1. Equivalent Fractions
Topic 8.1.2

2. Equivalent Fractions 8.1.2 1.1.1 California Standard:
Lesson
1.1.1
Topic
8.1.2
Equivalent Fractions
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.
Key words:
equivalent
rational
simplify
common factor

3. Topic
8.1.2
Lesson
1.1.1
Equivalent Fractions
Saying that two rational expressions are equivalent is just a way of saying that two fractions represent the same thing.

4. Lesson
1.1.1
Topic
8.1.2
Equivalent Fractions
Equivalent Fractions Have the Same Value
A ratio is a comparison of two numbers, often expressed by a fraction — for example, . a b
A proportion is an equality between two ratios.
Four quantities a, b, c, and d are in proportion if = . c d a b
Fractions like these that represent the same rational number or expression are often called equivalent fractions.

5. Topic
8.1.2
Lesson
1.1.1
Equivalent Fractions
Equivalent Fractions Have the Same Value 2 3 6 9 5x 6
10x 12 = =
You can determine whether two fractions are equivalent by using this rule: a b c d
The rational expressions and are equivalent if ad = bc.

6. So, the two rational expressions are equivalent.
Topic
8.1.2
Equivalent Fractions
Example 1
Prove that and are equivalent. 12
10x 6 5x
Solution
The rational expressions and are equivalent if ad = bc. c d a b
5x • 12 = 60x
This is ad in the rule above
and 6 • 10x = 60x
This is bc in the rule above
So, the two rational expressions are equivalent.
Solution follows…

7. Fractions and Rational Expressions
Lesson
1.1.1
Topic
8.1.1
Fractions and Rational Expressions
Guided Practice
Prove that the following pairs of rational expressions are equivalent.
1. and and and 6
54m 2
18m 3 1 6x 2x
3x – 9 12
x – 3 4
[ad] 54m × 2 = 108m [bc] 6 × 18m = 108m ad = bc
so the expressions are equivalent.
[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 – ad = bc so the expressions are equivalent.
Solution follows…

8. Lesson
1.1.1
Topic
8.1.2
Equivalent Fractions
Simplify Fractions by Canceling Common Factors
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:
6·13
6·11 78 66 = 13 11 1

9. Equivalent Fractions 8.1.2 56 64x
Topic
8.1.2
Equivalent Fractions
Example 2 56
64x
Reduce the expression to its lowest terms.
Solution
The greatest common factor (GCF) of 56 and 64 is 8. 1 56
64x
(8·8)x
7·8 7 8x
This means that: = = 1 56
64x 7 8x
So, and are equivalent fractions.
Solution follows…

10. Topic
8.1.2
Lesson
1.1.1
Equivalent Fractions
Simplify Fractions by Canceling Common Factors
Numbers are not the only things that can be canceled — variables can be canceled too. cv mc
c·v
m·c 1 v m
For example: = = 1

11. Fractions and Rational Expressions
Lesson
1.1.1
Topic
8.1.1
Fractions and Rational Expressions
Guided Practice
Reduce each of the following rational expressions to their lowest terms. 28 21 18 12
b2x bx 3 4 2 3 1 b 30
10d
10m3c2
4m2c –
(b – 3)
b(b – 3) d 3
5mc 2 – b
(5 + m)(5 – m)
(3 + m)(5 + m)
m2(m + 4)(m – 4)
m(m – 4)
x + 5
3x + 15 1 3
3 + m
5 – m
m(m + 4)
Solution follows…

12. 8.1.2 1.1.1 Equivalent Fractions Some Harder Examples to Think About
Topic
8.1.2
Lesson
1.1.1
Equivalent Fractions
Some Harder Examples to Think About
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. Equivalent Fractions 8.1.2 x2 – 9 6x – 18 Simplify the expression .
Topic
8.1.2
Equivalent Fractions
Example 3
x2 – 9
6x – 18
Simplify the expression
Solution
Factor the numerator and denominator, then cancel common factors:
x2 – 9
6x – 18 1
(x – 3)(x + 3)
6(x – 3)
x + 3 6 = = 1
Solution follows…

