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Basic Math Skills
2.4

Fractions-Multiplication and Division

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A fraction represents a part of the whole.

In the figure above, the four pieces of the pie represents the whole.   The shaded region represents the fractional part of the whole which is one piece out of four or .

The top of the fraction is called the numerator and the bottom of the fraction is called the denominator.

There are three forms of a fraction.

1.   Proper fractions:  

Proper fractions have the smaller number in the numerator.

2.   Improper fractions:  

Improper fractions have the smaller number in the denominator.

 

3.   Mixed fractions:  

Mixed fractions are improper fractions that are represented as a whole part and the fractional part.

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Changing Mixed Fractions Into Improper Fractions

 

Example 1:

Change into an improper fraction.

 

Solution:

 

Therefore the improper fraction is  

 

Check your answer:
The number 3 goes into 7 twice with one left over.  

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Example 2:

Change   into an improper fraction.

 

Solution:

To find the numerator, multiply (4)(5) and then add 3.   The denominator is the same denominator as before which is 5.

Therefore the improper fraction is  

 

Check your answer:   The number 5 goes into 23 four times with three left over.  

Student #1:

Change the mixed fraction into an improper fraction.

Answer

Student #2:

Change the mixed fraction into an improper fraction.

Answer

Multiplying Fractions

The easiest way to Add, Subtract, Multiply or Divide fractions is to work with improper fractions.

Rules for multiplying a fraction times another fraction

1.   Cancel.
2.   Multiply the numerators.   (multiply across the top)
3.   Multiply the denominators.   (multiply across the bottom)

Example 3:

Simplify:  

 

Solution:

Since this is in the form of a fraction times a fraction, the above rules apply.

First cancel like factors.

The second step is to multiply across the top of the fraction.

(1)(3)

Thus the numerator is 3.

The third step is to multiply across the bottom of the fraction.

(1)(4)

Thus the denominator is 4.

Therefore the product is  

Another way to think about canceling is as follows:

Simplify:  

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Example 4:

Simplify:  

 

Solution:

Step 1:   Cancel.

Step 2:   Multiply the numbers that remain in the numerator.

Step 3:   Multiply the numbers that remain in the denominator.

 

Another way to work this problem is by factoring:

Change 8 into (2)(2)(2).

Change 25 into (5)(5).

Change 16 into (2)(2)(2)(2).

Change 15 into (3)(5).

Therefore the simplified form is 1/3.

Most of the time a fraction is left in the improper fraction form instead of a mixed number.   This is because an improper fraction is easier to operate with then with a mixed number.




Student #3:

Simplify:  

3/2

2/3

3/7

4/9

Answer

Student #4:

Simplify:  

1/5

3/4

2/5

22/5

Answer

Student #5:

Simplify:  

22

16

5/8

12

Answer

Dividing Fractions

The easiest way to Add, Subtract, Multiply or Divide fractions is to work with improper fractions.

Rules for Dividing a fraction by another fraction

1.   Keep the first fraction the same.
2.   Change the operation of division into multiplication.
3.   Flip the second fraction.   (This is called the reciprocal.)
4.   Apply the rules for multiplying fractions.

Example 5:

Simplify:  

 

Solution:

Rewrite the problem as shown below:

Notice the first fraction remains the same, the operation is changed to multiplication, and the second fraction is changed to the reciprocal.

Simplify as before, using the rules of multiplication.

Therefore the answer is 1.





Student #6:

Simplify:  

2/3

32/15

16/3

10/3

Answer

Student #7:

Simplify:  

18/5

2/5

12/5

11/6

Answer

Complex Fractions

Complex fractions are fractions which have a fraction in the numerator and a fraction in the denominator.

Example 6:   Simplify:  

 

Solution:

The numerator is and the denominator is   This problem translates into:

Simplify as usual.

Therefore the answer is  

Student #8:

Simplify  

Answer

Homework Problems

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Student #1:

The answer is B:

Solution:

To find the numerator, multiply (5)(3) then add two.   The denominator is 3.

Therefore the simplified form is  

Go Back to Lesson

Student #2:

The answer is D:

Solution:

To find the numerator, multiply (10)(7) then add 5.

The denominator is 7.

Therefore the simplified form is  

Go Back to Lesson

Student #3:

The answer is A:

Solution:

Therefore the simplified form is 3/2.

Go Back to Lesson

Student #4:

The answer is C:

Solution:

Therefore the simplified form is 2/5.

Go Back to Lesson

Student #5:

The answer is B:

Solution:

To simplify fractions, change mixed numbers into improper fractions.

 

Therefore the simplified form is 16.

Go Back to Lesson

Student #6:

The answer is D:

Solution:

Rewrite as:   

Therefore the simplified form is 10/3 or

Go Back to Lesson

Student #7:

The answer is A:

Solution:

Change the mixed numbers to improper fractions.

 

Change division into multiplication, and flip the second fraction.

 

Simplify.

Therefore the simplified form is 18/5.

Go Back to Lesson

Student #8:

The answer is B:

Simplify  

Answer is B:

Solution:

Therefore the answer is

Go Back to Lesson