A alone and B alone can do a work in respectively 20 and 5 days more than both working together. Find the number of days required if both work together.

Let's denote:

1. The number of days A alone can do the work as 'a'
2. The number of days B alone can do the work as 'b'
3. The number of days A and B together can do the work as 'x'

From the problem, we know that:

a = x + 20 b = x + 5

We also know that the work done by A and B together in one day is equal to the sum of the work done by A alone in one day and B alone in one day. This can be written as:

1/x = 1/a + 1/b

Substituting the values of a and b from the first two equations into the third equation, we get:

1/x = 1/(x + 20) + 1/(x + 5)

To solve this equation for x, we first get rid of the fractions by multiplying through by x(x + 20)(x + 5), which gives:

(x + 20)(x + 5) = x(x + 5) + x(x + 20)

Expanding and simplifying, we get:

x^2 + 25x + 100 = 2x^2 + 50x

Rearranging terms, we get:

x^2 + 25x - 100 = 0

This is a quadratic equation in the form ax^2 + bx + c = 0, which can be solved using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / (2a). In this case, a = 1, b = 25, and c = -100, so the solutions are:

x = [-25 ± sqrt((25)^2 - 41(-100))] / (2*1) x = [-25 ± sqrt(625 + 400)] / 2 x = [-25 ± sqrt(1025)] / 2

Since x must be positive (it doesn't make sense to have a negative number of days), we only consider the solution with the plus sign in the square root:

x = [-25 + sqrt(1025)] / 2 x = [-25 + 32] / 2 x = 7/2 x = 3.5

So, A and B together can do the work in 3.5 days.