



If
you wanted to find out how many feet were in two yards you might say 


"Well
let's see, there's 3 feet in one yard so there must be 6 feet in 2
yards." 


That
works just fine. 





But
what if you wanted to find out how many inches there are in 2,500
miles. 


Or
find out how many feet per second there are in 55 miles per
hour. 


Then
we need a better plan. 





Good
news! We have one. 


Here
it is: 





Take
another look at that 


"How
many feet are in 2 yards?" problem. 





We
have 2 yards, and we want to have our answer in feet. 


To
get there, we need to do something very, very tricky. 


We
need to multiply ... 





by
1. 





OK,
OK, there really IS a trick. 


The
trick is the way we write the number 1 that we use to multiply. 


Any
time we have a fraction 


with
the same amount on the top and the bottom 


we
have a fraction equal to 1 (except when that amount is zero). 





Anyway, 


if
we have a fraction with 1 yard on the top and 1 yard on the
bottom, 


we
have a fraction equal to 1: 











We
also know that 1 yard is equal to 3 feet. 





So
instead of 1 yard, 


we
can write 3 feet for either the top part 


or
the bottom part of our fraction 


and
we haven't changed anything. 





Like
this: 











THIS
IS OUR BIG NEW TRICK ON THIS PAGE!






The
trick is the way we write the fraction that is equal to 1. 


We
write it with things that LOOK different on the top and the
bottom, 


but
are actually worth the same amount. 





Now
we have our new special name for 1. 


We
are ready to multiply the 2 yards: 











To
make things even out, we can put a denominator of 1 under the 2
Yards. 














OK,
We did it, but look! 


What
kind of a thing is "6 Yards x Feet" ??? 





Who
Knows! 





And
if we had to leave it looking like that 


we
wouldn't have anything too great. 





But
look! We have the word yards in the top 


and
the word yard (close enough) in the bottom part. 





We
can cancel them out. 





You're
kidding! 





Nope!
Works every time. 


The
only thing is that everything on top and everything on the
bottom 


must
be multiplied. No addition or subtraction. 





So
... 











That
means: 





2
Yards = 6 Feet






Hey,
we KNEW that 6 Feet was the answer all along. 


So
this is no shock. 





But
let's go over exactly how we come up with the funny name for 1. 





The
fraction gets units that are what we have to begin with 


and
what we want to end up with. 





We
had yards and wanted to get to feet. 


We
know we want feet for 1 number and yards for the other. 





Which
gets which? 





Do
we want: 











Here's
the deal. 


Put
the units you want on top and the units you have on the bottom. 





The
units you have go on the bottom so you can cancel them! 





We
had yards to begin with, so the one we want is: 











Try
another one ... 





Example: 





Suppose
we're really bored and want to find out 


how
many feet there are in 2500 miles. 


You
look up (or maybe you know) that there are 5,280 feet in a
mile. 





That
is: 





5,280
Feet = 1 Mile






Since
these 2 things are worth the same amount, 


we
can use them in a fraction as our names for 1. 


We
have miles and want feet, 


so
miles goes on the bottom and feet goes on the top: 











Now
we multiply 2500 miles by this fraction that's equal to 1. 


So
everybody has a denominator, we put a 1 under the 2500 miles 


(a
calculator really helps here): 











Since
we have miles on the top and mile on the bottom, 


and
everything is multiplied, 


we
can get rid of them. 











Hey
that's a lot of feet! 





OK,
One more and we're done. 





Example: 





Tennessee
Ernie Ford asks you to change 16 tons to
ounces. 





If
you look in a reference book, you will find that: 





1
ton = 2000 pounds






You
will also find that 





1
pound = 16 ounces






But
you probably won't find how many ounces are in a ton. 





So
here's the plan. 


First
we change 16 tons to some number of pounds. 


Then
we change that number of pounds to ounces. 





Ready? 


OK,
we have 16 tons and want pounds. 


That
means pounds goes
on the top and ounces goes on the bottom. 


2,000
Pounds = 1 Ton. 











We
have Tons on the top and Ton on the bottom. 


Everything
is multiplied. 


So
we can cancel tons
of stuff. (he
he he ... OK, it wasn't funny) 











One
part down and one to go. 


Now
we have pounds and want ounces. 


So
16 Ounces goes on top and 1 Pound goes on the bottom. 


1
Pound = 16 Ounces. 











We
have Pounds on the top and Pounds on the bottom. 


Everything
is multiplied. 


We
can cancel the Pounds. 











copyright 2005 Bruce Kirkpatrick 
