SOLUTION: it takes Tom and John 2 hours to do a certain job,it takes tom and harry 3 hours to do the same job and it takes john and harry 4 hours to do the same job. how long would it take

Question 464785: it takes Tom and John 2 hours to do a certain job,it takes tom and harry 3 hours to do the same job and it takes john and harry 4 hours to do the same job. how long would it take tom, john and harry to do the job if all 3 worked together?
Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!
it takes Tom and John 2 hours to do a certain job,it takes tom and harry 3 hours to do the same job and it takes john and harry 4 hours to do the same job. how long would it take tom, john and harry to do the job if all 3 worked together?

Make the chart below.  The number of jobs done in all cases is 1.
Let t = Tom's hours to do the 1 job alone
Let j = John's hours to do the 1 job alone
Let h = Harry's hours to do the 1 job alone
Let x = the answer, the number of hours it would take if
all three worked together.

                          number of
                          jobs done       time in hrs     rate in jobs/hr 
Tom only                      1                t               
John only                     1                j               
Harry only                    1                h                
Tom & John                    1                2               
Tom & Harry                   1                3               
John & Harry                  1                4               
Tom, John & Harry             1                x               
 
Now we fill in all the rates in jobs/hour by dividing the number of
jobs by the number of hours:
                          number of
                          jobs done       time in hrs     rate in jobs/hr 
Tom only                      1                t               1/t
John only                     1                j               1/j
Harry only                    1                h               1/h 
Tom & John                    1                2               1/2
Tom & Harry                   1                3               1/3
John & Harry                  1                4               1/4
Tom, John & Harry             1                x               1/x

Tom's rate + John's rate = Tom & John's rate together 
1/t + 1/j = 1/2

Tom's rate + Harry's rate = Tom & Harry's rate together
1/t + 1/h = 1/3

John's rate + Harry's rate = John & Harry's rate together
1/j + 1/h = 1/4

So we have this system of equations:
1/t + 1/j = 1/2
1/t + 1/h = 1/3
1/j + 1/h = 1/4

Let A = 1/t, B = 1/j,  C = 1/h
A + B = 1/2
A + C = 1/3
B + C = 1/4

Clear of fractions:

2A + 2B = 1
3A + 3C = 1
4B + 4C = 1

Line up like terms:

2A + 2B      = 1
3A      + 3C = 1
     4B + 4C = 1

Can you can solve that by elimination?  
If you don't know how, post again asking 
how to solve that system by elimination.
answers:

A = 7/24, B = 5/24, C = 1/24

A = 1/t = 7/24 jobs/hour is Tom's rate
B = 1/j = 5/24 jobs/hour is John's rate
C = 1/h = 1/24 jobs/hour is Harry's rate

Now we use the equation:

Tom's rate + John's rate + Harry's rate = Tom, John & Harry's combined rate


                     7/24 + 5/24 + 1/24 = 1/x
Multiply through by LCD of 24x
                            7x + 5x + x = 24
                                    13x = 24                                
                                      x = 24/13
                                      x = 1 11/13 hours

Edwin

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