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Topic 8.1.2Topic 8.1.2

Equivalent FractionsEquivalent Fractions

2

Lesson

1.1.1

California Standard:12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.

What it means for you:You’ll learn about equivalent fractions and how to simplify fractions to their lowest terms.

Equivalent FractionsEquivalent FractionsTopic

8.1.2

Key words:• equivalent• rational• simplify• common factor

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Lesson

1.1.1

Saying that two rational expressions are equivalent is just a way of saying that two fractions represent the same thing.

Equivalent FractionsEquivalent FractionsTopic

8.1.2

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Lesson

1.1.1

Equivalent Fractions Have the Same Value

Equivalent FractionsEquivalent FractionsTopic

8.1.2

A ratio is a comparison of two numbers, often expressed

by a fraction — for example, . ab

A proportion is an equality between two ratios.

Four quantities a, b, c, and d are in proportion if = . cd

ab

Fractions like these that represent the same rational number or expression are often called equivalent fractions.

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Lesson

1.1.1

Equivalent Fractions Have the Same Value

Equivalent FractionsEquivalent FractionsTopic

8.1.2

You can determine whether two fractions are equivalent by using this rule:

69

23

5x6

10x12==

The rational expressions and are equivalent if ad = bc.cd

ab

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Equivalent FractionsEquivalent Fractions

Example 1

Topic

8.1.2

Solution

5x • 12 = 60x

Solution follows…

So, the two rational expressions are equivalent.

Prove that and are equivalent.1210x

65x

This is ad in the rule above

This is bc in the rule aboveand 6 • 10x = 60x

The rational expressions and are equivalent if ad = bc.cd

ab

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Prove that the following pairs of rational expressions are equivalent.

1. and 2. and 3. and

Lesson

1.1.1

Guided Practice

Fractions and Rational ExpressionsFractions and Rational ExpressionsTopic

8.1.1

[ad] 54m × 2 = 108m[bc] 6 × 18m = 108m ad = bcso the expressions are equivalent.

Solution follows…

3x – 912

x – 34

654m

218m

31

6x2x

[ad] 1 × 6x = 6x [bc] 3 × 2x = 6x ad = bc so the expressions are equivalent.

[ad] 12 × (x – 3) = 12x – 36[bc] (3x – 9) × 4 = 12x – 36

ad = bc so the expressions are equivalent.

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Lesson

1.1.1

Simplify Fractions by Canceling Common Factors

Equivalent FractionsEquivalent FractionsTopic

8.1.2

A rational expression can be written in its lowest terms by reducing it to the simplest equivalent fraction.

This is done by factoring both the numerator and denominator and then canceling the common factors— that means making sure its numerator and denominator have no common factors other than 1.

For example:6·136·11

7866

=1311

=1

1

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Equivalent FractionsEquivalent Fractions

Example 2

Topic

8.1.2

Reduce the expression to its lowest terms.

Solution

The greatest common factor (GCF) of 56 and 64 is 8.

Solution follows…

So, and are equivalent fractions.

5664x

This means that:5664x

5664x

=(8·8)x7·8 7

8x=

1

1

78x

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Lesson

1.1.1

Simplify Fractions by Canceling Common Factors

Equivalent FractionsEquivalent FractionsTopic

8.1.2

Numbers are not the only things that can be canceled — variables can be canceled too.

For example: c·vm·c

= =1

1cvmc

vm

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Reduce each of the following rational expressions to their lowest terms.

4. 5. 6.

7. 8. 9.

10. 11. 12.

Lesson

1.1.1

Guided Practice

Fractions and Rational ExpressionsFractions and Rational ExpressionsTopic

8.1.1

Solution follows…

3010d

2821

1812

b2xbx

10m3c2

4m2c–(b – 3) b(b – 3)

(5 + m)(5 – m) (3 + m)(5 + m)

x + 53x + 15

m2(m + 4)(m – 4)

m(m – 4)

34

23

1b

d3

b

13

5mc2

–

m(m + 4)3 + m5 – m

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Lesson

1.1.1

Some Harder Examples to Think About

Equivalent FractionsEquivalent FractionsTopic

8.1.2

Factoring the numerator and denominator is the key to doing this type of question.

Breaking down a complicated expression into its factors means you can spot the terms that will cancel.

