Problem Solving

Vincen wrote:Amy and Emily need to build 2 identical houses. Amy, 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, Amy 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

B is the OA.

I am confused here. I don't know how to start. With the probability or the rates. Experts, can you show me how to solve this PS question. I'd really like to see the answer.

Executive decision: I'm changing the names here.

First, rephrase the question.

If Amy can do 1 house in 6 weeks, she can do 2 in 12 weeks. If Emily can do 1 house in 8 weeks, she can do 2 in 16 weeks. Neither alone can do 2 houses in 7 weeks.

Together: Amy's rate is 1/6 and Emily's rate is 1/8. Their combined rate is 1/6 + 1/8 = 14/48 = 7/24.
Rate and Time have a reciprocal relationship, so the time to do one house together would be 24/7. Thus, the time to do two houses together would be 48/7. This is less than 7.

So our rephrased question: what is the probability that Amy and Emily work together?

Desired outcomes (work together): Roll a 5 or 6 --> so 2 desired outcomes
Total possible outcomes: 6

Desired/Total = 2/6 = 1/3. The answer is B