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

Algebra ->  Rate-of-work-word-problems -> SOLUTION: A and B can do a piece of work in 42 days, B and C in 31 days, and A and C in 20 days. Working together, how many days can all of them finish the work?      Log On


   

Question 167380: A and B can do a piece of work in 42 days, B and C in 31 days, and A and C in 20 days. Working together, how many days can all of them finish the work?
Found 2 solutions by ptaylor, Edwin Parker:
Answer by ptaylor(2198) About Me  (Show Source):
You can put this solution on YOUR website!
A&B work at the rate of 1/42 job per day
B&C work at the rate of 1/31 job per day
A&C work at the rate of 1/20 job per day
Let t=time it takes all workers, working together, to do the job
Let 1/A=rate at which A works
1/B=rate that B works
1/C=rate that C works
Now we know that:
1/A + 1/B=1/42-------------------eq1
1/B + 1/C=1/31---------------------eq2
1/A + 1/C=1/20---------------------eq3
subtract eq3 from eq1:
1/B - 1/C=1/42 - 1/20------------------eq3a
add eq3a and eq2:
2/B=1/42 - 1/20 + 1/31 (13020 is LCM)
2/B=(310-651+420)/13020=79/13020
1/B=79/26040 job per day-------------------------rate at which B works
substitute the value for 1/B into eq1:
1/A + 79/26040=1/42
1/A=1/42 - 79/26040=(620-79)/26040=541/26040 job per day----rate at which A works
substitute the value for 1/A into eq3:
541/26040 + 1/C=1/20
1/C = 1/20 - 541/26040=(1302-541)/26040=761/26040 job per day----rate at which C works
Together, A, B, & C work at the the rate of:
541/26040 +79/26040 + 761/26040= 1381/26040 job per day
Now, our final equation to solve is:
(1381/26040)t=1 (1 job that is) multiply each side by 26040
(rate at which they work * time it take to complete the job=1 job)
1381t=26040 divide by 1381
t= 18.86 days----------------time it takes all the workers working together
CK
A: (541/26040)*18.86=0.3917 of the job
B: (79/26040)*18.86=0.0572 of the job
C: (761/26040)*18.86=0.5512 of the job
0.3917 + 0.0572 + 0.5512=1.000069
1~~~1
Hope this helps---sorry about the first effort----ptaylor

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