Expert: Josh - 1/17/2010

Question
QUESTION: Hello:

Here is my question:

If two house painters, working at the same rate, can paint one small house in one day, one house painter can paint the same size house in how many days, and likewise one house painter can paint how many houses of the same size in one day?

Since there is one painter compared to two, I would think it would take twice as long, two days, for one painter to paint the same size house, and perhaps one half of the house in one day. But I am not sure.


ANSWER: That is correct.

---------- FOLLOW-UP ----------

QUESTION: Hello:

I want to thank you for your reply.

How can I mathematically show that my answers are correct?  Can you think of a simple calculation?

Answer
As usual, we need to introduce each variable to set this up properly.

Definitions:

Let R(1) be the work rate of painter 1.
   R(2) be the work rate of painter 2.
   T be the time spent on the job.
   W be the amount of work to be completed (for one small house).

Initially,
W = [R(1)+ R(2)] * T
Simplification: Since their work rates are the same, we can let R(1)=R(2)=R.
W = 2*R*T .....[#1]

When painter 1 walks away, leaving the other to do the whole thing, the work rate of painter 2 remains constant. So, we have W = R(2)*t. More simply, we can write
W = R*t   .....[#2]
Substituting W=2RT from [#1] into the left hand side of [#2]
2*R*T = R*t

Thus, the time taken to complete the job with one painter (t) is given by:
t = 2*T (i.e., twice as long as before).

With time restriction, setting t=T, now the amount of work done (w) is allowed to vary, we see
w = R*T       ....comparing with initial situation where W = 2*(R*T), w = W/2.
This means only half the house gets painted.

For a question like this, it is not really worth all the trouble setting this up IMO. But if it's a mathematical argument that you are after, this is the way to do it. We can forget about it being any simpler. If we omit all the words and explanation, it's about as simple as it gets.

This type of setup is also used to explain the behavior of ideal gas in Chemistry. i.e., what happens to its volume when the pressure is halved...etc. It reminds me of Boyle's law.