14. Cancel the common factor (3 – m)
Topic
8.1.2
Equivalent Fractions
Example 4
9 – m2
m2 – m – 6
Simplify the expression
Solution
Factor the numerator and denominator completely.
9 – m2
m2 – m – 6
(3 – m)(3 + m)
(m – 3)(m + 2) =
(3 – m)(3 + m)
–1(3 – m)(m + 2) = 1 1
(3 + m)
–1(m + 2) =
Cancel the common factor (3 – m)
(3 + m)
(m + 2)
= –
Solution follows…

15. Cancel the common factors
Topic
8.1.2
Equivalent Fractions
Example 5
x3 – 2x2 – 15x
x3 + 10x2 + 21x
Reduce this expression to its lowest terms:
Solution
Factor both the numerator and denominator.
x3 – 2x2 – 15x
x3 + 10x2 + 21x
x(x2 – 2x – 15)
x(x2 + 10x + 21) =
x(x – 5)(x + 3)
x(x + 3)(x + 7) = 1 1
Cancel the common factors 1 1 =
x – 5
x + 7
Solution follows…

16. Equivalent Fractions 8.1.2 1.1.1 Guided Practice
Lesson
1.1.1
Topic
8.1.2
Equivalent Fractions
Guided Practice
13. Show how you can simplify the rational
expression
a2 + 5a + 6
a2 + 2a – 3
Write the numerator and denominator in factored form:
Cancel out the common factor a + 3, leaving the expression as:
Solution follows…

17. 8.1.2 1.1.1 Equivalent Fractions Guided Practice
Topic
8.1.2
Lesson
1.1.1
Equivalent Fractions
Guided Practice
Simplify the following rational expressions.
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
–2k
Solution follows…

18. Equivalent Fractions 8.1.2 Independent Practice 6k2 – 3ck – 2c2
Topic
8.1.2
Equivalent Fractions
Independent Practice
1. Simplify
6k2 – 3ck – 2c2
2c2 + 3ck – 2k2 –1
2. Simplify
(a + b)2 – (m + c)2
a + b – m – c
a + b + m + c
Solution follows…

19. Equivalent Fractions 8.1.2 Independent Practice
Topic
8.1.2
Equivalent Fractions
Independent Practice
Reduce each of the following rational expressions to their lowest terms. 7.
16x2 – v2
x2 – 2xv – 24x2
m + c
c – a
Cannot be reduced further
4x2 – 4x – 24
10x2 + 50x + 60
3y2 – 21y + 30
18v2 – 12y – 48
k3 + 10k2 + 21k
k2 + 2k – 35
Solution follows…

20. Equivalent Fractions 8.1.2 Independent Practice
Topic
8.1.2
Equivalent Fractions
Independent Practice
Reduce each of the following rational expressions to their lowest terms.
2x2 – 8y2
2x + 4y
x2 + 5xy – 14y2
x2 + 4xy – 21y2
x – 2y
3a2 – 12
a2 – 3a – 10
6a2 – 7ab – 5b2
6a2 – 13ab + 5b2
Solution follows…

21. Equivalent Fractions 8.1.2 Independent Practice
Topic
8.1.2
Equivalent Fractions
Independent Practice
12. Matthew simplified in this way:
2x2 – x – 21
2x + 6
(x + 3)(2x – 7)
2(x + 3)
2x – 7 2
= = x – 7
Explain the error that Matthew has made and then simplify the expression correctly.
Matthew has incorrectly assumed 2 to be a common factor of 2x and 7. Correct answer is x – 7 2
Solution follows…

22. Equivalent Fractions 8.1.2 Round Up
Topic
8.1.2
Equivalent Fractions
Round Up
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.