Work problems are definitely not as common on the GMAT as, say, solving simultaneous equations might be; but many test-takers are wary of these problems since they are not as commonly used in everyday life as averages are, for example. The key to most of these problems, though, is to know the work formula, and how to use it. Try the challenge problem below for an advanced twist that includes probability along with the work formula.

**Sample Problem:**

Mike and Emily need to build 2 identical houses. Mike, working alone, can build a house in 6 weeks. Emily, working alone, can build a house in 8 weeks. To determine who will do the building they will roll a fair six-sided die. If they roll a 1 or 2, Mike will work alone. If they roll a 3 or 4, Emily will work alone. If they roll a 5 or 6, they will work together and independently. What is the probability both houses will be completed after 7 weeks?

A) 0

B) 1/3

C) 1/2

D) 2/3

E) 1

**Solution:**

The first step in solving this problem is to determine how long it would take them to build one house working together. The fastest way to do this is to use the combined work formula, which is AB/(A+B), where A is the time it takes the first person working alone and B is the time it takes the second person working alone. In this problem, the equation gives us (6)(8)/(6+8) = 48/14.

48/14, however, is the amount of time it takes Mike and Emily to build one house together and the problem specifies that they must build two houses. To determine how much time it takes them to build two houses, simply double the time it takes to build one house. Thus, it takes them 2(48/14) = 96/14 weeks to complete both houses. 96/14 expressed as a mixed number is 6 6/7 or 6 weeks and 6 days.

Next, we need to consider the probability component of this problem. We have a 1/3 chance of Mike working alone, a 1/3 chance of Emily working alone and a 1/3 chance of them working together. If Mike works alone, two houses take 12 weeks to build; if Emily works alone, two houses take 16 weeks to build; and if they work together, two houses take 6 weeks, 6 days to build. Therefore, there is a 1/3 chance that both houses are completed in less than 7 weeks, which corresponds to choice (B).

~Bret Ruber

The question is: What is the probability both houses will be completed after 7 weeks? As a result, the answer is in fact, 2/3 (D).