Section

Objectives

2.2 Simplifying Fractions

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1. Factorizations and Divisibility2. Prime Factorization3.4.

Equivalent FractionsFind the Greatest Common Factor (GCF)

5. Simplifying Fractions to Lowest Terms6. Applications of Simplifying Fractions

Section 2.2 Simplifying Fractions

1. Factorizations and Divisibility

Slide 2Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

A factor of a number n is a nonzero whole number that divides evenly into n.

A factorization of a number n is a product of factors that equals n.

Example 1 Finding Factorizations of a Number

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Find four different factorizations of 12.

Find four different factorizations of 18

Example

Find four different factorizations of 24.

2 You Try

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PROCEDURE Divisibility Rules for 2, 3, 5, and 10

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• Divisibility by 2. A whole number is divisible by 2 if it is an even number.That is, the ones-place digit is 0, 2, 4, 6, or 8.

Examples: 26 and 384• Divisibility by 3. A whole number is divisible by 3

if the sum of its digits is divisible by 3.Example: 312 (sum of digits is 3 + 1 + 2 =

6, which is divisible by 3)

PROCEDURE Divisibility Rules for 2, 3, 5, and 10

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• Divisibility by 5. A whole number is divisible by 5 if its ones-place digit is 5 or 0.

Examples: 45 and 260• Divisibility by 10. A whole number is divisible by

10 if its ones-place digit is 0.Examples: 30 and 170

Example 3 Applying the Divisibility Rules

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c. 240 d. 570

Example

1. 45 2. 241

Determine whether the given number is divisible by 2,3, 5, or 10.

4 You Try

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Section 2.2 Simplifying Fractions

2. Prime Factorization

Two important classifications of whole numbers areprime numbers and composite numbers.

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DEFINITION Prime and Composite Numbers

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• A prime number is a whole number greater than 1 that has only two factors (itself and 1).

• A composite number is a whole number greater than 1 that is not prime. That is, a composite number will have at least one factor other than 1 and the number itself.

Note: The whole numbers 0 and 1 are neither prime nor composite.

Example 5 Identifying Prime and Composite Numbers

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Determine whether the number is prime, composite, or neither.

d. 17 e. 29 f. 153

Example

1. 21 2. 0 3. 57

Determine whether the number is prime, composite,or neither.

6 You Try

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DEFINITION Prime Factorization

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The prime factorization of a number is the factorization in which every factor is a prime number.Note: The order in which the factors are written does not affect the product.

Example 7 Determining the Prime Factorization of a Number

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Find the prime factorization of 220.

Example

4. Find the prime factorization of 96

8 You Try

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Section 2.2 Simplifying Fractions

2. Prime Factorization

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Another technique to find the prime factorization of a number is to divide the number by the smallest known prime factor of the number. Then divide the quotient by its smallest prime factor. Continue dividing in this fashion until the quotient is a prime number. The prime factorization is the product of divisors and the final quotient.

Example Determining Prime Factorizations

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9

Example

1. 126 2. 260

Find the prime factorization of the givennumber.

10 You Try

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Section 2.2 Simplifying Fractions

3. Equivalent Fractions

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Fractions are equivalent if they all represent the same portion of a whole.

Example

One method to show that two fractions areequivalent is to calculate their cross products.For example, to show that , we have

Determining whether two fractions are

equivalent.

11 Equivalent Fractions

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3 2

6 4

3 2

6 4

3 4 6 2

12 12

Example 12 Determining Whether Two Fractions Are Equivalent

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Example

1. 2.

Fill in the blank with or .

13 You Try

2

34

5

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3

5 12

15

Section

The greatest common factor (GCF) of two ormore numbers is the largest number that will divide each of the given numbers evenly.A common factor is a number that divide two ormore numbers evenly.

2.2 Simplifying Fractions

4. Find the Greatest Common Factor of two or more numbers.

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PROCEDURE

1. Write the prime factorization for each of the numbers in the group.2. Locate the prime factors that are common to all the numbers.3. The greatest common factor will be the product of all the common prime factors.

Finding the greatest common factor (GCF) of two are more numbers

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Example

28 and 24 35 and 26

14 Find the GCF

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Example

15, 45, and 90

15 Find the GCF

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Example

12 and 48 18 and 54

16 You Try

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Find the GCF.

Example

Find the GCF.36, 72, and 144

17 You Try

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Section 2.2 Simplifying Fractions

5. Simplifying Fractions to Lowest Terms

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A fraction is said to be in lowest terms if the numerator and denominator share no common factors other than 1.

To simplify a fraction, we begin by factoring the numerator and denominator into prime factors. This will help identify the common factors.

For example; 20 2 2 5 5

24 2 2 2 3 6

PROPERTY Fundamental Principle of Fractions

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Suppose that a number, c, is a common factor in the numerator and denominatorof a fraction. Then

Example 18 Simplifying a Fraction to Lowest Terms

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Example 19 Simplifying Fractions to Lowest Terms

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Simplify the fraction. Write the answer as a fraction or whole number.

Example 20 Simplifying Fractions by 10, 100, and 1000

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Simplify each fraction to lowest terms by first reducing by 10, 100, or 1000. Write the answer as a fraction.

Example

Simplify to lowest terms. Write the answer as afraction or whole number.

Simplify to lowest terms.

21 You Try

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1. 2.25

65

150

105

3. 4.54

6

35

105

Example

1. 2.

Simplify to lowest terms by first reducing by 10, 100, or1000.

22 You Try

160

120

5100

30,000

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Example 23 Simplifying Fractions in an Application

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Madeleine got 28 out of 35 problems correct on an algebra exam. David got 27 out of 45 questions correct on a different algebra exam.

What fractional part of the exam did eachstudent answer correctly?

6. Applications of Simplifying Fractions

Example

Joanne planted 77 seeds in her garden and 55 sprouted. Geoff planted 140 seeds and 80sprouted.What fractional part of the seeds sprouted forJoanne and what fractional part sprouted forGeoff?

24 You Try

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1. Factorizations and Divisibility2. Prime Factorization3.4.

Equivalent FractionsFind the Greatest Common Factor (GCF)

5. Simplifying Fractions to Lowest Terms6. Applications of Simplifying Fractions

A review of the objectives you are responsible for learning.

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