So, I was looking at G's math homework, and there's
this word problem that reads:
"Susan can paint her living room in 2 hours. Jaime can
paint the same room in 3 hours. How long will it
take both of them to paint the room together?"
First of all, who can paint a living room in 2 or 3 hours?
We must suspend disbelief. Okay. I get it.
Now, the first impulse is to average their times,
but that means that it would take Susan 30 minutes
longer to paint the room with help, than doing it
alone, and while that is possible (and not unheard of)
we are going to hope that Jaime is not a toddler,
and is actually really helpful.
So, let's imagine we cut the room in half, and
let each of them paint half of the room.
After one hour, Susan is done with her half,
but Jaime still has a third of his half to do.
This is the personality test portion of this word
problem. Is Susan the kind of person who will
sit on her tush and drink whipped coffee drinks
while Jaime finishes his part of the room?
Or will she help him?
If she goes the blended drink route, it will take
1 1/2 hours to paint the room. If she decides to
help him.. well.. that's an interesting idea.
(This is the part of my thinking about the word
problem where Big E began to groan and shake his
So, Jaime has 1/3 of his half of the room left
to paint (this would be 1/6 of the total room).
If we cut this portion in half, (1/12 of the total
room) it would take Susan 10 minutes to paint
her half, but Jaime would still have 1/3 of the
1/12 left unpainted (or 1/36th of the entire room).
You can see that in this method of calculation,
the room would theoretically never be painted.
Some infinitesimal space would be left undone.
Happily, in reality, Susan would just push Jaime
aside at a certain point and say, "Dude, we're done."
With all of the reasonable times added up, it would
take them about 1 hour and 12 minutes.
But, there is another way of looking at the problem.
If we combine their rates, instead of dividing the
room, we can cut out a lot of circular work.
If we consider that Susan paints 1/2 of a living room
per hour, and Jaime paints 1/3 of a living room per
hour, we can combine their rates: 1/3 + 1/2=5/6.
So, together they paint 5/6 of a room in an hour.
This means that in 6/5 of an hour, they paint a room.
Or, in normal time speak, 1 hour and 12 minutes.
Of course, this is the type of word problem that
gave me fits in school. Which answer is right?
Is working with Jaime worse than working alone?
Is Susan the kind of person that will just do her work
and then leave Jaime to finish up on his own?
Or, will they work, efficiently and kindly together?
Ah, the complexity of life.