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Summer 2016 - Session 2
Math 1300
FUNDAMENTALS OF MATH
Section #16535
Monday - Friday, 10am-12pm
Instructor: Dr. Angelynn Alvarez
[email protected]
06/07/2016
Section 1.3 - Fractions
Section 1.3 – Fractions
Section 1.1 consists of two parts:
Part I: The Greatest Common Divisor and Least Common Multiple
Part II: Addition/Subtraction, Multiplication/Division of Fractions
Part I: The Greatest Common Divisor and Least Common Multiple
Tip: Remind yourself of the divisibility rules. It will be extremely
helpful to know for this section.
The Greatest Common Divisor (GCD)
In this section, we will discuss how to find the Greatest Common
Divisor, or GCD, of any two natural numbers.
A factor or divisor of a natural number, say n, is a number that you
multiply with another number to get n.
Examples:
 6 is a factor or divisor of 18.
 7 is a factor or divisor of 70.
Let m and n be two natural numbers. Then the greatest common
divisor (GCD) of m and n is the largest natural number that divides
both m and n.
The method to find the GCD of two natural numbers m and n:
(1)
Find the prime factorization of m and the prime
factorization of n.
(2)
Put all primes factors in the following table:
Prime
factors of m
Prime
factors of n
*Align common factors in the same column, and place
uncommon terms to the right. *
(3)
The common factors are the numbers which appear in
every row.
If there is only one common factor, it is the GCD.
If there is more than one common factor, multiply the
common factors together to get the GCD.
Example: Find the GCD of 30 and 25.
Prime
factors of
30
Prime
factors of
25
Example: Find the GCD of 36 and 24.
Prime
factors of
36
Prime
factors of
24
Example: Find the GCD of 54, 243, 810.
Prime
factors of
54
Prime
factors of
243
Prime
factors of
810
The Least Common Multiple (LCM)
A multiple of a natural number, say n, is a number obtained by
multiplying n with another number.
Examples:
 12 is a multiple of 4 because 12=4(3)
(You multiply 4 with 3 to get 12.)
 16 is a multiple of 8 because 16=8(2).
(You multiply 8 with 2 to get 16.)
Let m and n be two natural numbers. Then the least common
multiple (LCM), of m and n is the smallest natural number that is a
multiple of both m and n.
The method to find the GCD of two natural numbers m and n:
(1)
Find the prime factorization of m and the prime
factorization of n.
(2)
Put all primes factors in the following table:
Prime
factors of m
Prime
factors of n
*Align common factors in the same column, and place
uncommon terms to the right. *
(3)
Multiply the numbers across in a “descending staircase
style” to get the LCM.
Example: Find the LCM of 16 and 18.
Prime
factors of
16
Prime
factors of
18
Example: Find the LCM of 6 and 15.
Prime
factors of
6
Prime
factors of
15
Note: For the LCM of 3 numbers, you can leave your answers as
products of the prime factors (USING EXPONENTS)---so you do not
need to multiply everything out.
Example: Find the LCM of 24, 400, 250.
Prime
factors of
24
Prime
factors of
400
Prime
factors of
250
Example: Find the LCM of 12, 135, 250.
Prime
factors of
12
Prime
factors of
135
Prime
factors of
250
Part II: Addition/Subtraction, Multiplication/Division of Fractions
Addition/Subtraction of Fractions
When adding and subtracting fractions, we must pay attention to the
denominators (the numbers on the bottom of the fraction).
There are two cases:
(1)
(2)
Adding and subtracting fractions with the same
denominators
Adding and subtracting fractions with different
denominators.
Adding/Subtracting Fractions with The Same Denominators
To add and subtract two fractions whose denominators are the
same, add and subtract the numerators and keep the denominator--that is,
𝑎 𝑏 𝑎+𝑏
+ =
,
𝑐 𝑐
𝑐
𝑎 𝑏 𝑎−𝑏
− =
𝑐 𝑐
𝑐
Examples: Add or subtract the following fractions and put the
answer in simplest form.
7 6
+
9 9
4 2
−
3 3
Adding/Subtracting Fractions with DIFFERENT Denominators
To add/subtract two fractions whose denominators are different, we
first must rewrite the fractions so they have the same denominator.
The least common denominator (LCD) is the denominator that both
fractions will have before being added together (or subtracted).
*The least common denominator (LCD) is the least common
multiple (LCM) of the two denominators.
Steps to add/subtract fractions with different denominators:
(1)
(2)
(3)
(4)
Find the LCD (which is the LCM of the 2 denominators)
Determine what number you need to multiply each
denominator by to get the LCD.
Multiply the top and bottom of each fraction by the
number found in step (2)
The denominators should be the same, so just add (or
subtract) the numerators and keep the denominator.
Example: Add
7
24
+
5
.
12
Prime factors
of 24
Prime factors
of 12
Example: Add
Prime factors
of 20
Prime factors
of 5
1
20
4
+ .
5
5
7
8
12
Example: Subtract −
Prime factors
of 8
Prime factors
of 12
1
4
9
15
Example: Subtract −
Prime factors
of 9
Prime factors
of 15
.
Multiplication of Fractions
When multiplying fractions, we luckily do not need to worry about
common denominators.
To multiply two fractions, multiply the numerators together and then
multiply the denominators together. Then place the product of the
numerators on top of the product of the denominators----that is,
𝑎 𝑐 𝑎∙𝑐
∙ =
𝑏 𝑑 𝑏∙𝑑
*Do not forget the multiplication rules about the signs! +/-
Example: Multiply the following fractions.
6 3
− ∙
7 8
3
2
∙−
196
15
−
12 1
∙
75 14
8
5
4
∙2
6
9
(To multiply 2 mixed numbers, convert the mixed number into a
single fraction. Then multiply. You can leave it as a single fraction.)
9
1
6
∙4
8
7
Division of Fractions
Like multiplication, we luckily also do not need to worry about
common denominators.
The reciprocal of a fraction
𝑎
𝑏
𝑏
is the fraction .
𝑎
Example:
 The reciprocal of
4
7
7
is .
4
5
3
3
5
 The reciprocal of − is − .
To divide two fractions, multiply the first fraction by the reciprocal of
the second fraction---that is,
𝑎 𝑐 𝑎 𝑑
÷ = ∙
𝑏 𝑑 𝑏 𝑐
Note:
𝑎
(
)
𝑎 𝑐
𝑏
÷ is the same as 𝑐
𝑏 𝑑
( )
𝑑
Example: Divide the following fractions. Write your answer in
simplest form.
5 4
− ÷
2 7
3
11
− ÷(− )
2
4
3
( )
2
6
(− )
7
2
( )
9
4
( )
5