Let the number of days taken by B to finish the work alone be denoted as x. Therefore, the number of days taken by A to finish the work alone will be x – 6 (since A takes 6 days less than B).
When A and B work together, their combined work rate can be expressed as:
- Work rate of A = (1 / (x – 6)) (work per day)
- Work rate of B = (1 / x) (work per day)
Thus, their combined work rate is:
Combined work rate = (1 / (x – 6)) + (1 / x)
Since A and B together can finish the work in 4 days, their combined work rate is also equal to:
1/4 (work per day)
Equating the two expressions, we get:
(1 / (x – 6)) + (1 / x) = 1/4
Now, to solve for x, we’ll find a common denominator on the left side:
x(x – 6) is the common denominator. Thus, we have:
(x + (x – 6)) / (x(x – 6)) = 1/4
This simplifies to:
(2x – 6) / (x^2 – 6x) = 1/4
Cross-multiplying gives us:
4(2x – 6) = x^2 – 6x
Expanding both sides produces:
8x – 24 = x^2 – 6x
Rearranging leads to:
x^2 – 14x + 24 = 0
This quadratic equation can be factored as:
(x – 12)(x – 2) = 0
Setting each factor to zero gives the possible solutions:
- x – 12 = 0 → x = 12
- x – 2 = 0 → x = 2
Since x represents the number of days taken by B, we can only consider positive and reasonable solutions. The first solution x = 12 is valid, while x = 2 is not feasible since A would take -4 days to complete the work, which doesn’t make sense.
Therefore, B takes 12 days to finish the work alone.