Answer to the May 10, 2004 Problem of the Week

House Painting Problem

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Akash, Bella, Cassie, and David are house painters. The team of Akash, Bella, and Cassie can paint my house in 3 days; if Akash and David work as a pair it takes 6 days; the pair of Cassie and David takes 8 days, and the team of Bella, Cassie, and David can finish in 5 days. How long would it take the whole 4-person crew to paint my house? (Answer to the nearest minute if necessary)

 

Solution to the Problem:

The answer is 2 and 6/7 days, which is equivalent to 2 days, 20 hours, 34 minutes.
Let A = days Akash would take to paint the house himself,
B = days for Bella alone,
C = days for Cassie alone,
and D = days for David alone.

Then Akash can do 1/A of the house in a day.
If Akash, Bella, and Cassie take 3 days,
then the fractions for one day give : (1/A) + (1/B) + (1/C) = 1/3 ;
letting small letters a, b, c, d replace 1/A, 1/B, 1/C, 1/D :
we get the linear system
{a + b + c = 1/3 ,
  a + d = 1/6 ;
  c + d = 1/8 ;
  b + c + d = 1/5}.

The solution is : a = 3/20 , b = 3/40 , c = 13/120 , d =1/60 ;
so that all four together would paint a + b + c + d = 7/20 of the house in a day;
it would take 20 / 7 days which is 2 and 6/7 days.
This is 2 days, 20 hours, 34 minutes and change.

James Alarie responded, "You asked for the answer to the nearest minute, but 'nearest' is ...34 minutes and the job would not be complete. I think that we had better allow them ...35 minutes."