1 Topic 8.1.2 Equivalent Fractions. 2 Lesson 1.1.1 California Standard: 12.0 Students simplify...
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Transcript of 1 Topic 8.1.2 Equivalent Fractions. 2 Lesson 1.1.1 California Standard: 12.0 Students simplify...
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
3
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
4
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.
5
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
6
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
7
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.
8
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
9
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
10
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
11
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
12
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.
13
(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
14
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:
17
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
18
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
19
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
20
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
21
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
22
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.