frac{1}{36}$$​​

Work rate of C + A =$$\frac{1}{30}$$​​

Work rate of A + B + C + D =$$\frac{1}{18}$$​​

Now, 

(A + B) + (B + C) + (C + A) =$$\frac{1}{25} + \frac{1}{36} + \frac{1}{30}$$​

2A + 2B + 2C =$$\frac{1}{25} + \frac{1}{36} + \frac{1}{30}$$​​

2(A + B + C) =$$\frac{72}{1800} + \frac{50}{1800} + \frac{60}{1800} = \frac{182}{1800}$$​

$$A + B + C= \frac{1}{2} \times \frac{182}{1800} = \frac{91}{1800}$$........(1)

A + B + C + D $$= \frac{1}{18}$$........(2) 

Subtracting (1) and (2), we get

D =$$\frac{1}{18} - \frac{91}{1800}$$

D = $$\frac{100}{1800} - \frac{91}{1800} = \frac{9}{1800} = \frac{1}{200}$$​

The rate of work of D is $$\frac{1}{200}$$​, meaning D can complete the work in 200 days.

Alternate Method:

​Efficiency of A + B + C =$$\frac{36+25+30}2=\frac{91}2=45.5$$

​​Efficiency of D = 50 - 45.5 = 4.5

Work done by D =$$\frac{900}{4.5}=200$$ days​