Anjali can do a certain piece of work in 24 days. Anjali and Ayushi can together do the same work in 13 days, and Anjali, Ayushi and Ankita can do the same work together in 8 days. In how many days can Anjali and Ankita do the same work?

Correct option is D

Given:

Anjali can complete the work in 24 days.

Anjali and Ayushi together can complete the work in 13 days.

Anjali, Ayushi, and Ankita together can complete the work in 8 days.

Formula Used:

Work done per day =$$\frac{1}{\text{Time taken to complete the work}}.$$​​

Solution:

​Let the work done by Anjali, Ayushi, and Ankita per day be represented as A, Y, and K respectively.

Since Anjali completes the work in 24 days, the work done by Anjali per day is:

A =$$\frac{1}{24}$$​

Since Anjali and Ayushi can complete the work in 13 days, the combined work rate of Anjali and Ayushi is:

A + Y = $$\frac{1}{13}$$​

Since Anjali, Ayushi, and Ankita can complete the work in 8 days, the combined work rate of Anjali, Ayushi, and Ankita is:

A + Y + K $$= \frac{1}{8}$$​

Subtract the equation for Anjali and Ayushi from the equation for Anjali, Ayushi, and Ankita:

(A + Y + K) - (A + Y) =$$\frac{1}{8} - \frac{1}{13}$$​

K $$= \frac{1}{8} - \frac{1}{13} = \frac{13 - 8}{104} = \frac{5}{104}$$​

So, Ankita's work rate is $$\frac{5}{104}.$$​​

A + K =$$\frac{1}{24} + \frac{5}{104}$$​

A + K $$= \frac{13}{312} + \frac{15}{312} = \frac{28}{312} = \frac{7}{78}$$​

Time taken by Anjali and Ankita to complete the work:
The time taken to complete the work is the reciprocal of their combined work rate:

​$$\text{Time} = \frac{1}{\frac{7}{78}} = \frac{78}{7}$$ days

Alternate Method:

Take LCM of 24, 13, and 8 = 312 (Assume total work = 312 units).

Anjali's 1-day work = 312 ÷ 24 = 13 units.

(Anjali + Ayushi)'s 1-day work = 312 ÷ 13 = 24 units.

(Anjali + Ayushi + Ankita)'s 1-day work = 312 ÷ 8 = 39 units.

Calculate individual contributions:

Ankita's 1-day work = 39 - 24 = 15 units.

Anjali + Ankita's 1-day work = 13 + 15 = 28 units.

Time taken by Anjali and Ankita = 312 ÷ 28.

Thus, the required answer = $$\frac{78}7$$​ days.