Pat, Tracy and Adrian, when all working together can paint a room in 2 hours. Pat painting alone can do the entire job in 5 hours. Tracy when working alone can do it in 6. Working alone, how long will it take Adrian to do the entire job?

Question

Pat, Tracy and Adrian, when all working together can paint a room in 2 hours.  Pat painting alone can do the entire job in 5 hours.  Tracy when working alone can do it in 6.  Working alone, how long will it take Adrian to do the entire job?

anonymous · Answer

i will keep quiet

hero · Answer

p = time pat can paint the room alone
t = time tracy can paint the room alone
a = time adrian can paint the house alone
h = hours they can paint the house together

Represented by the following formula
$$\frac{pta}{pt + pa + ta} = h$$

hero · Answer

If p = 5, t = 6 and h = 2, then:

$$\frac{(5)(6)a}{(5)(6) + 5a + 6a} = 2$$

hero · Answer

$$\frac{30a}{30 + 11a} = 2$$

hero · Answer

From here you can solve for a

zarkon · Answer

I would write it as 

$$\frac{1}{5}+\frac{1}{6}+\frac{1}{a}=\frac{1}{2}$$

(using work rates)

then solve for a

anonymous · Answer

a = 15/2 hours for both solutions.

anonymous · Answer

Thanks everyone for the help. Figured it out after receiving the formula.