To find the time taken by the faster man to complete the work, let's first assign variables to the unknown quantities in the problem.
Let:
x = the number of days it takes for the second man to complete the job
x - 20 = the number of days it takes for the faster man to complete the job (since he completes it 20 days faster than the second man)
Since it takes both men together 24 days to complete the job, we can create an equation based on their combined work rates:
1/x + 1/(x - 20) = 1/24
To solve this equation, we can first find a common denominator. In this case, the common denominator is 24x(x - 20). Therefore, we multiply each term by this denominator to eliminate the fractions, resulting in:
24(x - 20) + 24x = x(x - 20)
Expanding and simplifying the equation:
24x - 480 + 24x = x^2 - 20x
48x - 480 = x^2 - 20x
Rearranging the equation to form a quadratic equation:
x^2 - 68x + 480 = 0
To solve this quadratic equation, we can factor it (if possible) or use the quadratic formula.
In this case, the quadratic equation doesn't factor easily, so we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -68, and c = 480. Substituting these values into the quadratic formula:
x = (-(-68) ± √((-68)^2 - 4(1)(480))) / (2(1))
Simplifying:
x = (68 ± √(4624 - 1920)) / 2
x = (68 ± √2704) / 2
x = (68 ± 52) / 2
Solving for both possible values of x:
x1 = (68 + 52) / 2 = 120 / 2 = 60
x2 = (68 - 52) / 2 = 16 / 2 = 8
Since the number of days cannot be negative, we discard x2 = 8 as the extraneous solution.
Therefore, the second man takes 60 days to complete the job, and since the faster man is 20 days faster, he takes 60 - 20 = 40 days to complete the work.
Therefore, the time taken by the faster man to complete the work is 40 days.