SOLUTION: Andrew, Beth, and Carlos are painting a house. If Andrew and Beth paint the house together, it will take 10 days. If Beth and Carlos paint the house together , it will take 6 days.

Algebra ->  Rate-of-work-word-problems -> SOLUTION: Andrew, Beth, and Carlos are painting a house. If Andrew and Beth paint the house together, it will take 10 days. If Beth and Carlos paint the house together , it will take 6 days.      Log On


   

Question 731272: Andrew, Beth, and Carlos are painting a house. If Andrew and Beth paint the house together, it will take 10 days. If Beth and Carlos paint the house together , it will take 6 days. If Andrew, Beth and Carlos all paint together, it will take 5 days. How long will it take Carlos to paint the house alone?
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Add rates of painting to get rate of
painting together
Let +a+ = Andrew's time in days to paint house alone
Let +b+ = Beth's time in days to paint house alone
Let +c+ = Carlos's time in days to paint house alone
For example:
( 1 house ) / ( a days ) + ( 1 house ) / ( b days ) = ( 1 house ) / ( 10 days )
(1) +1%2Fa+%2B+1%2Fb+=+1%2F10+
(2) +1%2Fb+%2B+1%2Fc+=+1%2F6+
(3) +1%2Fa+%2B+1%2Fb+%2B+1%2Fc+=+1%2F5+
---------------------------
make these substitutions:
+1%2Fa+=+R%5Ba%5D+
+1%2Fb+=+R%5Bb%5D+
+1%2Fc+=+R%5Bc%5D+
-------------
(1) +R%5Ba%5D+%2B+R%5Bb%5D+=+1%2F10+
(2) +R%5Bb%5D+%2B+R%5Bc%5D+=+1%2F6+
(3) +R%5Ba%5D+%2B+R%5Bb%5D+%2B+R%5Bc%5D+=+1%2F5+
----------------------------
(1) +R%5Ba%5D+=+1%2F10+-+R%5Bb%5D+
and
(2) +R%5Bc%5D+=+1%2F6+-+R%5Bb%5D+
------------------------
By substitution:
(3) +1%2F10+-+R%5Bb%5D+%2B+R%5Bb%5D+%2B+1%2F6+-+R%5Bb%5D+=+1%2F5+
(3) +R%5Bb%5D+=+1%2F10+%2B+1%2F6+-+1%2F5+
(3) +R%5Bb%5D+=+3%2F30+%2B+5%2F30+-+6%2F30+
(3) +R%5Bb%5D+=+2%2F30+
(3) +R%5Bb%5D+=+1%2F15+
and
(2) +R%5Bc%5D+=+1%2F6+-+R%5Bb%5D+
(2) +R%5Bc%5D+=+1%2F6+-+1%2F15+
(2) +R%5Bc%5D+=+15%2F90+-+6%2F90+
(2) +R%5Bc%5D+=+9%2F90+
(2) +R%5Bc%5D+=+1%2F10+
It will take Carlos 10 days to paint the house alone
check:
(1) +R%5Ba%5D+=+1%2F10+-+R%5Bb%5D+
(1) +R%5Ba%5D+=+1%2F10+-+1%2F15+
(1) +R%5Ba%5D+=+3%2F30+-+2%2F30+
(1) +R%5Ba%5D+=+1%2F30+
------------------
(3) +R%5Ba%5D+%2B+R%5Bb%5D+%2B+R%5Bc%5D+=+1%2F5+
(3) +1%2F30+%2B+1%2F15+%2B+1%2F10+=+1%2F5+
(3) +1%2F30+%2B+2%2F30+%2B+3%2F30+=+1%2F5+
(3) +6%2F30+=+1%2F5+
(3) +1%2F5+=+1%2F5+
OK