# A takes 6 days less than time taken by B to finish the piece of work. if both A and B together can finish the work in 4 days find the time taken by B to finish the work

Let B alone finish the work in x days.

Therefore, A alone can finish the work in (x – 6) days.

Given, (A + B) can finish the work in 4 days

x ≠ 2, ∵ if x = 2, then A alone can finish the work in = (2 – 6) = –4 days, which is not possible.

x = 12,  i.e. B alone can finish the work in 12 days and A alone will finish the work in  12 - 6 = 6 days.

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Assume B alone takes x days to finish the work. Then, A alone can finish it in (x - 6) days.
Now,
(A's one day's work) + (B's one days's work) = 1/x + 1/(x - 6)
and,
(A + B)'s one day's work = 1/4.
Therefore,
1/x + 1/(x - 6) = 1/4
(x - 6 + x)/[x (x - 6)] = 1/ 4
(2x - 6)/(x2 - 6x) = 1/4
8x - 24 = x2 - 6x
x2 - 14x + 24 = 0
x2 - 12x - 2x + 24 = 0
(x -12) (x - 2) = 0
x = 12 or x = 2.
But x cannot be less than 6. So, x = 12.
Hence, B alone can finish the work in 12 days.

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