SOLUTION: Arthur can arrange a number of books in the bookshelf in x hours. Charlie can do the same job in y hours. If they work together,they will finish the same job in 6 hours. Arthur's r

Question 154708: Arthur can arrange a number of books in the bookshelf in x hours. Charlie can do the same job in y hours. If they work together,they will finish the same job in 6 hours. Arthur's rate is one-third of Charlie's. Find the number of hours each can do the job alone?
Found 3 solutions by stanbon, ptaylor, checkley77:
Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
Arthur can arrange a number of books in the bookshelf in x hours.
Arthur rate = 1/x job/hr.
----------------------------
Charlie can do the same job in y hours.
Charlie rate = 1/y job/hr.
----------------------------
If they work together,they will finish the same job in 6 hours.
Together rate = 1/6 job/hr.
----------------------------
Arthur's rate is one-third of Charlie's.
1/x = (1/3)(1/y) = 1/3y
------------------------------
Find the number of hours each can do the job alone?
EQUATIONS:
x = 3y
1/x + 1/y = 1/6
----------------
Substitute to solve for "y":
1/3y + 1/y = 1/6
Multiply thru by 6y to get:
2 + 6 = y
y = 8 hrs. (# of hours it would take Charlie to complete the job)
---------------
x = 3y, so x = 24 hrs (time it would take Arthur to complete the job)
===================
Cheers,
Stan H.

Answer by ptaylor(2198)   (Show Source): You can put this solution on YOUR website!
Arthur does the job at the rate of 1/x books per hour
Charlie does the job at the rate of 1/y books per hour
Together do the job at the rate of 1/x + 1/y =(y+x)/xy books per hour
Now we are told the following:
1/x=(1/3)*1/y=1/3y or
1/x=1/3y multiply each side by 3xy
3y=x---------------eq1
and
((y+x)/xy)*6=1 (1 job, that is); multiply each side by xy
6(x+y)=xy--------------eq2
substitute x=3y from eq1 into eq2
6(3y+y)=3y^2
24y=3y^2 divide each side by 3y
y=8 hours---------time it takes Charlie
substitute into eq1
x=24 hours-----------time it takes Authur
CK
1/24=(1/3)1/8
1/24=1/24
and
1/24+1/8=1/24+3/24=4/24=1/6
(1/6)*x=1
x=6 hours time it takes both working together
Hope this helps----ptaylor
Answer by checkley77(12844)   (Show Source): You can put this solution on YOUR website!
XY/(X+Y)=6
Y=X/3
(X*X/3)/(X+X/3)=6
X^2/3=6(3X+X)/3
X^2/3=2*4X
X^2=3*8X
X^2=24X
X^2-24X=0
X(X-24)=0
X=24 HOURS FOR ARTHUR WORKING ALONE.
24/3=8 HOURS FOR CHARLIE WORKING ALONE.
PROOF:
24*8/(24+8)=6
192/32=6
6=6
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