Two workers finished a job in 12 days. How long would it take each worker to do the job by himself if one of the workers needs 10 more days to finish the job than the other worker?
To solve this problem, we can use a system of equations.
Let's assume that one worker takes x days to finish the job by himself. According to the given information, the other worker takes x + 10 days to finish the job.
Now, let's calculate the rate at which each worker completes the job. The rate is usually measured as the amount of work completed per unit of time. In this case, the amount of work completed is the whole job, which we can represent as 1.
Worker 1's rate = 1 job / x days
Worker 2's rate = 1 job / (x + 10) days
Since they both worked together and finished the job in 12 days, their combined rate is:
Combined rate = 1 job / 12 days
Now, we can form the equation:
Worker 1's rate + Worker 2's rate = Combined rate
1/x + 1/(x + 10) = 1/12
To simplify this equation, we can multiply every term by 12x(x + 10) to eliminate the denominators:
12(x + 10) + 12x = x(x + 10)
Now, let's solve this equation:
12x + 120 + 12x = x^2 + 10x
24x + 120 = x^2 + 10x
Rearranging the equation and making it equal to zero:
x^2 - 14x - 120 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, factoring won't work, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Applying the formula with a = 1, b = -14, and c = -120:
x = (-(-14) ± √((-14)^2 - 4(1)(-120))) / 2(1)
Simplifying:
x = (14 ± √(196 + 480)) / 2
x = (14 ± √676) / 2
x = (14 ± 26) / 2
This gives us two possible values for x:
x1 = (14 + 26) / 2 = 40 / 2 = 20
x2 = (14 - 26) / 2 = -12 / 2 = -6
Since time cannot be negative, we reject x2 = -6. Therefore, the first worker takes 20 days to finish the job by himself.
Given that the other worker takes x + 10 days, we can substitute x = 20 into the equation and find the second worker's time:
x + 10 = 20 + 10 = 30
Therefore, the second worker takes 30 days to finish the job by himself.
In conclusion, one worker takes 20 days to finish the job alone, and the other worker takes 30 days.
rate of one worker --- 1/x
rate of the other ---- 1/(x+10)
1/x + 1/(x+10) = 1/12
times 12x(x+10)
12(x+10) + 12x = x(x+10)
12x + 120 + 12x = x^2 + 10x
x^2 - 14x - 120 = 0
(x - 20)(x + 6) = 0
x = 20 or x = -6, clearly x > 0
State your conclusion.