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Chapter 2
Fractions
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2.2
Simplifying
Fractions
Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc.
Prime Numbers
A prime number is a whole number greater than 1
that cannot be evenly divided except by 1 and itself.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
The first 10 prime numbers
A composite number is a whole number greater
than 1 that can be evenly divided by whole numbers
other than 1 and itself.
24 = 2  12
24 = 3  8
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24 = 4  6
Divisibility Tests
1. A number is divisible by 2 if the last digit is 0, 2,
4, 6, or 8.
2. A number is divisible by 3 if the sum of the digits
is divisible by 3.
3. A number is divisible by 5 if the last digit is 0 or 5.
Example:
The number 450 is divisible by 2. (It ends in 0.)
The number 450 is divisible by 3. (4 + 5 + 0 = 9 and 9 is
divisible by 3.)
The number 450 is divisible by 5. (It ends in 0.)
Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc.
Prime Factorization
Example:
Write the number 24 as a product of primes.
24 = 4  6
22 23
24 = 2  2  2  3
Write 24 as the product of any two factors.
If the factors are not prime, they must
be factored.
When all of the factors are prime, the
number has been completely factored.
Instead of writing 2  2  2  3, we can also write 23  3.
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The Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic
Every composite number can be written in exactly
one way as a product of primes.
24 = 4  6
24 = 3  8
22 23
24 = 2  2  2  3
24
22
24 = 3  2  2  2
The order of the prime factors is not important because
multiplication is commutative.
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Equivalent Fractions
Equivalent fractions can be written in more than one
way. The value of the fractions is the same.
3 is shaded.
6
1 is shaded.
2
Equivalent fractions
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Simplest Form
Any nonzero number divided by itself is equal to 1.
2
43
b


 1
2
43
b
1
3
3


2
3
6
1
4
4


2
4
8
1
50
50


2
50
100
1 3 4
50
, , , and
are equivalent fractions.
2 6 8
100
A fraction is in simplest form when the numerator and
denominator have no common factors.
Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc.
Common Factors
When a fraction is not in simplest form, it can be reduced.
To reduce a fraction, find a common factor in the
numerator and the denominator and divide it out.
25
5
5


40
5
8
25
5

40
8
5 is the common factor.
A fraction is called simplified, reduced, or in lowest terms
if the numerator and the denominator only have 1 as a
common factor.
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Equal Fractions
If two fractions are equivalent, their diagonal products or
cross products will be equal.
1 4
?=
2 8
2 4  8
18  8
The products are equal,
1 4
therefore  .
2 8
Equality Test for Fractions
For any two fractions where a, b, c and d are whole
numbers and b  0, d  0, if a  c , then a  d = b  c.
b
d
Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc.