SOLUTION: A and B can do a piece of work in 42 days, B and C in 31 days and C and A in 20 days. In how many days can all of them do the work together?

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Question 658846: A and B can do a piece of work in 42 days, B and C in 31 days and C and A in 20 days. In how many days can all of them do the work together?
Found 3 solutions by Edwin McCravy, AnlytcPhil, Edwin Parker:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
A and B can do a piece of work in 42 days, B and C in 31 days and C and A in 20 days. In how many days can all of them do the work together?

A and B can do a piece of work in 42 days,
So A's and B's combined rate is 1 job per 42 days or %281_job%29%2F%2842_days%29 or 1%2F42jobs%2Fday

B and C in 31 days
So B's and C's combined rate is 1 job per 31 days or %281_job%29%2F%2831_days%29 or 1%2F31jobs%2Fday

C and A in 20 days
So C's and A's combined rate is 1 job per 31 days or %281_job%29%2F%2831_days%29 or 1%2F31jobs%2Fday

Suppose A's rate working alone is is 1 job per x days or %281_job%29%2F%28x_days%29 or 1%2Fxjobs%2Fday.

Suppose B's rate working alone is is 1 job per y days or %281_job%29%2F%28y_days%29 or 1%2Fyjobs%2Fday.

Suppose C's rate working alone is is 1 job per z days or %281_job%29%2F%28z_days%29 or 1%2Fzjobs%2Fday.

Suppose their combined rate is 1 job per d days or %281_job%29%2F%28d_days%29 or 1%2Fdjobs%2Fday.


The four equations come from:

      %28matrix%284%2C1%2C%0D%0A%0D%0A%22A%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22B%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 = %28matrix%285%2C1%2C%0D%0A%0D%0ATheir%2C+combined%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29

            1%2Fx + 1%2Fy = 1%2F42

      %28matrix%284%2C1%2C%0D%0A%0D%0A%22B%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22C%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 = %28matrix%285%2C1%2C%0D%0A%0D%0ATheir%2C+combined%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29

            1%2Fy + 1%2Fz = 1%2F31

      %28matrix%284%2C1%2C%0D%0A%0D%0A%22C%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22A%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 = %28matrix%285%2C1%2C%0D%0A%0D%0ATheir%2C+combined%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29

            1%2Fz + 1%2Fx = 1%2F20



      %28matrix%284%2C1%2C%0D%0A%0D%0A%22A%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22B%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22C%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 = %28matrix%285%2C1%2C%0D%0A%0D%0ATheir%2C+combined%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29

            1%2Fx + 1%2Fy + 1%2Fz = 1%2Fd

1%2Fx + 1%2Fy = 1%2F42
1%2Fy + 1%2Fz = 1%2F31
1%2Fz + 1%2Fx = 1%2F20
1%2Fx + 1%2Fy + 1%2Fz = 1%2Fd

Now we must find their combined rate which is

So we line up the first three equations like this and add them all:

          1%2Fx + 1%2Fy         = 1%2F42
              1%2Fy + 1%2Fz     = 1%2F31
          1%2Fx     + 1%2Fz     = 1%2F20
         ---------------------
          2%2Fx + 2%2Fy + 2%2Fz     = 1%2F42%2B1%2F31%2B1%2F20

          2%2Fx + 2%2Fy + 2%2Fz     = 1381%2F13020

Dividing both side by 2

          1%2Fx + 1%2Fy + 1%2Fz     = 1381%2F26040

And since the fourth equation is

          1%2Fx + 1%2Fy + 1%2Fz = 1%2Fd

Since things equal to the same thing are equal to each other,

           1%2Fd = 1381%2F26040

Cross-multiplying:

       1381d = 26040
           d = 26040%2F1381
           d = 18.85590152 days

Edwin

Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
A and B can do a piece of work in 42 days, B and C in 31 days and C and A in 20 days. In how many days can all of them do the work together?

A and B can do a piece of work in 42 days,
So A's and B's combined rate is 1 job per 42 days or %281_job%29%2F%2842_days%29 or 1%2F42jobs%2Fday

B and C in 31 days
So B's and C's combined rate is 1 job per 31 days or %281_job%29%2F%2831_days%29 or 1%2F31jobs%2Fday

C and A in 20 days
So C's and A's combined rate is 1 job per 31 days or %281_job%29%2F%2831_days%29 or 1%2F31jobs%2Fday

Suppose A's rate working alone is is 1 job per x days or %281_job%29%2F%28x_days%29 or 1%2Fxjobs%2Fday.

Suppose B's rate working alone is is 1 job per y days or %281_job%29%2F%28y_days%29 or 1%2Fyjobs%2Fday.

Suppose C's rate working alone is is 1 job per z days or %281_job%29%2F%28z_days%29 or 1%2Fzjobs%2Fday.

Suppose their combined rate is 1 job per d days or %281_job%29%2F%28d_days%29 or 1%2Fdjobs%2Fday.

We wish to find d.


