SOLUTION: Randy and Paul can do a piece of work in 10 days ;Randy and Guy can do it in 8 days ;Paul and Guy can do it in 12 days. How long will it take to each to do it alone ?

Algebra ->  Rate-of-work-word-problems -> SOLUTION: Randy and Paul can do a piece of work in 10 days ;Randy and Guy can do it in 8 days ;Paul and Guy can do it in 12 days. How long will it take to each to do it alone ?      Log On


   

Question 937145: Randy and Paul can do a piece of work in 10 days ;Randy and Guy can do it in 8 days ;Paul and Guy can do it in 12 days. How long will it take to each to do it alone ?
Found 2 solutions by josmiceli, ptaylor:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +R%5Br%5D+ = Randy's rate of working
Let +R%5Bp%5D+ = Paul's rate of working
Let +R%5Bg%5D+ = Guy's rate of working
---------------------------------
Add their rates of working to get
their rate working together
(1) +R%5Br%5D+%2B+R%5Bp%5D+=+1%2F10+
(2) +R%5Br%5D+%2B+R%5Bg%5D+=+1%2F8+
(3) +R%5Bp%5D+%2B+R%5Bg%5D+=+1%2F12+
-------------------------
Subtract (3) from (2)
(2) +R%5Br%5D+%2B+R%5Bg%5D+=+3%2F24+
(3) +-R%5Bp%5D+-+R%5Bg%5D+=+-2%2F24+
--------------------------
+R%5Br%5D+-+R%5Bp%5D+=+1%2F24+
Add this result to (1)
+R%5Br%5D+%2B+R%5Bp%5D+=+1%2F10+
+R%5Br%5D+-+R%5Bp%5D+=+1%2F24+
+2R%5Br%5D+=+12%2F120+-+5%2F120+
+2R%5Br%5D+=+7%2F120+
+R%5Br%5D+=+7%2F240+
-------------------
and
(2) +R%5Br%5D+%2B+R%5Bg%5D+=+1%2F8+
(2) +7%2F240+%2B+R%5Bg%5D+=+30%2F240+
(2) +R%5Bg%5D+=+23%2F240+
---------------------
and
(1) +R%5Br%5D+%2B+R%5Bp%5D+=+1%2F10+
(1) +7%2F240+%2B+R%5Bp%5D+=+24%2F240+
(1) +R%5Bp%5D+=+17%2F240+
---------------------
check answers:
(1) +R%5Br%5D+%2B+R%5Bp%5D+=+1%2F10+
(1) +7%2F240+%2B+17%2F240+=+24%2F240+
(1) +24%2F240+=+24%2F240+
and
(2) +R%5Br%5D+%2B+R%5Bg%5D+=+1%2F8+
(2) +7%2F240+%2B+23%2F240+=+30%2F240+
(2) +30%2F240+=+30%2F240+
and
(3) +R%5Bp%5D+%2B+R%5Bg%5D+=+1%2F12+
(3) +17%2F240+%2B+23%2F240+=+40%2F240+
---------------------------------
I made a mistake somewhere. The method is
right, but this last step is wrong. I'm
too tired to find the problem. Just convert
fractions to mixed numbers and you
will have answers ( after finding problem )


Answer by ptaylor(2198) About Me  (Show Source):
You can put this solution on YOUR website!
Let R= amount of time it takes Randy to do the work alone
Then Randy works at the rate of 1/R of the job per day
Let P= amount of time it takes Paul to do the work alone
Then Paul works at the rate of 1/P of the job per day
Let G=amount of time it takes Guy to do the work alone
Then Guy works at the rate of 1/G of the job per day
Sooooo our equations to solve are:
1/R+1/P=1/10---------------------eq1
1/R+1/G=1/8------------------------eq2
1/P+1/G=1/12--------------------------eq3
Subtract eq2 from eq1 and we get:
1/P-1/G=(1/10)-(1/8)=(4/40)-(5/40)=-1/40 add this to eq3
2/P=(1/12)-(1/40)=(10/120)-(3/120)=7/120
2/P=7/120
1/P=7/240 substitute this into eq1-----------------EQ1a
1/R=(1/10)-(7/240)=(24/240)-(7/240)=17/240
1/R=17/240 substitute this into eq2-----------------EQ2a
17/240 + 1/G=1/8
1/G=(1/8)-(17/240)=(30/240)-(17/240)=13/240
1/G=13/240------------------------------------------EQ3a
From eq2a: (you can divide these out but it's easier to work in fractions when checking your work)
17R=240
R=240/17 days
From eq1a:
7P=240
P=240/7 days
From eq3a:
13G=240
G=240/13 days
CK
Randy and Paul
(17/240)+(7/240)=????1/10
24/240=1/10
1/10=1/10
Randy and Guy
(17/240)+(13/240)=????1/8
30/240=1/8
1/8=1/8
Paul and Guy
(7/240)+(13/240)=????1/12
20/240=1/12
1/12=1/12
Does this help?---ptaylor