Mathcounts problem...it's tough?

Question

Julian and Vino working together can paint a room in 11 hours. Vino and Ted working together can paint the same room in 10 hours, and ted and Julian working together can paint the room in 11 hours. What is the number of hours that it will take Julian, Vino and Ted working together to paint 31 identical rooms?

Answer 1

Looks like Vino and Ted oaint 1/20-th of a room /hours when working alone, now let's figure out how much time Julian takes:
11(J+1/20)=1
J=1/11-1/20=9/220

Now we can go get the answer:
H(1/20+1/20+9/220)=31
H*31/220=31
H=220 hours

Answer 2

Try considering "painting" as a unit of work.

First, lets figure out how much work each person generates. We know that Vino and Ted generate the same amount of work, because they both take the same amount of time to finish a room with Julian's help.

Assume that painting a room takes 100 arbitrary "units" of work. Vino and Ted together accomplish this task in 10 hours, meaning that they generate 10 units/ hour together. Because we know that they are equal, we can say that they generate 5 units/ hour each.

How about Julian? It takes eleven hours for him to finish a room with the help of either other person. Since we know that the other person will generate 55 units in 11 hrs, Julian must produce the other 45 units. So, Julian's rate of production is 45/11. That's a messy fraction, so let's just leave that as it is for now. Leaving the 11 as the denominator, we can say that all three painters, working together, can generate 155 units in 11 hours.

We need 3100 units of work to finish our entire project (31 rooms * 100 units per room). We can figure out the amount of time as follows:

hrs =
3100 units / 155 units / 11hrs =
(3100 units * 11 hrs) / 155 units =
220 hrs

Answer 3

J = Julian      V = Vino      T = Ted

J + V = 1 / 11
V + T = 1 / 10
T + J = 1 / 11

J + V = T + J      V = T
2 V = 1 / 10      V = 1 / 20
J + 1/20 = 1/11      J = 9 / 220

Julian can do 9/220 rooms per hour.
Vino can do 1/20 (or 11/220) rooms per hour.
Ted can do 1/20 (or 11/220) rooms per hour.
Together, they can do (9+11+11) / 220 rooms per hour.

(9+11+11) / 220 = 31 rooms in 220 hours

Answer 4

h stands for hours, r for rooms, Julian is J, Vino is V and Ted is T.

J & V = 11 hours
V & T = 10 hours
T & J = 11 hours

This means that V and T have the same "speed" and if together thay can paint a room in 10 hours, means that their speed is 1 room in 10 hours (0.1 r/h) each.

Knowing that let's calculate J speed:

(J speed + V speed)*hours=rooms => J speed=rooms/hours - V speed

J speed= 1r/11h - 0.1 r/h => J speed= (1/11-1/10)r/h

J speed= -1/110 (amazing, he unpaints! XD)

Using the same equations above:

(J speed + V speed + T speed)*hours = rooms

Then:

hours=rooms/(V speed + Y speed + J speed)

Hours= 31r /(1/10+1/10-1/110) r/h

Hours=148,26 hours

Answer 5

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Answer 6

Approximately 155 hours?

Answer 7

220 hours.