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2.1
LEAST COMMON MULTIPLE & GREATEST COMMON FACTOR
Objective A - Finding the Least Common Multiple
 The multiples of a number are the products of that number if you multiply it by 1, 2, 3,
4, 5, 6...
Ex. The multiples of 5 are:
Ex. The multiples of 7 are:
 When you compare the multiples of two (or more) numbers, if they have any in
common, they are called a common multiple of those numbers.
 The Least Common Multiple (referred to as the LCM) is the smallest common multiple
of two (or more) numbers.
 Looking at the multiples of 5 and 7 above, what are the common multiples listed.
How to Find the LCM?
 A method to find the LCM without listing all the multiples as above:
1) Find the prime factorization of each number.
2) List the prime factorizations of each number in rows in a table with a prime
number listed as the heading of each column.
3) Circle the largest product in each column.
4) The LCM is that largest product - from all circled numbers.
Ex. Find the LCM of 6 and 8.
2
3
6 =
8 =
Therefore, the LCM is:
Ex. Find the LCM of 14 and 42.
Ex. Find the LCM of 120 and 160.
1
Objective B - Finding the Greatest Common Factor
 Recall that a number that divides a second number evenly is called a factor of the
second number.
 A number is a common factor of two (or more) numbers if it’s a factor of those
numbers.
o Factors of 12 are: 1, 2, 3, 4, 6, and 12
o Factors of 40 are: 1, 2, 4, 5, 8, 10, 20, and 40
o The common factors of 12 and 40 are:
 The Greatest Common Factor (referred to as the GCF) is the largest common factor
of two or more numbers.
o Looking at the factors of 12 and 40 above, what is the GCF?
 A method to find the GCF without listing all of the factors as above:
1) Find the prime factorization of each number.
2) List the prime factorizations of each number in rows in a table with a prime
number listed as the heading of each column.
3) Circle the smallest product in each column that does not have a blank.
4) The GCF is that smallest product - from all circled numbers.
Ex. Find the GCF of 18 and 24.
2
3
18 =
24 =
Therefore, the GCF is:
Ex. Find the GCF of 15 and 28.
Ex. Find the GCF of 24, 36, and 48.
2
2.2
INTRODUCTION TO FRACTIONS
Objective A - Write a Fraction That Represents Part of a Whole
 A fraction represents a certain number of equal parts of a whole item.
o Divide the circle into 4 equal parts. Shade 3 of them.
3
o This represents of that shape.
4
o The 3 is in the numerator.
o The 4 is in the denominator.
 A proper fraction is less than one. That means the numerator is less than the
denominator. For example, ¾ is a proper fraction because it is < 1.
 A mixed number is a number that is greater than 1 that has a whole number part and a
fractional part. For example, 1¾ is a mixed number because it is > 1.
 An improper fraction is greater than or equal to 1. Meaning, the numerator is greater
than or equal to the denominator. Note: an improper fraction is not a “bad fraction.” In
algebra, fractions that are greater than one are most often left in improper fraction form.
For example, 7/3 is an improper fraction because it is ≥ 1.
Objective B - Writing an Improper Fraction as a Mixed Number or a Whole Number
 Process to change from an improper fraction to a mixed number:
1) Divide the denominator into the numerator until you get a final remainder.
2) The whole number part of the mixed number is the whole number in the quotient.
The fraction part of the mixed number is the remainder over the divisor.
16
as a mixed number.
3
45
Ex. Write
as a mixed or whole number.
9
Ex. Write
Ex. Write
Ex.
3
19
as a mixed number.
7
9
as a mixed or whole number.
1
Representing a Mixed Number as an Improper Fraction
 Process to change a mixed number to an improper fraction:
1) Multiply the whole number times the denominator.
2) Add that to the numerator.
3) Write that number “over” the denominator that was there.
5
2
Ex. Write 3 as an improper fraction.
Ex. Write 7 as an improper fraction.
11
7
2.3
WRITING EQUIVALENT FRACTIONS
Objective A - Finding an Equivalent Fraction By Rising to Higher Terms
