Anjali can do a certain piece of work in 16 days. Anjali and Ayushi can together do thesame work in 10 days, and Anjali, Ayushi and Ankita can do the same work together in 8days. In how many days can Anjali and Ankita do the same work?

Correct option is A

​Given:

Anjali can complete the work in 16 days.

Anjali and Ayushi together can complete the work in 10 days.

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

Solution:

Let the work done by each person per day be their rate of work.

Anjali's rate of work = $$\frac{1}{16}$$​​

Anjali and Ayushi's combined rate of work = $$\frac{1}{10}$$​​

Anjali, Ayushi, and Ankita's combined rate of work = $$\frac{1}{8}$$​​

Let Ayushi's rate of work be  $$\frac{1}{x}$$​​
So,

​$$\frac{1}{16} + \frac{1}{x} = \frac{1}{10}$$​​

​$$\frac{1}{x} = \frac{1}{10} - \frac{1}{16} = \frac{16 - 10}{160} = \frac{6}{160} = \frac{3}{80}$$​​

Thus, Ayushi's rate of work is $$\frac{1}{\frac{80}{3}} = \frac{3}{80}$$​ per day.

Let Ankita's rate of work be $$\frac{1}{y}$$​​

So,

​$$= \frac{1}{16} + \frac{3}{80} + \frac{1}{y} = \frac{1}{8}$$​​

​$$= \frac{1}{10} + \frac{1}{y} = \frac{1}{8}$$​​

​$$\frac{1}{y} = \frac{1}{8} - \frac{1}{10} = \frac{10 - 8}{80} = \frac{2}{80} = \frac{1}{40}$$​​

So, Ankita's rate of work is$$\frac{1}{40}$$​ per day.

the combined rate of work for Anjali and Ankita:

​$$= \frac{1}{16} + \frac{1}{40}$$​​

​$$= \frac{5}{80} + \frac{2}{80} = \frac{7}{80}$$​​

Thus, the time taken for Anjali and Ankita to complete the work together is:

Time = $$\frac{1}{\frac{7}{80}} = \frac{80}{7} \, \text{days}$$​​

Alternate Solution: 

Let the total work be the lcm of 16, 10, 8 = 80 units

Anjali's Work rate =$$\frac{80}{16}$$​ = 5 units/day

Anjali + Ayushi Combined work rate =$$\frac{80 }{10}$$​= 8 units/day

Ayushi work rate = 8 - 5 = 3 units/day

Anjali + Ayushi + Ankita combined work rate = $$\frac{80}{8}$$​ = 10 units/day

Ankita work rate = 10 - 8 = 2 units/day

Combined Work Rate of Anjali and Ankita = 5 + 2 = 7 units/day

Time taken by Anjali and Ankita =$$\frac{80}{7} = \frac{80}{7}$$​ days