Anjani can do a certain piece of work in 18 days. Anjani and Khushbu can together dothe same work in 14 days, and Anjani, Khushbu and Sushmita can do the same worktogether in 9 days. In how many days can Anjani and Sushmita do the same work?

Correct option is C

Given:

Anjani can complete the work in 18 days.

Anjani and Khushbu together can complete the work in 14 days.

Anjani, Khushbu, and Sushmita together can complete the work in 9 days.

Formula Used:

Work rate = $$\frac{1}{\text{Number of days}}$$​​

Solution:

If A, K, and S are the rates of work for Anjani, Khushbu, and Sushmita respectively, then:

A = $$\frac{1}{18}$$​​

A + K = $$\frac{1}{14}$$​​

A + K + S = $$\frac{1}{9}$$​​

Now,

​$$(A + K + S) - (A + K) = \frac{1}{9} - \frac{1}{14}$$​​

$$S = \frac{1}{9} - \frac{1}{14}$$​​

​$$S = \frac{14 - 9}{9 \times 14}$$​​

​$$S = \frac{5}{126}$$​​

Now, 

$$A + S = \frac{1}{18} + \frac{5}{126}$$​​

​$$A + S = \frac{7}{126} + \frac{5}{126}$$​​

​$$= \frac{12}{126} = \frac{2}{21}$$​​

The time taken by Anjani and Sushmita together to complete the work is the reciprocal of their work rate:

Time = $$\frac{21}{2}\text{ days}$$ 

Alternate Solution: 

Total work = LCM of 18, 14, 9 = 126

Efficiency of Anjani = $$\frac{126}{18} = 7$$

​​Efficiency of Anjani + Khushbu = $$\frac{126}{14} = 9$$

Efficiency of Anjani + Khushbu + Sushmita = $$\frac{126}{9} = 14$$ 

Efficiency of Khushbu = 9 - 7 = 2 

Efficiency of Anjani + Sushmita = 14 - 2 = 12 

Time taken to complete the work by Anjani + Sushmita;

= $$\frac{126}{12} = \frac{21}2 \ days$$​​