The first time I saw a worker rate problem was within the movie “Little Big League.” I don’t really like how it was solved in the movie, but there are definitely some humorous moments in the clip:
“If Joe can paint a house in 3 hours and Sam can paint the same house in 5 hours, how long does it take for them to do it together?”
The key to solving a worker rate problem is to convert the word problem into single hour (hourly) rates.
“Joe can paint a house in 3 hours,” in the form of an hourly rate, means “Joe can paint 1/3 of a house in 1 hour.”
“Sam can paint the same house in 5 hours” means “Same can paint 1/5 of the same house in 1 hour.”
When we add 1/3 and 1/5, we get the combined hourly rate of both Joe and Sam working together. Since their least common multiple is 15, we’ll add 5/15 to 3/15.
5/15 + 3/15 = 8/15 of a house per hour (combined rate of Joe and Sam)
Now we go back and reference the question to see exactly what they are asking for, as 8/15 is not the final answer. We want to know how many hours it will take to finish painting a single house. Using our combined rate, mathematically that would be:
( 8/15 house per hour) x (number of hours) = 1 house
To get to the “number of hours” variable, we simply divide “1 house” by the combined hourly rate: 8/15.
This gives us a final answer of 15/8 or 1 and 7/8.
Be weary of variations the GMAT may try to throw at you. For example, if the question were to state, “Joe can paint 3 houses in one hour,” the fraction for an hourly rate would be 3/1. If the question were to state Joe can paint 2 houses in 3 hours,” the fraction for an hourly rate would be 2/3.
Also make sure to check carefully what the question is asking for at the end. A GMAT question like this could easily try to throw you off at the end by asking how many hours does it take to paint 3 houses.
This can be dealt with easily by simply changing the right hand side of the previous formula from 1 house to 3 houses.
( 8/15 house per hour) x (number of hours) = 3 houses
That gives us a final answer of 45/8 or 5 and 5/8.
