The Painting Triplets

Problem

Dave, Sarah and Brian are triplets. Dave, can paint the TV room by himself in 3 hours. His sister, Sarah can do it in 4 hours. Their brother Brian would take 6 hours on his own.

If they all work together and don’t get in each other’s way, how long will the job take?

Teaching sequence

1. You might like to introduce the class by asking them the "trick" problem:
   If it takes three students 5 hours to dig a hole, how long does it take 4 students to empty it?
2. Pose the problem to the class and ask the students to identify the approaches they think might be useful in solving the problem.
3. As the students work (in pairs or small groups) check that they are expressing the statements algebraically.
   What variables are in this problem?
   What do we know about the triplets and their painting?
   How could you express these?
4. If the some of the students are having difficulty get others to share the way that they started the problem.
5. Share solutions requiring the students to explain their reasoning to the others.
6. Look at and discuss the different ways that the students have expressed the equations and their methods of solution.

Extension to the problem

Suppose Dave can do another job in 3 hours and Sarah can do the same job in 4 hours. Working together, the triplets can now do the whole task in an hour because Brian has had some help on his technique.

How long would it take Brian to do the job on his own?

Solution

Suppose that Dave paints a fraction d of the room, that Sarah paints a fraction s of the room and Brian paints a fraction b of the room. Then

d + s + b = 1.

But this is one equation with three unknowns. What else do we know that can reduce the number of variables for us? Well we know the relative speeds at which the triplets work. So maybe we can get a relation between d and s.

Let’s look at Dave and Brian first because the numbers are simpler there. For every 3 hours that Dave paints, Brian paints 6 hours. So while Dave is doing one room, Brian is only doing half of a room. If Dave paints half a room then Brian paints a quarter of the room. So Brian’s fraction is always half as big as Dave’s. This means that d/2 = b.

If we compare Dave and Sarah we can see that, no matter what job of painting they are doing, for every room that Dave paints, Sarah will paint ¾ of a room. (This is because in 3 hours, Dave will paint a whole room.) So if Dave paints d of a room, then Sarah paints 3d/4 of the room. So s = 3d/4.

So d + 3d/4 + d/2 = 1 or 9/4d = 1 or d = 4/9.

But how does this help? Well, Dave takes 3 hours to do the room on his own. This means that he takes 4/9 of 3 hours when he working with his siblings. So he (and the whole job) takes 4/3 of an hour or 1 hr and 20 minutes.

Solution to the extension:

In 1 hour Dave can do 1/3 of the job and Sarah can do 1/4 of the job. So Brian does the rest. His fraction is therefore 1 - 1/3 - 1/4 = 5/12 of the job. So if Brian does 5/12 of the job in an hour he can do the whole job in 12/5 hours. That is, he can do it in 2 hours and 24 minutes.