Problem:
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted of a house, quitting at PM. On Tuesday, when Paula wasn't there, the two helpers painted only of the house and quit at PM. On Wednesday Paula worked by herself and finished the house by working until PM. How long, in minutes, was each day's lunch break?
Answer Choices:
A. 30
B.
C.
D.
E.
Solution:
Let the length of the lunch break be minutes. Then the three painters each worked minutes on Monday, the two helpers worked minutes on Tuesday, and Paula worked minutes on Wednesday. If Paula paints of the house per minute and her helpers paint a total of of the house per minute, then
Adding the last two equations gives , and subtracting this equation from the first one gives , so . Substitution into the first equation then leads to the system
The solution of this system is and . Note that .
The problems on this page are the property of the MAA's American Mathematics Competitions