A and B can complete a piece of work in 10 days and 12 days, respectively. If they work on alternate days beginning with A, then in how many days will the work be completed?

Correct option is D

Given:

A can complete the work in 10 days.

B can complete the work in 12 days.

They work on alternate days, starting with A.

Solution:

A's work rate $$\frac{1}{10}$$ of the work per day.

​B's work rate $$\frac{1}{12}$$ of the work per day.

Working on Alternate Days:

Work done by A in 1 day: $$\frac{1}{10}$$ 

​Work done by B in 1 day: $$\frac{1}{12}$$

​​Work done in 2 days $$\frac{1}{10}$$ + $$\frac{1}{12}$$ = $$\frac{11}{60}$$

​​In every 2 days, they complete $$\frac{11}{60}$$ of the work

2 - day cycles are needed to complete the entire work:

After 5 cycles (10 days):

Total work done = 5 $$\times$$ $$\frac{11}{60}$$ = $$\frac{55}{60}$$ = $$\frac{11}{12}$$

​​Remaining work after 10 days:

Remaining work = 1- $$\frac{11}{12}$$ = $$\frac{1}{12}$$

​​On the 11th day, A will work-

A's work rate for 1 day = $$\frac{1}{10}$$

​​Since the remaining work $$\frac{1}{12}$$ is less than $$\frac{1}{10}$$, A will complete the task on the 11th day.

​Time taken by A to complete $$\frac{1}{12}$$ of the work

Time = $$\frac{\frac{1}{12}}{\frac{1}{10}}$$ = $$\frac{10}{12}$$ = $$\frac{5}{6}$$ day

​The work will be completed in 10 $$\frac{5}{6}$$ days ​