# Mike, Joe, and Bill are painting a fence. The painting can be finished if Mike and Joe work together for 4 hoursand Bill works alone for 2 hours; or if Mike and Joe work together for 2 hours and Bill works alone for 5 hours;or if Mike works alone for 6 hours, Joe works alone for 2 hours, and Bill works alone for 1 hour. How much timedoes it take for each man working alone to complete the painting?

Question

Solve the problem.

Step 1

Let M, J, B = fences per hour for Mike, Joe, Bill respectively.

Equations:

1. 4 (M + J) + 2 B = 1 (i)
2. 2 (M + J) + 5 B = 1    (ii)
3. 6 M + 2 J + 1 B = 1   (iii)

Let's Franklin be the person can work as fast as Mike and Joe together.

So, F = M + J
From the first equation, we get
4 F + 2 B = 1 fence
From the second equation, we get
2 F + 5 B = 1 or
4 F + 10 B = 2

Subtracting equation (i) from equation (ii) , we get
8 B = 1
B = 1/8
So, Bill paints the fence in 8 hours.

Step 2

From equation (iii) we have
6 M + 2 J + 1/8 = 1
6 M + 2 J = 7/8            (iv)

From equation (ii) we have
2 M + 2 J = 3/8            (v)
Subtracting equation (iv) f...

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