SOLUTION: Anne, Lucy, and Phoebe can do a piece of work in 2 2/3 days when they work together. If Anne and Phoebe work for 3 days and Lucy for 2 days, they will finish the work. If Anne work

Algebra ->  Rate-of-work-word-problems -> SOLUTION: Anne, Lucy, and Phoebe can do a piece of work in 2 2/3 days when they work together. If Anne and Phoebe work for 3 days and Lucy for 2 days, they will finish the work. If Anne work      Log On


   

Question 329354: Anne, Lucy, and Phoebe can do a piece of work in 2 2/3 days when they work together. If Anne and Phoebe work for 3 days and Lucy for 2 days, they will finish the work. If Anne works for 2 days and Lucy and Phoebe each 1 day, the work will be only 13/24 finished. In how many days can each do the work alone?
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
In these kind of problems, you add the
RATES that each works to get their rates
working together.
(portion of job completed by #1)/(time to complete that portion) +
(portion of job completed by #2)/(time to complete that portion) +
(portion of job completed by #3)/(time to complete that portion) =
(portion of job completed by all working together)/(time to complete that portion)
Let r[a] = Anne's rate of working in (portion of job)/(days to do that portion)
Let r[l] = Lucy's rate of working in (portion of job)/(days to do that portion)
Let r[p] = Phoebe's rate of working in (portion of job)/(days to do that portion)
given:
(1) r%5Ba%5D+%2B+r%5Bl%5D+%2B+r%5Bp%5D+=+1%2F%288%2F3%29 (note that 1 means the whole job)
I can also say:
r%5Ba%5D%2Ax equals Anne's rate x days worked = portion of job done by her
r%5Bl%5D%2Ay equals Lucy's rate y days worked = portion of job done by her
r%5Bp%5D%2Az equals Phoebe's rate z days worked = portion of job done by her
given:
(2) r%5Ba%5D%2A3+%2B+r%5Bp%5D%2A3+%2B+r%5Bl%5D%2A2+=+1 (1 means whole job)
(3) r%5Ba%5D%2A2+%2B+r%5Bl%5D%2A1+%2B+r%5Bp%5D%2A1+=+13%2F24
-------------------------------
The problem wants to find rate for each = (1 job)/(time to do 1 job)
There are 3 equations and 3 unknowns, so it's solvable
First, I divide both sides of (2) by 3
(2) r%5Ba%5D%2A3+%2B+r%5Bp%5D%2A3+%2B+r%5Bl%5D%2A2+=+1
(2) r%5Ba%5D+%2B+r%5Bp%5D+%2B+%282%2F3%29%2Ar%5Bl%5D+=+1%2F3
Now, I subtract this from (1)
(1) r%5Ba%5D+%2B+r%5Bl%5D+%2B+r%5Bp%5D+=+3%2F8
(2) -r%5Ba%5D+-%282%2F3%29%2Ar%5Bl%5D+-+r%5Bp%5D+=+-1%2F3
----------------------------------------------
%281%2F3%29%2Ar%5Bl%5D+=+3%2F8+-+1%2F3
r%5Bl%5D+=+3%2A%289%2F24+-+8%2F24%29
r%5Bl%5D+=+3%2F24
r%5Bl%5D+=+1%2F8
Now I put this value back into (1)
(1) r%5Ba%5D+%2B+1%2F8+%2B+r%5Bp%5D+=+1%2F%288%2F3%29
(1) r%5Ba%5D+%2B+r%5Bp%5D+=+3%2F8+-+1%2F8
(1) r%5Ba%5D+%2B+r%5Bp%5D+=+1%2F4
Now I subtract this from (3)
(3) r%5Ba%5D%2A2+%2B+r%5Bl%5D%2A1+%2B+r%5Bp%5D%2A1+=+13%2F24
(1) -r%5Ba%5D+-+r%5Bp%5D+=+-6%2F24
---------------------------------
r%5Ba%5D+%2B+r%5Bl%5D+=+7%2F24
r%5Ba%5D+%2B+3%2F24+=+7%2F24
r%5Ba%5D+=+4%2F24
r%5Ba%5D+=+1%2F6
Now I put r%5Ba%5D and r%5Bl%5D back into (2)
(2) %281%2F6%29%2A3+%2B+r%5Bp%5D%2A3+%2B+%281%2F8%29%2A2+=+1
(2) 1%2F2+%2B+r%5Bp%5D%2A3+%2B+1%2F4+=+1
(2) r%5Bp%5D%2A3+=+4%2F4+-+3%2F4
r%5Bp%5D+=+1%2F12
Anne can do the job in 6 days working alone
Lucy can do the job in 8 days working alone
Phoebe can do the job in 12 days working alone
check answer:
(3) %281%2F6%29%2A2+%2B+%281%2F8%29%2A1+%2B+%281%2F12%29%2A1+=+13%2F24
(3) 8%2F24+%2B+3%2F24+%2B+2%2F24+=+13%2F24
13%2F24+=+13%2F24
OK