Let’s assume that A and B together can complete the work in x days. So, the amount of work done by A and B together in one day is:
Work done by A and B together in one day = 1/x
As per the given information, A and B work together for 2 days, so the amount of work done by them in these two days is:
Work done by A and B together in 2 days = 2/x
Now, the remaining work is completed by A alone in 2 more days. Let’s assume that A alone can complete the work in y days. So, the amount of work done by A alone in one day is:
Work done by A alone in one day = 1/y
Since A worked alone for 2 more days, the total amount of work done by A in those 2 days is:
Total work done by A in 2 days = 2/y
Therefore, the total amount of work done in the first 4 days (2 days by A and B together and 2 days by A alone) is:
Total work done in first 4 days = 2/x + 2/y
As per the given information, the work is completed in 5 days in total. Therefore, the amount of work done by B alone in 3 days is:
Work done by B alone in 3 days = 1 – (2/x + 2/y)
Now, we know that B worked for only 2 days. Therefore, the total amount of work done by B in those 2 days is:
Total work done by B in 2 days = 2/x – Work done by A and B together in 2 days
Total work done by B in 2 days = 2/x – 2/x = 0
Since the work done by B alone in 3 days is equal to the remaining work after 4 days, we can say that:
Work done by B alone in 3 days = Work done by A alone in 2 days
Therefore, we can equate the above two expressions and get:
1 – (2/x + 2/y) = 2/y
Simplifying the above equation, we get:
2/x = 1/y
Multiplying both sides by yx, we get:
2 = xy
We are given that A and B together can complete the work in 3 days. Therefore:
1/x = Work done by A and B together in one day = 1/3
Substituting the above value of x in the equation 2 = xy, we get:
2 = y/3
Multiplying both sides by 3, we get:
y = 6
Therefore, B alone can complete the work in 6 days.