 Two fractions can be equal but look different → draw two circles:
o Divide one into halves and the other into fourths.
o Shade 1 2 of the first circle and 2 4 of the second circle.
o Notice the two regions you shaded are the same.
 You can take one fraction and make an equivalent fraction by multiplying both the
numerator and denominator by the same number.
o This is → raising a fraction to higher terms.
3
Ex. Write a fraction that is equivalent to and has a denominator of 21.
7
Ex. Write a fraction that is equivalent to
2
and has a denominator of 55.
11
Note: You can multiply both the numerator and denominator by the same number because
you are really just multiplying by the number ___. (Place Answer Here!)
Ex. Write the equivalent fraction of 2 3 by using the Multiplication Property of One:
4
Objective B - Writing a Fraction in Simplest Form (or Simplest Terms)
 This is really the reverse process of Objective A.
o A fraction in simplest form is when the numerator and denominator have no
common factors between them.
1
2
o Just as you could raise to by multiplying both the numerator and
2
4
2
1
denominator by 2, so you can also simplify to by dividing both the
4
2
numerator and denominator by 2.
o To do this, you divide both the numerator and denominator by a factor that divides
evenly into both. The GCF is the best one to use because that will be the only
number by which you need to divide.
o Sometimes that number will be obvious to you; other times you will need to do
some work. When it’s not obvious, the method that always works is this:
1) Factor the numerator into its prime factors.
2) Factor the denominator into its prime factors.
3) Cancel any factors that are the same.
Ex. Write
12
in simplest form. (Also called reducing the fraction.)
36
Ex. Write
50
in simplest form.
75
2.4
Ex. Write
48
in simplest form.
14
ADDITION OF FRACTIONS AND MIXED NUMBERS
Objective A - Adding Fractions with the Same Denominator
Process:
1. Add the numerators.
2. Write that sum over the same denominator.
Ex.
3 4
+
11 11
Ex.
6 12
+
17 17
Ex.
5
8
3
4
+
+
25 25 25
Objective B - Adding Fractions With the Unlike Denominators
Process:
1. Find the LCM of the denominators.
2. Write each fraction as an equivalent fraction with the LCM as the denominator.
3. Add the fractions once they have the same denominator.
Ex.
Ex.
3 2
+
10 5
1 3
+
2 29
Ex.
5
7
+
12 30
Ex.
2 5 7
+ +
3 6 12
Objective C - Adding Whole Numbers, Mixed Numbers, and Fractions
Case 1:
To add a whole number and a fraction, you end up with a mixed number.
3
Ex. 5 + =
7
Case 2:
To add a whole number and a mixed number, you add the whole numbers and write the
fraction part next to the sum so you have a mixed number.
(Note: It can be helpful to rewrite the problem vertically.)
6
Ex. 9 + 2
=
13
Case 3:
To add two mixed numbers, add the fractional parts and then add the whole numbers.
Always reduce the fraction part to simplest form.
(Note: It can be helpful to rewrite the problem vertically.)
3
3
Ex. 2 + 5
8
7
Ex. 4
5
11
+6
12
18
5
5
5
Ex. 6 + 6 + 2
9
12
18
6
2.5
SUBTRACTION OF FRACTIONS AND MIXED NUMBERS
Objective A - Subtracting Fractions with the Same Denominator
Process:
1. Subtract the numerators.
2. Write that sum over the same denominator.
Ex.
11 5
19 19
Ex.
48 13
55 55
Objective B - Subtracting Fractions With the Unlike Denominators
Process:
1. Find the LCM of the denominators.
2. Write each fraction as an equivalent fraction with the LCM as the denominator.
3. Subtract the fractions once they have the same denominator.
Ex.
7 5
8 16
Ex.
19 3
40 16
Ex.
7 1
9 6
Ex. What is
11
3
less than ?
5
12
Objective C - Subtracting Whole Numbers, Mixed Numbers, and Fractions
Case 1:
Subtracting mixed number without borrowing.
(Note: It can be helpful to write the problem vertically.)
7
10
3
-2
10
8
Ex.
Ex.
7
10
- 2
8
Case 2:
Subtracting a whole number from a mixed number: you to borrow 1 from the whole
number. Note: It can be helpful to write the problem vertically.
7
10
Ex.
-
2
9
Case 3:
Subtracting two mixed numbers when you do need to borrow: this happens when the
