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.
Question
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.
Solution
Let's denote:
- The number of days A alone can do the work as 'a'
- The number of days B alone can do the work as 'b'
- 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.
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