Ch1-Section 1.3
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Transcript Ch1-Section 1.3
§ 1.3
Fractions
Numerators and Denominators
A quotient of two numbers is called a fraction.
The fraction 14 represents the
shaded part of the circle. 1 out of
1
4 pieces is shaded. 4 is read “onefourth.”
1
4
numerator
denominator
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Simplifying Fractions
To simplify fractions we can simplify the numerator and
the denominator.
2 · 5 = 10
factors
product
A fraction is said to be simplified or in lowest terms
when the numerator and denominator have no factors in
common other than 1.
2
3
17
23
1
9
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Prime and Composite Numbers
A prime number is a natural number, other than 1,
whose only factors are 1 and itself.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
The first 10 prime numbers
A natural number, other than 1, that is not a prime
number is called a composite number. Every
composite number can be written as a product of
prime numbers
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Product of Primes
Example:
Write the number 24 as a product of primes.
24 = 4 6
22 23
24 = 2 2 2 3
Write 24 as the product of any two
whole numbers.
If the factors are not prime, they
must be factored.
When all of the factors are prime, the
number has been completely factored.
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The Fundamental Principal of Fractions
The Fundamental Principal of Fractions
If a is a fraction and c is a nonzero real number, then
b
ac a
bc b
Example:
25
Write the fraction
in lowest terms.
40
25
55
5
5
40 2 2 2 5 2 2 2 8
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Multiplying Fractions
To multiply two fractions, multiply numerator times
numerator to obtain the numerator of the product.
Multiply denominator times denominator to obtain the
denominator of the product.
Multiplying Fractions
a c a c
, if b 0 and d 0
b d bd
3 2
6
7 5
35
3 2 6
7 5 35
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Multiplying Fractions
12 3
Example: Multiply.
17 24
12 3
12 3
36
17 24 17 24 408
Multiply numerators.
Multiply denominators.
Simplify the product by dividing the numerator and
the denominator by any common factors.
36
2 2 33
3
408 2 2 2 3 17 34
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Dividing Fractions
Two fractions are reciprocals of each other if their
product is 1.
3 4
1
4 3
3
4
and are reciprocals.
4
3
Dividing Fractions
a c a d
, if b 0 and d 0
b d b c
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Dividing Fractions
3 1
Example: Divide.
4 4
3
1
3 4
12 3
4
4
4 1
4
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Fractions with the Same Denominator
To add or subtract fractions with the same denominator,
combine numerators and place the sum or difference
over the common denominator.
2
1
3
4
4
4
Adding and Subtracting Fractions with the Same Denominator
a c ac
, if b 0
b b
b
a c ac
, if b 0
b b
b
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Equivalent Fractions
Equivalent fractions are fractions that represent the
same quantity.
3 is shaded.
6
1 is shaded.
2
Equivalent fractions
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Equivalent Fractions
3
Example: Write as an equivalent fraction with a
4
denominator of 20.
5
Since 4 · 5 = 20, multiply the fraction by .
5
3 3 5
3 5 15
4 4 5
4 5 20
5
Multiply by or 1.
5
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Fractions without the Same Denominator
To add or subtract fractions without the same denominator,
first write the fractions as equivalent fractions with a
common denominator
The least common denominator (LCD) is the smallest
number both denominators will divide evenly into.
Example: Add.
3
1
8
6
3 3
9
8 3
24
1 4
4
6 4
24
LCD = 24
9
4
13
24
24
24
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Fractions without the Same Denominator
Example: Subtract.
5
7
12
30
LCD = 60
5
5
25
12 5
60
7
2
14
30 2
60
25
14
11
60
60
60
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