Download Example: Reducing a Rational Number

The Rational Numbers
Notes and Examples
for 8/15/16
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Learning Targets
I will define the rational numbers.
I will reduce rational numbers.
I will convert between mixed numbers and improper fractions.
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Defining the Rational Numbers
• The set of rational numbers is the set of all numbers
a
which can be expressed in the form , where a and b
b
are integers and b is not equal to 0.
• The integer a is called the numerator.
• The integer b is called the denominator.
The following are examples of rational numbers:
¼, ½, ¾, 5, 0
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Reducing a Rational Number
a
b
• If is a rational number and c is any number other
than 0,
ac a

bc b
• The rational numbers a and a  c are called equivalent
b
bc
fractions.
• To reduce a rational number to its lowest terms,
divide both the numerator and denominator by their
greatest common divisor.
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Example: Reducing a Rational Number
Reduce
130
455
to lowest terms.
Solution: Begin by finding the greatest common divisor
of 130 and 455.
Thus, 130 = 2 · 5 · 13, and 455 = 5 · 7 · 13. The greatest
common divisor is 5 · 13 or 65.
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Example: Reducing a Rational Number
(continued)
Divide the numerator and the denominator of the given
rational number by 5 · 13 or 65.
130 2  5  13 2


455 5  7  13 7
or
130 130  65 2


455 455  65 7
There are no common divisors of 2 and 7 other than 1.
2
Thus, the rational number 7 is in its lowest terms.
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Mixed Numbers and Improper Fractions
• A mixed number consists of the sum of an integer and
a rational number, expressed without the use of an
addition sign.
Example:
• An improper fraction is a rational number whose
numerator is greater than its denominator.
Example: 19
19 is larger than 5
5
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Example 2: Converting a Positive Mixed Number to an
Improper Fraction
Example: Convert
to an improper fraction.
Solution:
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Converting a Positive Improper Fraction to a Mixed
Number
1. Divide the denominator into the numerator. Record
the quotient and the remainder.
2. Write the mixed number using the following form:
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Example 3: Converting from an Improper Fraction to a
Mixed Number
42
Convert
to a mixed number.
5
Solution: Step 1 Divide the denominator into the
numerator.
Step 2 Write the mixed number using
Thus,
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