Fraction Multiplication

Motivation Problem

 Solve the following problem.

Pat used three-fourths of a bag of flour that was two-thirds full. How much of a full bag of flour did Pat use.

Here, we solve this problem by the use of an area model where we will use a rectangle to represent a bag of flour. We shade  of the rectangle to show that we have two-thirds of a bag of flour.

Next, since Pat used of the bag of flour, we shade of the of the rectangle.

We now have  of the rectangle that has been shaded twice; this is the region that represents the amount of flour used by Pat. Therefore, Pat used one-half of a full bag of flour. 

Note that this problem is an example of . We could have also illustrated the problem with the following model which builds on the whole number area model for multiplication.

The diagram has  shaded horizontally and shaded vertically. The solution is the amount of area shaded both horizontally and vertically.

Note that we have completed the multiplication problem: .

Multiplication of Fractions

Example: A recipe calls for   cup of flour. If you are making only the recipe, how much flour do you use? 

We can draw a diagram to work this problem, but it will take us several steps to work it out.

We begin with a rectangle as our whole to represent a full cup of flour.

Then to represent  of a cup of flour we divide the rectangle into 4 equal pieces and shade 3 of them. The shaded part is of the whole.

So how do we take half of this shaded part of the rectangle when there are 3 shaded parts?

One way would be to split the rectangle in half horizontally. Notice that we still have the same part of the rectangle shaded, but it is now divided into eighths.

Finally, since we want one-half of the partial cup of flour, we choose half of the shaded boxes. This leaves of the rectangle shaded.  

We conclude that of is .

A Word of Caution

It is important to notice that in this problem we have taken half of and have not divided by. Remember that of is usually translated as multiplication. What we have computed is .

Rule for Multiplying Fractions

A more efficient way of multiplying two fractions is to realize that , that is, we can multiply the numerators to get the numerator of the product and we can multiply the denominators to get the denominator of the product. Even better, if we remove all the common factors from the numerator and denominator before we multiply; our final answer will be in simplest form. That is, we can simplify by dividing common factors where for any common factor one is divided from the numerator and one is divided from the denominator.

Example: Evaluate .

Example: Evaluate .

Methods of Simplifying Multiplication

Two Methods of Simplifying

One method is to use prime factorization and then divide out the common factors between the numerator and denominator.

Example:

A second method is to divide out the common factors without completing the prime factorization of each term.

Example:

Two Final Reminders about Simplifying by Dividing Common Factors

1. We may only simplify by dividing common factors when multiplying fractions. We cannot simplify by dividing common factors when adding or subtracting fractions. We use basic examples with models to help understand this rule.
   '
   Example:
   Note here when we are multiplying, we may simplify by dividing common factors.
   
   Note here when we are adding, we cannot simplify by dividing common factors.
   
   

    

2. We must always use one factor in the numerator and one factor in the denominator when dividing common factors because it is the division bar (fraction bar) that we are interpreting as divided by.

Multiplication of Mixed Numbers

To multiply a fraction or a mixed number by a mixed number or whole number, we usually rewrite each term that is a mixed number or whole number as an equivalent improper fraction.

Examples:

Mixed Numbers Versus Improper Fractions

When we added mixed numbers, we found that it is usually easier to leave each addend as a mixed number and not change to improper fractions (see the previous session 21); whereas, when multiplying fractions it is usually easier to change to improper fractions and not leave as mixed numbers. We illustrate with the examples below. 

Example: Compute .

We use the distributive property of multiplication over addition to compute the multiplication without changing the mixed numbers to improper fractions.

Now we rework the problem by first changing each mixed number into an improper fraction.

Note the fewer number of operations needed. Also, note that with the mixed number form we needed to find a common denominator to add the fractions. For some problems, the mixed number form is easier to use when making mental computations, but usually it is easier to work multiplication problems involving mixed numbers by changing to improper fractions. The opposite was true when adding and subtracting mixed numbers.

Example:  Evaluate .

Again, we use the distributive property of multiplication over addition to compute the multiplication without changing the mixed numbers to improper fractions.

Next, we rework the problem by first changing each mixed number into an improper fraction.

Summary:

1. It is usually easier to leave the terms as mixed numbers when adding or subtracting mixed numbers.
   
2. But, it is usually easier to change each mixed number to an improper fraction when multiplying mixed numbers.

Properties for Fractions

Commutative Property for Fraction Addition   and Multiplication  .

Associative Property for Fraction Addition   and Multiplication  

Inverse Property for Fraction Multiplication where a and b are nonzero. The fraction   is called the multiplicative inverse of   (or reciprocal) and vice versa.

Joke or Quote

How do we know that the following fractions are in Europe? A/C, X/C and W/C ?  Answer.