A can do a piece of work in 13 days while B can do it in 39 days. A works on the first day B works on the second day and thus the functioning goes on and the work ends. How many days did it take to complete this work?
Answers
Answer:
It will take 19 days to finish the work
Step-by-step explanation:
A can do the full work in 13 days
Fraction of work that A can do in 1 day = 1/13
B can do the full work in 39 days
Fraction of work that B can do in 1 day = 1/39
Fraction of work done by A in 1 day = 1/13 = 3/39
On the first day, only A is working
=> Fraction of work completed = 3/39
On the second day, only B is working
=> Fraction of work completed = 1/39
..and so on.
Let A work for "x" days and B work for "y" days to complete the work
Fraction of work done by A in "x" days = 3x/39
Fraction of work done by B in "y" days = y/39
Fraction of work done by A and B in this period = (3x + y)/39
We want that work should get completed. This means that the fraction of work completed should be ONE.
=> (3x + y)/39 = 1
=> (3x + y)/39 = 39/39
=> 3x + y = 39 ...(i)
We can get "x" and "y" by solving this equation.
However, as there are two unknowns and just one equation, this will not have a unique solution.
But we know that A and B are working alternately. This means that there one of the following three possibilities will hold true:
1. x = y
2. x = y - 1
3. x = y + 1
This means that A and B will either work for same number of days OR there will be a difference of ONE day between the number of days they have each worked.
Assuming x = y, (i) will be:
3y + y = 39 => y = 39/4
Assuming x = (y - 1), (i) will be:
3(y - 1) + y = 39
=> 3y - 3 + y = 39
=> 4y = 42
=> y = 42/4
Assuming x = (y+1), (i) will be:
3(y + 1) + y = 39
=> 3y + 3 + y = 39
=> 4y = 36
=> y = 9
As we are expecting "x" and "y" to be natural numbers (being number of full working days), we can accept the value y = 9
Substituting for y = 9 in (i), we get:
3x + 9 = 39
3x = 39 - 9
3x = 30
x = 10
Hence, A works for 10 days and B works for 9 days to complete the work.
The total number of days taken = (10 + 9) = 19
Answer: It will take 19 days to complete the work