Section 6.3 – Multiplication and Division with Fractions
Models for Multiplication of Fractions
I.
Product of a whole number factor and a fractional factor. (Repeated addition)
Sydney made five trips carrying a pail that was one-third full of sand. How much sand did Sydney carry
altogether?
1
Use fraction strips to find 5  .
3
5
1
3
5
2
1
3
3
=
Sydney carried one and two-thirds pails of sand altogether.
Chris ran three-fourths of a mile each of the past three days. How far did Chris run altogether?
3
Use a number line to find 3  .
4
3
4
3
4
0
1
3
4
3
2 9
4
4
5
3 9
1
 2
4 4
4
Chris ran two and one-fourth miles over the three days.
3
State a rule that determines the product of a whole number factor and a fractional factor.
Sample response: The new numerator is the product of the whole number and the numerator of the fraction.
Keep the same denominator.
II. Product of a fractional factor and a whole number factor.
Lynn plans on giving two-fifths of his fifteen rabbits away. How many rabbits will Lynn give away?
2
Use a discrete model to find  15 .
5
2
 15  2  3  6
5
Lynn will give six of his rabbits away.
-1-
Jamie has worked two-thirds of the fifteen hours she needs to work this week. How many hours has Jamie worked?
2
Use a bar diagram to find  15 . (Variation of fraction strips.)
3
15
5
5
5
2
 15  2  5  10
3
Jamie has worked ten of her fifteen hours this week.
Find
1
 5 . (Use the commutative property.)
4
1
1 5
1
 5  5   1
4
4 4
4
Find each of the following products.
2
of 28
7
28 
3
4
1
of 24
6
3
 27
8
State a rule that determines the product of a fractional factor and a whole number factor.
Sample response: The product is found one of the two following ways:
(1) Divide the whole number by the denominator of the fraction and then multiply the quotient by the
numerator of the fraction; or
(2) The new numerator is the product of the whole number and the numerator of the fraction. Keep the
same denominator.
III. Product of two fractional factors. We use a variation of the area model that we used when multiplying whole
numbers.
Review example. Illustrate 3 × 4 = 12.
Pat used two-thirds of a bag of flour that was three-fourths full. How much of a full bag of flour did Pat use.
Find
2 3
 .
3 4
2 3 6 6 1 1
 


3 4 12 6  2 2
Pat use one-half of a full bag of flour.
Model each problem and find each product.
3 1

7 2
3 2

4 5
-2-
State a rule that determines the product of two fractional factors
The numerator of the product is the product of the numerators and the denominator of the product is the
product of the denominators.
Properties of Multiplication of Rational Numbers.
a c
 is a unique rational number.
b d
a c c a
Commutative Property for Rational Number Multiplication.
  
b d d b
a c e a c e 
Associative Property for Rational Number Multiplication.         
b d f b d f 
Closure Property for Rational Number Multiplication.
a
a a
 1  1 
b
b b
a b b a
    1 where a and b are nonzero. The rational number
Inverse Property for Rational Number Multiplication.
b a a b
b
a
a
is called the multiplicative inverse of
(or reciprocal). Also, the rational number
is called the multiplicative
a
b
b
b
inverse of (or reciprocal).
a
a c e  a c a e 
           .
Distributive Property of Multiplication over Addition of Rational Numbers.
b d f  b d b f 
Identity Property for Rational Number Multiplication.
IV. Product of mixed number factors.
1 1
Use the properties to find 2  1 .
3 4
1 1  1  1
2 1   2   1  
3 4  3  4
 1
 1 1
  2  1  2  
 3
 3 4
1
2
Use the properties to find 3  2 .
2
3
1
2  1 
2
3 2  3 2 
2
3  2 
3
 1
 1 2
 3 2 3 
 2
 2 3
1
1 1


  2   1  2   
3
3 4



1
1 1 1
 2 1 1 2   
3
4 3 4
1 1 1
 2  
3 2 12
4 6 1
 2  
12 12 12
11
2
12
1
1 2


 3  2 3 
2
2 3


1
2 1 2
 3 2   2  3  
2
3 2 3
1
 6  1 2 
3
1
9
3
Convert to improper fractions and use the rule for multiplying fractions to find:
1 1
1
2
2 1
3 2
3 4
2
3
4
1 1 7 5 35
11
2 1   
2
3 4 3 4 12
12
1
2 7 8 28
1
3 2   
9
2
3 2 3
3
3
1
-3-
Which of the two methods do you prefer? Is the choice different for multiplication than for addition?
-4-
IV. Find the reciprocal (multiplicative inverse) for each and justify.
3
7
0
4
4
since
3
1
2
3
1
since
7
no reciprocal
3
since
14
1 7
7  1
7 7
1
is undefined
0
2 3 14 3 1
4  

