SOLUTION: Working together, Pat, Lindsey, and Robin can dig a drainage ditch in 2 hours. Working alone, Robin can do it in 6 hours, while Pat can do it 2 1/4 hours faster than Lindsay. How l

Algebra ->  Rate-of-work-word-problems -> SOLUTION: Working together, Pat, Lindsey, and Robin can dig a drainage ditch in 2 hours. Working alone, Robin can do it in 6 hours, while Pat can do it 2 1/4 hours faster than Lindsay. How l      Log On


   

Question 553883: Working together, Pat, Lindsey, and Robin can dig a drainage ditch in 2 hours. Working alone, Robin can do it in 6 hours, while Pat can do it 2 1/4 hours faster than Lindsay. How long will it take Lindsay to dig the ditch by herself? (Round your answer to two decimal places.)
Please help me solve this problem. Thank you very much!
Answer by AnlytcPhil(1810) About Me  (Show Source):
You can put this solution on YOUR website!
Working together, Pat, Lindsey, and Robin can dig a drainage ditch in 2 hours. Working alone, Robin can do it in 6 hours, while Pat can do it 2 1/4 hours faster than Lindsay. How long will it take Lindsay to dig the ditch by herself? (Round your answer to two decimal places.)
Please help me solve this problem. Thank you very much!

Make this chart:

                              Number of                   Digging rate
                              ditches        Hours            in
                                dug         Required      ditches/hour
Pat digging alone

Lindsey digging alone

Robin diggin alone

All three digging together

Let the answer be x, which is the number of hours required for Lindsay
to dig ONE ditch, so we put x for Lindsay's hours required and 1 for
the number of ditches.

                              Number of                   Digging rate
                              ditches        Hours            in
                               to dig         Required    ditches/hour
Pat digging alone

Lindsey digging alone            1             x

Robin diggin alone

All three digging together

>>...Pat can do it 2%261%2F4 hours faster than Lindsay...<<
So we put Pat's time as x - 2%261%2F4 and 1 ditch to dig. We SUBTRACT
because "faster" means LESS time.

                              Number of                   Digging rate
                              ditches        Hours            in
                               to dig         Required    ditches/hour
Pat digging alone                1           x-2%261%2F4 
Lindsey digging alone            1            x

Robin diggin alone

All three digging together


Similar, since all the other information is about digging ONE ditch,
fill in 1 for the number of ditches to dig in the other two cases.  
Also fill in the given number of hours required in each case:

                              Number of                   Digging rate
                              ditches        Hours            in
                               to dig       Required      ditches/hour
Pat digging alone                1           x-2%261%2F4 
Lindsey digging alone            1            x

Robin diggin alone               1            6

All three digging together       1            2
 
Next we fill in the digging rates in ditches/hour by dividing the
number of ditches by the hours:

                              Number of                   Digging rate
                              ditches        Hours            in
                               to dig       Required      ditches/hour
Pat digging alone                1           x-2%261%2F4           1%2F%28x%2B2%261%2F4%29 
Lindsay digging alone            1            x              1%2Fx         
Robin diggin alone               1            6              1%2F6
All three digging together       1            2              1%2F2 


The equation comes from:

            %28matrix%285%2C1%2C%22Pat%27s%22%2C+digging%2C+rate%2C+working%2C+alone%29%29 + %28matrix%285%2C1%2C%22Lindsay%27s%22%2C+digging%2C+rate%2C+working%2C+alone%29%29 + %28matrix%285%2C1%2C%22Robins%27s%22%2C+digging%2C+rate%2C+working%2C+alone%29%29 =             

              1%2F%28x-2%261%2F4%29 + 1%2Fx + 1%2F6 = 1%2F2

              Let's first simplify 1%2F%28x-2%261%2F4%29
  
           Change the mixed fraction to an improper fraction 1%2F%28x-2%261%2F4%29 = 1%2F%28x-9%2F4%29.  
           Multiply top and bottom by 4:
           4%2F%284x-9%29

So the equation becomes:

              4%2F%284x-9%29 + 1%2Fx + 1%2F6 = 1%2F2
  

Solve that by clearing of fractions by multiplying through by the 
LCD of 6x(4x-9) 

    You will get 7.329001404 which we round to 7.33 hours

It takes Robin approximately 7.33 hours to dig one ditch.

Edwin