The four equations come from:

      %28matrix%284%2C1%2C%0D%0A%0D%0A%22A%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22B%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 = %28matrix%285%2C1%2C%0D%0A%0D%0ATheir%2C+combined%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29

            1%2Fx + 1%2Fy = 1%2F42

      %28matrix%284%2C1%2C%0D%0A%0D%0A%22B%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22C%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 = %28matrix%285%2C1%2C%0D%0A%0D%0ATheir%2C+combined%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29

            1%2Fy + 1%2Fz = 1%2F31

      %28matrix%284%2C1%2C%0D%0A%0D%0A%22C%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22A%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 = %28matrix%285%2C1%2C%0D%0A%0D%0ATheir%2C+combined%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29

            1%2Fz + 1%2Fx = 1%2F20



      %28matrix%284%2C1%2C%0D%0A%0D%0A%22A%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22B%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22C%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 = %28matrix%285%2C1%2C%0D%0A%0D%0ATheir%2C+combined%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29

            1%2Fx + 1%2Fy + 1%2Fz = 1%2Fd

1%2Fx + 1%2Fy = 1%2F42
1%2Fy + 1%2Fz = 1%2F31
1%2Fz + 1%2Fx = 1%2F20
1%2Fx + 1%2Fy + 1%2Fz = 1%2Fd

So we line up the first three equations like this and add them all:

          1%2Fx + 1%2Fy         = 1%2F42
              1%2Fy + 1%2Fz     = 1%2F31
          1%2Fx     + 1%2Fz     = 1%2F20
         ---------------------
          2%2Fx + 2%2Fy + 2%2Fz     = 1%2F42%2B1%2F31%2B1%2F20

          2%2Fx + 2%2Fy + 2%2Fz     = 1381%2F13020

Dividing both side by 2

          1%2Fx + 1%2Fy + 1%2Fz     = 1381%2F26040

And since the fourth equation is

          1%2Fx + 1%2Fy + 1%2Fz = 1%2Fd

Since things equal to the same thing are equal to each other,

           1%2Fd = 1381%2F26040

Cross-multiplying:

       1381d = 26040
           d = 26040%2F1381
           d = 18.85590152 days

Edwin

Answer by Edwin Parker(36) About Me  (Show Source):
You can put this solution on YOUR website!
A and B can do a piece of work in 42 days, B and C in 31 days and C and A in 20 days. In how many days can all of them do the work together?

A and B can do a piece of work in 42 days,
So A's and B's combined rate is 1 job per 42 days or %281_job%29%2F%2842_days%29 or 1%2F42jobs%2Fday

B and C in 31 days
So B's and C's combined rate is 1 job per 31 days or %281_job%29%2F%2831_days%29 or 1%2F31jobs%2Fday

C and A in 20 days
So C's and A's combined rate is 1 job per 31 days or %281_job%29%2F%2831_days%29 or 1%2F31jobs%2Fday

Suppose A's rate working alone is is 1 job per x days or %281_job%29%2F%28x_days%29 or 1%2Fxjobs%2Fday.

Suppose B's rate working alone is is 1 job per y days or %281_job%29%2F%28y_days%29 or 1%2Fyjobs%2Fday.

Suppose C's rate working alone is is 1 job per z days or %281_job%29%2F%28z_days%29 or 1%2Fzjobs%2Fday.

Suppose their combined rate is 1 job per d days or %281_job%29%2F%28d_days%29 or 1%2Fdjobs%2Fday.


The four equations come from:

      %28matrix%284%2C1%2C%0D%0A%0D%0A%22A%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22B%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 = %28matrix%285%2C1%2C%0D%0A%0D%0ATheir%2C+combined%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29

            1%2Fx + 1%2Fy = 1%2F42

      %28matrix%284%2C1%2C%0D%0A%0D%0A%22B%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22C%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 = %28matrix%285%2C1%2C%0D%0A%0D%0ATheir%2C+combined%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29

            1%2Fy + 1%2Fz = 1%2F31

      %28matrix%284%2C1%2C%0D%0A%0D%0A%22C%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22A%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 = %28matrix%285%2C1%2C%0D%0A%0D%0ATheir%2C+combined%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29

            1%2Fz + 1%2Fx = 1%2F20



      %28matrix%284%2C1%2C%0D%0A%0D%0A%22A%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22B%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 + %28matrix%284%2C1%2C%0D%0A%0D%0A%22C%27s%22%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29 = %28matrix%285%2C1%2C%0D%0A%0D%0ATheir%2C+combined%2C+rate%2C+in%2C+%22jobs%2Fday%22%29%29

            1%2Fx + 1%2Fy + 1%2Fz = 1%2Fd

1%2Fx + 1%2Fy = 1%2F42
1%2Fy + 1%2Fz = 1%2F31
1%2Fz + 1%2Fx = 1%2F20
1%2Fx + 1%2Fy + 1%2Fz = 1%2Fd

Now we must find their combined rate which is

So we line up the first three equations like this and add them all:

          1%2Fx + 1%2Fy         = 1%2F42
              1%2Fy + 1%2Fz     = 1%2F31
          1%2Fx     + 1%2Fz     = 1%2F20
         ---------------------
          2%2Fx + 2%2Fy + 2%2Fz     = 1%2F42%2B1%2F31%2B1%2F20

          2%2Fx + 2%2Fy + 2%2Fz     = 1381%2F13020

Dividing both side by 2

          1%2Fx + 1%2Fy + 1%2Fz     = 1381%2F26040

And since the fourth equation is

          1%2Fx + 1%2Fy + 1%2Fz = 1%2Fd

Since things equal to the same thing are equal to each other,

           1%2Fd = 1381%2F26040

Cross-multiplying:

       1381d = 26040
           d = 26040%2F1381
           d = 18.85590152 days

Edwin