 1
3 14
3 14 1
1
1
3 4 1
  1
4 3 1
1
4
1
1
1
1
Division of Fractions
Many find the division of fractions to be one of the most difficult concepts to teach and for people to understand,
often due to the difficulty in creating meaningful concrete models. We begin with some examples of "real-world" division
problems.
Example Problems.
1.
Your mother told you to get $5 in quarters. How many quarters should you bring back?
Model:
1
 20
4
You would bring twenty quarters back.
5
This is the repeated subtraction model for division that we used with whole numbers. In this case where we are
dividing by a fraction, some people call this model shattering since we shatter five objects (dollars) into quarters.
2.
Dad told you to give the chickens four pails of feed. You are only strong enough to carry half a pail at a time.
How many trips do you need to make?
Model this problem and write a mathematical sentence.
Model:
1
8
2
You would need to make eight trips.
4
Also, compare the problem to the situation where you are able to carry two pails at a time.
Model:
4÷2=2
You would need to make two trips.
3.
You need to lay tile to create a frieze above a doorway. The tile measures 2
1
inches wide, how many pieces of tile are needed?
2
Write a mathematic expression for this problem.
doorway is 35
-5-
7
7
inches by 2 inches. If the
8
8
1
7
35  2
2
8
Algorithms for Dividing Fractions
Here are three different approaches for justifying the traditional invert-the-divisor-and-multiply algorithm.
I.
Divide Numerators and Denominators Approach
Consider that you have just completed learning how to multiply fractions, how would you naturally want to divide
fractions?
Try dividing the following fractions by using the approach used in multiplying fractions, that is, divide numerators
and divide denominators.
Examples.
8 2

9 9
8 2 82 4
 
 4
9 9 99 1
4 2

15 3
18 6

35 7
4 2 42 2
 

15 3 15  3 5
18 6 18  6 3
 

35 7 35  7 5
This method is easy and consistent with multiplication, but what problems may occur?
Use the concept of equivalent to fix the problem with the divide numerators and denominators approach.
Examples.
5 3

7 4
5 3 5  12 3 5  4 20
 
 

7 4 7  12 4 7  3 21
2 5

3 11
1 3

4 5
2 5 2  55 5 2  11 22
7
 
 

1
3 11 3  55 11 3  5 15
15
1 3 1 15 3 1 5 5
 
 

4 5 4  15 5 4  3 12
Note how the method leads to the traditional algorithm of invert-the-divisor-and-multiply approach.
II. Compensation Property for Division Approach
Given any division problem, we may multiply both the dividend and the divisor by the same nonzero number and the
quotient will remain the same. Symbolically, a  b  (a  c)  (b  c) where c 0.
Example. 24  6 = (24  2)  (6  2) = 48  12 = 4.
We use this concept to develop another method to motivate the invert-the-divisor-and-multiply approach. We use
three properties: Compensation Property, Inverse Property, and Identity Property. The approach is based on asking
the question: "What is the easiest number to divide by?”
3 2

4 5
5 3

7 4
2 5

3 11
-6-
3 2  3 5 2 5
      
4 5  4 2  5 2
 3 5
    1
 4 2
3 5 15
7
  
1
4 2 8
8
5 3  5 4  3 4
      
7 4 7 3  4 3
 5 4
    1
 7 3
5 4 20
  
7 3 21
2 5  2 11   5 11 
      
3 11  3 5   11 35 
 2 11 
    1
3 5 
2 11 22
7
  
1
3 5 15
15
Note how the method leads to the traditional algorithm of invert-the-divisor-and-multiply approach.
-7-
III. Missing-Factor Definition for Division Approach
5 3

7 4
2 5

3 11
1 3

4 5
5 3
 N
7 4
3
5
N 
4
7
4 3 
  N  
3 4 
4 3
   N 
3 4
4 5
1 N  
3 7
4 5
N 
3 7
5 4
N 
7 3
2 5
 N
3 11
5
2
N 
11
3
11  5
 11 2
  N   
5  11  5 3
11 2
 11 5 
   N  
5 3
 5 11 
11 2
1 N  
5 3
11 2
N 
5 3
2 11
N 
3 5
1 3
 N
4 5
3
1
N 
5
4
5 3  5 1
  N   
3 5  3 4
5 1
5 3
   N  
3 4
3 5
5 1
1 N  
3 4
5 1
N 
3 4
1 5
N 
4 3
Missing-Factor Def.
4 5

3 7
4 5

3 7
Multi. Prop. of Equality
Associative Property
Inverse Property
Identity Property
Commutative Property
Hence we have
5 3 5 4
  
7 4 7 3
2 5 2 11
  
3 11 3 5
1 3 1 5
  
4 5 4 3
Note how the method leads to the traditional algorithm of invert-the-divisor-and-multiply approach.
Division with Mixed Numbers
1 2
2 1
5 3
5
3
1 2
8
4
1 2 11 5 11 3 33
8
2 1     
1
5 3 5 3 5 5 25
25
5
3 13 11 13 4 13
1 2     
8
4 8 4
8 11 22
1
2
-8-
Problems and Exercises
1.
1 1

4 8
2.
5
 24
8
3.
2 1

5 4
4.
9
 12
10
5.
7 4

12 5
6.
8
7.
2 5

3 12
8.
7 7

8 32
9.
3 3
2 
5 4
3
8
1
2
10. 4  2
2
3
5
4
11. 3  2
8
5
12.
3 4 5
 
8 5 6
1
3 1
13. 2  3  5
2
4
3
7 15
14. 1 
8 32
15.
7
3
5
10
5
3 1
16. 6  1
4 8
1
3
17. 4  4
6
8
4
1
18. 5  2
5
2
19. Three-fourths of a pan of brownies was setting on the kitchen table. Jerry and Terry ate two-thirds of that partial pan
of brownies. What portion of a pan of brownies did they eat?
20. You need to lay tile to create a frieze above a doorway. The tile measures 2
doorway is 35
7
7
inches by 2 inches. If the
8
8
1
inches wide, how many pieces of tile are needed?
2
21. A recipe calls for two-thirds of a cup of sugar. You find that you only have one-half of a cup of sugar left.
If the recipe calls for one and one-fourth cups of flour, how much flour will you use?
22. The area of a floor that is fourteen and one-half feet wide is two hundred thirty-nine and one-fourth square feet.
Find the perimeter of the floor.
-9-