Pre Algebra Multiplicative Inverse as Division
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No More Division
Multiplicative Inverse as Division

 
 Most people like multiplying more than they like dividing. 
 It would be great if we could change division into multiplication.
 
 WE CAN!
 
 Say we have 21 and we want to divide it by 7:
 

 21 7

 
 Instead of dividing by 7 
 we need to find something to multiply by instead.
 
 What?
 
 Here's the deal, instead of dividing by 7 we multiply by 1/7
 We multiply by the thing that times 7 equals 1. 
 It is called the Multiplicative Inverse of 7. 
 
 Step 1:
 Give the number a denominator of 1 ...
 

 
 Now we multiply this by 1/7
 Multiplying fractions together is pretty easy. 
 To multiply them, 
 you just multiply the numerators (numbers on top) 
 and the denominators (numbers on the bottom).
 

 

 Now simplify if possible. 

 21 is equal to 7 x 3 so: 
 

 
 Example:
 
 Divide 28 by 4.
 

 28 4 = ?

 
 The take the 4 and flip it over to get 1/4 and multiply:
 

 
 Now simplify if possible. 
 28 = 4 x 7 so:
 

 
 We could have broken those 4's into 2 x 2 
 to get all the way to prime factors before simplifying if we wanted to.
 
 OK, so far this is pretty easy. 
 In fact, we have seen it all on earlier pages in a slightly different way. 
 
 But here's where we really need this stuff:
 
 Divide 3 by 2/3 .
 
 Icky! Icky! Nasty! Nasty! 
 What do we do? 
 
 Don't sweat it. 
 Just flip over the thing you are dividing by and multiply:
 

 
 We can write the 3 as 3/1 to make things look nice if we want.
 

 
 Always look to see if you can simplify these. 
 This time, the number on top breaks into prime factors as 9 = 3 x 3 
 (hmm, I think we saw those 3's a second ago). 
 There are no 3's that are in that denominator of 2. 
 That means we can't simplify this one.
 
 Here's another one ...
 
 Example:
 
 Divide 3/5 by 2/3 :
 

 
 and
 

 
 The prime factors in the top part are 3 x 3. 
 The prime factors in the bottom part are 5 x 2. 
 No numbers show up in both places so there's nothing to cancel.
 
 OK, One More ...
 
 Example:
 
 Divide - 3 by 3/4 :
 
 OH NO! NOT A NEGATIVE NUMBER!
 
 Yeah a negative number. No big deal. 
 The negative number doesn't change a thing. 
 Do the problem the same way. 
 Put a 1 under the -3 as a denominator and we're off:
 

 
 Now multiply:
 

 
 Try to simplify by breaking both parts into prime factors.
 
 The prime factors of - 12 are -1 x 2 x 2 x 3
 
 The prime factors of 3 are, hmm, 3. 
 So
 

 

   copyright 2005 Bruce Kirkpatrick

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