numerator of the “top” fraction is smaller than the numerator of the “lower” fraction.
Note: It can be helpful to write the problem vertically.
4
25
9
Ex.
7
- 16
9
Ex. In this example, you need to find the LCM for the denominators first.
2
16
5
4
-8
9
2.6
MULTIPLICATION OF FRACTIONS AND MIXED NUMBERS
Objective A - Multiplying Fractions
Procedure:
1. Multiply the numerators to find the new numerator.
2. Multiply the denominators to find the new denominator.
3. You can write the prime factorization of both the numerator and denominator.
4. Cancel all common factors between the numerator and denominator – this will
reduce the final fraction to simplest terms.
Ex.
3 4
•
7 11
Ex.
8
5 6
•
18 25
Objective B - Multiplying Whole Numbers, Mixed Numbers, and Fractions
Procedure:
1. If there is a whole number, write it “over 1” so it looks like a fraction.
2. If there is a mixed number, change it to an improper fraction.
3. Multiply the two fractions like you did in Objective A.
Ex. 5 •
3
13
1 4
Ex. 9 •
2 11
Ex.
7
• 12
16
2 3
Ex. 5 •
3 17
Ex. 5
3
1
•5
16 3
3
3
Ex. 3 • 2
4 20
2.7
DIVIDING FRACTIONS AND MIXED NUMBERS
Objective A Dividing Fractions
Reciprocal: The reciprocal of a fraction means that the numerator and denominator have
been ____________________________.
2
What is the reciprocal of ?
3
What is the reciprocal of 4?
5
What is the reciprocal of ?
2
1
What is the reciprocal of ?
7
 Division is actually easy for fractions – you just change every division problem into a
multiplication problem. (“Flip and Multiply”)
Ex.
3 2 3 3
÷ = • =
5 3 2 5
Notice that it is the second fraction (the divisor fraction) that gets “flipped.”
9
Ex.
11 5
÷
15 22
Ex.
4 1
÷
9 9
Objective B - Dividing Whole Numbers, Mixed Numbers, and Fractions
Procedure:
1. If there is a whole number, write it “over 1” so it looks like a fraction.
2. If there is a mixed number, change it to an improper fraction.
3. Divide the two fractions like you did in Objective A.
Ex. 22 ÷
Ex.
2.8
3
11
Ex.
3
1
÷2
8
4
7
÷ 14
8
5
Ex. 3 ÷ 32
9
ORDER, EXPONENTS, AND ORDER OF OPERATIONS AGREEMENT
Objective A - Identifying the Order Relation between Two Fractions
 Just as you can graph whole numbers on a number line, so you can also graph fractions
on a number line. You place a solid dot one the number line where the fraction is
positioned.
2
Ex. Graph .
5
 Instead of dividing each interval into fifths, divide into any set of equal increments.
1
7
 Divide the following number line into thirds and graph the fractions and .
3
3
 When you compare fractions on the number line, the smaller fraction is on the
__________, and the larger fraction is on the _______________________.
10
 If two fractions have the same denominator, it is easy to determine which the smaller
fraction is: Compare the numerators and the fraction with the smaller numerator is the
smaller fraction.
Ex. Place the correct inequality symbol (< or >) between these fractions:
14 8
9 9
 If two fractions have different denominators, you first need to find the LCM and write
them with the same denominator. Then you can compare the numerators and the fraction
with the smaller numerator is the smaller fraction.
 Place the correct inequality symbol (< or >) between these fractions:
Ex.
6 7
12 15
Ex.
3
10
5
15
Ex.
9
8
12
9
Objective B - Fractions with Exponents
You can write 5 • 5 • 5 • 5 • 5 • 5 • 5 with an exponent – what is it?
So too, it is possible to write a fraction with an exponent – the exponent just means that you
are multiplying it by itself that many times.
Write the following with an exponent:
2 2 2 2
a. • • • =
3 3 3 3
b.
1 1 1 1 1
• • • •
4 4 4 4 4
To simplify an exponential expression like one of these, you need to multiply the numbers,
cancel any common factors, and simplify the resulting fraction.
 4
Ex. Simplify  
 5
2
 10 
 
4
3
2
1  6  2
Ex. Simplify        
 6  7  3
2
 1  3
Ex. Simplify 3     
 3  5
3
11
Objective C - Using the Order of Operations Agreement with Fractions
 Remember PEMDAS from Chapter #1.
 Simplify the following:
Ex.
2 3 2
 
5 10 3
Ex.
11  3 
7
  
16  4  12
3
Ex.
3
 3
  
 5  25
Ex.
7  2 5
  
12  3 9 
2
12