Working alone at a constant rate if A takes 'a' days to complete a task, A will complete of the task in a day.
For e.g, If A takes 20 days to complete a task and B takes 30 days to complete a task, it will be INCORRECT to frame an equation a + b = 50
The correct way to express this information is:
A will complete th of the task in a day.
B will complete th of the task in a day.
Working together they will complete of the task in day.
i.e., they will complete of the task in day.
So, they will complete the task in 12 days.
Let us look at another example.
A takes 'a' days to complete a task and B takes 'b' days to complete the same task. If they work together they complete the task in 10 days.
It is INCORRECT to frame the equation as a + b = 10.
The correct expression is
A will complete of the task in a day.
Therefore, in 2 days A will complete of the task in a day.
Similarly, A will complete of the task in 5 days.
A starts and works for 2 days. So, A will work on day 1 and day 2.
Then B will work for the next 2 days. B will work on day 3 and day 4.
A will continue for the next 2 days. i.e., on day 5 and day 6.
B will work on day 7 and day 8.
A will for the last 2 days i.e., day 9 and day 10.
Therefore, A will work on day 1, day 2, day 5, day 6, day 9, and day 10. i.e., for 6 days.
And B will work on day 3, day 4, day 7, and day 8. i.e., for 4 days.
In 6 days A will complete of the task.
In 4 days B will complete of the task.
With A working 6 days and B working 4 days, the task is completed.
i.e., + = 1 .... eqn (1)
Therefore, B will work on day 1, day 2, day 5, day 6, day 9, and day 10. i.e., for 6 days.
And A will work on day 3, day 4, day 7, day 8 and half a day on day 11. i.e., for 4.5 days.
In 6 days B will complete of the task.
In 4.5 days A will complete of the task.
With B working 6 days and A working 4.5 days, the task is completed.
i.e., + = 1 .... eqn (2)
+ = 1 .... eqn (1)
+ = 1 .... eqn (2)
Solving the two equations we get a = 9 days and b = 12 days.
Working together A and B will complete th of the task in a day.
Hence, they will complete the task in days.