



Here's a
real cute type of problem that you might see: 





Fred can
mow the yard in 3 hours. 


Fred Jr
can mow the yard in 6 hours. 


How long
will it take if they both work on it? 





Well in
real life there are all kinds of questions like 


"Will
they get in each other's way?" and 


"Do
they have two mowers?" and stuff like that. 


For these
problems, we just figure that they 


have all
of the equipment that they need, 


and that
they don't get in each other's way. 





Here's
how we solve it. 





It takes
Fred 3 hours to mow the entire yard 


so in 1
hour, he will be able to mow ^{1}/3 of the yard. 








It takes
Fred Jr. 6 hours to mow the entire yard 


so in 1
hour, he will be able to mow ^{1}/6 of the yard. 





So in 1
hour together, they mow: 











So here
comes the big trick of these problems ... 





If they
do 1/X of the job in 1 hour, 


then the
whole job takes X hours! 





So the
"set up" on this type of problem is:












X is how
long it takes to do the entire job 


with both
of them working on it. 





The first
thing we need to do to solve this, 


is to get
a common denominator on the left side. 











We want
to wind up with "X = STUFF" 


so the
first thing to do is get X out of the denominator. 





If we
multiply both sides by 2X (the product of the 2 denominators), 


we get
rid of both denominators at the same time. 


Tricky,
eh? 











So
working together, it takes Fred and Fred Jr 


2 hours
to mow the lawn. 





Example: 





Pump 1
can fill the pool in 6 hours by itself. 


Pump 2
can fill the pool in 4 hours by itself. 


Pump 3
can fill the pool in 8 hours by itself. 





How long
does it take if all 3 pumps are used? 





OK
FREEZE, STOP, HALT, TIME OUT ... 





Let's
think about this for a second. 


How long
would you guess it will take? 





1 hour?
10 hours? what? 





Well, if
pump 2 did all the work 


and the
other two pumps just stood around, 


it would
take 4 hours. 





So as
long as pump 2 is on the job, 


it can't
take any longer than that.. 





If there
was one other pump and it was as fast as pump 2 


they
would both do half the job and be done in half the time. 


That
would be 2 hours. 





Here we
have 2 other pumps besides pump 2. 


They are
both slower than pump 2, 


but maybe
between the two of them they are as good 


as
another pump 2 would be. 





That
means that a guess of 2 hours for the 3 pumps 


might be
close to right.. 


Lets see
... 





Set up
the problem: 











Find a
common denominator. 





6 can be
broken into 3 x 2 


4 can be
broken into 2 x 2 


8 can be
broken into 2 x 2 x 2 





The least
common denominator is made up 


of the
most times any factor appears in any denominator 


so we
have one 3 in the first denominator 


and three
2's in the third denominator. 


That is
... 


3 × 2 × 2
× 2 = 24






So we
multiply each fraction by another name for 1 


that
contains the factors it's denominator is missing. 





For
example, 6 contains a 2 and a 3, 


so it's
missing two of the 2' (2 x 2 = 4) 











Now get
this to look like X = stuff 


We need
to multiply by X and by 24 


to get
rid of the denominators. 





We can do
it all at once like we did last time. 


Or take
them one at a time. 


Let's do
that this time. 





Multiply
each side by X and simplify ... 











Multiply
each side by 24 and simplify ... 











Now
divide by 13 to get X by itself ... 











Now use a
calculator to find 


a decimal
equivalent to ^{24}/13 ... 





X = 1.846154
hours 





Convert
fractional hours to minutes ... 











Last
Example: 





Joe can
paint the barn in 4 hours by himself. 


Steve has
never timed himself painting a barn before 


so we
don't know how fast he is. 





Working
together, the two finish the job in 2.4 hours. 


How long
would it take Steve to paint the barn by himself? 





The
formula for doing this type of problem is: 











In the
other problems we've done 


we did
not know the total time. 





Here we
do know the total time for the job, 


but we
don't know the "time for the second person alone." 


We have
... 











This will
take a bit longer to solve, 


but it's
really no big deal. 





First get
a common denominator on the left. 


The
factors are 2 x 2 x X = 4X. 











Multiply
both sides by 4X, and simplify ... 











Multiply
both sides by 2.4 and simplify. 


A
calculator really helps here ... 











Subtract
2.4X from each side ... 











Divide
both sides by 1.6 and simplify ... 











So it
would take Steve 6 hours 


to paint
the barn by himself. 





copyright 2005 Bruce Kirkpatrick 
