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Question 329503: A and B can do a piece of work in 12 days, B and C can do it in 15 days and C and A can do the same work in 20 days. How long would each take to complete the job
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Stay with me. This is going to get very ugly before I'm done.
If A can do a job in x time periods, then A can do  of the job in 1 time period. Likewise, if B can do the same job in y time periods, then B can do  of the job in 1 time period.
So, working together, they can do
of the job in 1 time period.
Therefore, they can do the whole job in:
time periods. So now we have an expression that relates two people working at different rates to the amount of time it takes them to do a job working together.
Let  represent the number of days it takes A to do the job, and let  represent the number of days it takes B to do the job, and finally, let  represent the number of days it takes C to do the job.
From the information we are given, we can write the following three relationships:
A little algebra on the first one gets us:
)
\ =\ -12y)
And performing the same sort of work on the second equation, we arrive at:
Now substitute these two expressions in for and in the third equation.
\left(\frac{-15y}{15\ -\ y}\right)}{\left(\frac{-12y}{12\ -\ y}\right)\ +\ \left(\frac{-15y}{15\ -\ y}\right)}\ =\ 20)
A little simplifying music, Sammy...
\left(\frac{-15y}{15\ -\ y}\right)\ =\ 20\left[\left(\frac{-12y}{12\ -\ y}\right)\ +\ \left(\frac{-15y}{15\ -\ y}\right)\right])
\left(15\ -\ y\right)}\ =\ 20\left[\left(\frac{-12y}{12\ -\ y}\right)\ +\ \left(\frac{-15y}{15\ -\ y}\right)\right])
\left(15\ -\ y\right)}\ =\ \left(\frac{-12y}{12\ -\ y}\right)\ +\ \left(\frac{-15y}{15\ -\ y}\right))
\left(15\ -\ y\right)}\ =\ \frac{-12y\left(15\ -\ y\right)\ -\ 15y\left(12\ -\ y\right)}{\left(12\ -\ y\right)\left(15\ -\ y\right)})
Notice that the denominators are the same. Multiply by the denominator.
\ -\ 15y\left(12\ -\ y\right))
\ -\ 5y\left(12\ -\ y\right))
Hence,  , which is absurd and can be discarded, or  . B would take 20 days to do the job by him/herself.
Substituting back into the first equation:
And substituting back into the second equation:
I'll leave those last two for you to solve to determine the times for A and C.
John
My calculator said it, I believe it, that settles it
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