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(x – 3)(x + 3) 6(x – 3)

Equivalent FractionsEquivalent Fractions

Example 3

Topic

8.1.2

Simplify the expression .

Solution

Factor the numerator and denominator, then cancel common factors:

Solution follows…

=1

1

=x2 – 9

6x – 18x + 3

6

x2 – 9 6x – 18

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Equivalent FractionsEquivalent Fractions

Example 4

Topic

8.1.2

Simplify the expression .

SolutionFactor the numerator and denominator completely.

Solution follows…

1

Cancel the common factor (3 – m)

9 – m2 m2 – m – 6

9 – m2 m2 – m – 6

(3 – m)(3 + m) (m – 3)(m + 2)

=

(3 – m)(3 + m) –1(3 – m)(m + 2)

=

(3 + m) –1(m + 2)

=

(3 + m) (m + 2)

= –

1

15

x(x – 5)(x + 3)x(x + 3)(x + 7)

=

Equivalent FractionsEquivalent Fractions

Example 5

Topic

8.1.2

Reduce this expression to its lowest terms:

Solution

Factor both the numerator and denominator.

Solution follows…

Cancel the common factors

1

1

1

x3 – 2x2 – 15x x3 + 10x2 + 21x

x3 – 2x2 – 15x x3 + 10x2 + 21x

x(x2 – 2x – 15)x(x2 + 10x + 21)

=

=x – 5x + 7

1

16

13. Show how you can simplify the rational

expression .

Lesson

1.1.1

Guided Practice

Equivalent FractionsEquivalent FractionsTopic

8.1.2

Solution follows…

a2 + 5a + 6 a2 + 2a – 3

Cancel out the common factor a + 3, leaving the expression as:

Write the numerator and denominator in factored form:

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Lesson

1.1.1

Guided Practice

Equivalent FractionsEquivalent FractionsTopic

8.1.2

Solution follows…

Simplify the following rational expressions.

14. 15.

16. 17.

18. 19. –2k

4 – k k2 – 16

m2 – c2 c2 – mc

(k – 2)2 4 – k2

(m – 3)(m + 5) 3(3 – m)(m + 5)

m2 + mk – 6k2 2k2 + mk – m2

20k3 + 26k2 – 6k 3 – 13k – 10k2

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Equivalent FractionsEquivalent Fractions

Independent Practice

Solution follows…

Topic

8.1.2

–1

a + b + m + c

1. Simplify .6k2 – 3ck – 2c2

2c2 + 3ck – 2k2

2. Simplify .

(a + b)2 – (m + c)2

a + b – m – c

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Equivalent FractionsEquivalent Fractions

Independent Practice

Solution follows…

Topic

8.1.2

Reduce each of the following rational expressions to their lowest terms.

3. 4.

5. 6.

7.

Cannot be reduced further

16x2 – v2 x2 – 2xv – 24x2

m + c c – a

3y2 – 21y + 3018v2 – 12y – 48

4x2 – 4x – 2410x2 + 50x + 60

k3 + 10k2 + 21kk2 + 2k – 35

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Equivalent FractionsEquivalent Fractions

Independent Practice

Solution follows…

Topic

8.1.2

Reduce each of the following rational expressions to their lowest terms.

8. 9.

10. 11.

x – 2y2x2 – 8y2 2x + 4y

x2 + 5xy – 14y2

x2 + 4xy – 21y2

3a2 – 12a2 – 3a – 10

6a2 – 7ab – 5b2

6a2 – 13ab + 5b2

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Equivalent FractionsEquivalent Fractions

Independent Practice

Solution follows…

Topic

8.1.2

Matthew has incorrectly assumed 2 to be a common factor of 2x and 7.

Correct answer is x – 72

12. Matthew simplified in this way:2x2 – x – 21

2x + 6

Explain the error that Matthew has made and then simplify the expression correctly.

(x + 3)(2x – 7) 2(x + 3)

2x – 7 2

= = x – 7

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Topic

8.1.2

Round UpRound Up

Equivalent FractionsEquivalent Fractions

If you were to condense everything from this Section into a couple of points, they would be:

• Rational expressions are the same as fractions and are undefined if the denominator equals zero.

• A rational expression can be reduced to its lowest equivalent fraction by dividing out common factors of the numerator and denominator.