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Lesson 5-1 Warm-Up

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“Comparing and Ordering Rational Numbers” (5-1)

What is the a “multiple”?

What is the the “least common multiple” or LCM?

How do you find the LCM?

Multiple: The multiple of a number is the product of that number and any nonzero number (when you count by a number, you are finding its multiples)

Example: Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36,….

Example: Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54,….

Least Common Multiple (LCM): the smallest multiple shared by all of the numbers

Example: Common Multiples of 4 are 6 are 12, 24, and 36. The smallest multiple of both numbers , or Least Common Multiple (LCM) is 12.

To find the LCM: 1. list the multiples of both numbers until you find the first one that they share in common, or 2. multiply the greatest power of the factors the numbers.

Example: Find the LCM of 18, 27, and 36.

Method 1: List the multiples of each number until you find a common one.

Multiples of 18 are 18, 36, 54, 72, 90, 108,….

Multiples of 27 are 27, 54, 81, 108,…

Multiples of 36 are 36, 72, 108,….

The LCM of 18, 27, and 36 is 108.

Find the multiple of each number. Stop when you find a multiple the numbers share in common.

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“Comparing and Ordering Rational Numbers” (5-1)

Method 2: Multiply the greatest power of all factors together.

Example: Find the LCM of 18, 27, and 36.

18 = 3 • 3 • 2 = 32 • 2

27 = 3 • 3 • 3 = 33

36 = 3 • 3 • 2 • 2 = 32 • 22

33 • 22 = 27 • 4 = 108

Example: Find the LCM of 6a2 and 18a3.

LCM of 6a2 and 18a3 is 18a3.

Create a factor tree number to find the prime factors of each number.

Write each number in prime factorization. form

Multiply the greatest powers of all factors together.

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Today, the school’s baseball and soccer teams had

games. The baseball team plays every 7 days. The soccer

team plays every 3 days. When will the teams have games

on the same day again?

7, 14, 21, 28, 35, 42, . . . List the multiples of 7.

3, 6, 9, 12, 15, 18, 21, . . . List the multiples of 3.

The LCM is 21. In 21 days both teams will have games again.

Comparing and Ordering Rational NumbersLESSON 5-1

Additional Examples

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Find the LCM of 16 and 36.

= 144 Multiply.

16 = 24

36 = 22 • 32Write the prime factorizations.

The LCM of 16 and 36 is 144.

LCM = 24 • 32 Use the greatest power of each factor.

Comparing and Ordering Rational NumbersLESSON 5-1

Additional Examples

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Find the LCM of 5a4 and 15a.

5a4 = 5 • a4

15a = 3 • 5 • a Write the prime factorizations.

= 15a4 Multiply.

The LCM of 5a4 and 15a is 15a4.

LCM = 3 • 5 • a4 Use the greatest power of each factor.

Comparing and Ordering Rational NumbersLESSON 5-1

Additional Examples

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“Comparing and Ordering Rational Numbers” (5-1)

How do you compare fractions?

To compare fractions, you can: 1. use a number line (numbers to the right are greater than numbers to the left), or 2. compare the numerators (number of parts) if the denominator (size of the parts) are equal. So, if the denominators aren’t the same, you need to change one or more of the fractions into equivalent fractions with a common denominator.

Method 1: Use a number line.

Example: Compare and .

is on the left of , so .

Method 1: Compare the numerators.

Example: Compare and .

3 • 4 = 12 Multiply the denominators together to find a common denominator

Write equivalent fractions with a denominator of 12 and compare the numerators (Hint: Notice that you multiply the each fraction by the other fractions denominator)

Since , then .

2 3

3 4

2 • 4 = 8_ 3 • 4 = 12 3 • 3 = 9_ 4 • 3 = 12

9 12

8 12

3 4

2 3

- 1 2

- 1 10 - 1

2 - 1 10

- 1 10

- 1 2

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Graph and compare the fractions in each pair.

is on the left, so < .38

38

78

b. – 13

, – 16

is on the right, so > .– 16

– 16

– 13

a. 78

38

,

38

78

– 13

– 16

Comparing and Ordering Rational NumbersLESSON 5-1

Additional Examples

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“Comparing and Ordering Rational Numbers” (5-1)

What is the “least common denominator” (LCD)?

Least Common Denominator (LCD): the LCM of two or more denominators (in other words, the smallest common denominator)

Example:

List the multiples of each denominator until you find a multiple that is shared by both numbers (LCM). LCM = 36

Rewrite the fractions into equivalent fractions with a denominator of 36 (The LCD is 36). Then, compare thenumerators.

Since , then .16 36

15 36

4 9

5 12

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The softball team won of its games and the

hockey team won of its games. Which team won the

greater fraction of its games?

677

9

Step 1  Find the LCM of 7 and 9.7 = 7 and 9 = 32

LCM = 7 • 32 = 63Step 2  Write equivalent fractions with a denominator of 63.

6 • 97 • 97 • 79 • 7

=54634963

=

Step 3  Compare the fractions.5463

4963

67

79

> >, so

The softball team won the greater fraction of its games.

Comparing and Ordering Rational NumbersLESSON 5-1

Additional Examples

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Order , , and from least to greatest.37

14

23

37

14

23

3 • 127 • 12

1 • 214 • 21

2 • 283 • 28

3684

2184

5684

=

=

=

=

=

=

The LCM of 7, 4, and 3 is 84.Use 84 as the common denominator.

2184

3684

5684

14

37

23

< < , so < < .

Comparing and Ordering Rational NumbersLESSON 5-1

Additional Examples

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Find the LCM of each pair of numbers.

1. 8, 6 2. 12, 16

3. Compare and order , – , and – from least to greatest. 316

810

316

24 48

810

– –< <316

316

Lesson Quiz

Comparing and Ordering Rational NumbersLESSON 5-1