fractions and variables - diablo valley...

Free Pre-Algebra Lesson 28 page 1

© 2010 Cheryl Wilcox

Lesson 28

Fractions and Variables

In this lesson we work out the details of using variables in fractions. Since variables stand for numbers, everything we’ve learned so far applies. Multiplying and Dividing Algebraic Fractions

Since these operations don’t require a common denominator, they are the easiest fraction operations. You’ve already simplified and multiplied fractions with variables (Lesson 13). When working with algebraic fractions we never use mixed numbers, so even if the number in the numerator is larger than the number in the denominator, don’t try to convert.

Example: Simplify.

14a

7a2

2 • 7 • a

7 • a • a

2 • 7

1

• a

1

71

• a1

• a

2

a

15a

2b2•

14b

5a

3 • 5 • a

2 •b • b•

2 • 7 • b

5 • a

21

2b

15a

2b2

14b

5a

15a

2b2•

5a

14b

3 • 5 • a

2 •b •b•

5 • a

2 • 7 •b

75a2

28b3

30xy

z

15x

yz

30xy

z•

yz

15x

2 • 3 • 5 • x • y

z•

y • z

3 • 5 • x

2xy 2

12xy 2

Fractions with Exponents Whenever you work with exponents, you can figure out what to do by translating to a multiplication.

Example: Simplify.

5a

2

5a 5a 5 • 5 a • a

25a2

1

5a

2

1

5a•

1

5a

1

25a2

4

5a

2

4

5a•

4

5a

16

25a2

x

y

3

x

y•

x

y•

x

y

x 3

y 3

2x

5y

3

2x

5y•

2x

5y•

2x

5y

8x 3

125y 3

2x

5y

4

2x

5y•

2x

5y•

2x

5y•

2x

5y

16x 4

625y 4



Fractions with Fractions Since a fraction bar is also a division symbol, fractions stacked in fractions are really just (intimidating) division problems.

Example: Simplify.

7

5

2

7

52

7

5•

1

2

7

10

a

2

a

a2

a

a

1•

a

2

a2

2

4x

5

10x

3

4x

5

10x

3

4

2

x

5•

3

105

x

6

25

Adding and Subtracting Algebraic Fractions To add or subtract expressions with fractions, you need a common denominator. The prime factorization technique is easily adapted to fractions involving variables.

Example: Find equivalent fractions with a common denominator.

7

6

7

2 • 3•

3

3

21

18

8

9

8

3 • 3•

2

2

16

18

7

6a

7

2 • 3 • a•

3

3

21

18a

8

9

8

3 • 3•

2 • a

2 • a

16a

18a

7

6a

7

2 • 3 • a•

3 • a

3 • a

21a

18a2

8

9a2

8

3 • 3 • a • a•

2

2

16

18a2

To add

7

6

8

9, we convert to equivalent fractions with a common denominator and add the numerators:

21

18

16

18

37

18.

I can add 21 and 16 because both are just numbers. But if I want to add the fractions in the second box,

7

6a

8

9,

converting to a common denominator

21

18a

16a

18a means that I have to add 21 16a . Since 21 and 16a are not like

terms, they can’t be combined. The result looks very odd at first sight: (It is conventional to write the term with the variable first.)

21

18a

16a

18a

16a 21

18a

At this point it’s important to remember the cautions about cancelling. Since 16a and 21 are terms that are added, they cannot cancel factors in the denominator. So avoid temptation – you can’t simplify here. This ungainly beast actually is the answer.



Example: Find equivalent fractions with a common denominator, then add.

2

5x

2

5 • x•

2

2

4

10x

3

10

3

2 • 5•

x

x

3x

10x

2

5x

3

10

4

10x

3x

10x

3x 4

10x

If there are like terms, though, they can be combined.

Example: Find equivalent fractions with a common denominator, then add.

x

3y

x

3 • y•

2

2

2x

6y

x

2y

x

2 • y•

3

3

3x

6y

x

3y

x

2y

2x

6y

3x

6y

5x

6y

Rules for working with subtraction and negatives stay the same.

Example: Use the equivalent fractions found earlier to add or subtract.

x

3y

x

2y

2x

6y

3x

6y

x

6y

x

2y

x

3y

3x

6y

2x

6y

x

6y

x

2y

x

3y

3x

6y

2x

6y

x

6y

Mixed Operations The order of operations is of course the same.

Example: Simplify.

5

1

2• 6

51

2• 6 5 3 2

51

2

2

51

2

2

51

4

44

4

1

44

3

4

3

2•

1

5

7

10• 9

3

2•

1

5

7

10• 9

3

10

63

10

60

106



Simplifying Algebraic Expressions It’s the same, the same, the same, the same, the same, thesamesamesame…as it ever was.

Example: Simplify.

Example: Simplify.

103

5x

1

2

10

23

5x 10

51

2

6x 5

20x

5

3

4

20

4x

520

53

4

4x 15

One of These Things is Not Like the Others

3

4x

3x

4

3

4x

Since

3

4x means

3

4• x

3

4x

3

4•

x

1

3x

4

But

3

4x

3

4•

1

x

3

4x

3x

4

3

4x

1

2x

7

4x

x

2

7x

4

Common denominator:

x

2

x

2•

2

2

2x

4

7x

4

7x

2 • 2

7x

4

Add:

2x

4

7x

4

5x

4



Evaluating Algebraic Formulas and Expressions It’s a little more work to evaluate expressions when you substitute a fraction, but essentially the same process as with integers.

Example: Find the height of the object at the given times.

An orange is thrown straight up at 72 ft/sec from the roof of a 63-foot building and falls until it hits the ground (h = 0) below. The height t seconds after falling is given by the equation

h 16t2 72t 63

Find the height after 1/4 second.

h 161

4

2

721

463

16

1

1

16

72

1

1

463

1 18 63 80 feet above ground

Find the height after 51/4 seconds.

51

4

21

4

h 1621

4

2

7221

463

16

1

441

16

72

1

21

463

441 378 63 0 feet.

It hits the ground after 51

4 seconds.

Example: Evaluate the expression when x = –2/3.

4 x1

3

42

3

1

34

3

34 1

4 • 1 4

x

2

1

6

2

3

2

1

6

2

32

1

6

2

3•

1

2

1

6

2

6

1

6

1

6



Lesson 28: Fractions and Variables

Worksheet Name __________________________________________

1. Simplify

60xy

144x 2.

2. Multiply

45a

8b•

56b

60

3. Divide

90m

77

63m

121.

4. Change to a multiplication and simplify

4a

5

2

5. Simplify

9a

3

2

6. Simplify

72

9•

9

7

2

7. Find equivalent fractions with a common denominator.

7

4a

5

6

8. Add

7

4a

5

6.


7x

9

5x

6

10. Subtract

7x

9

5x

6.



11. Simplify

5

4

1

2

2

12. Simplify

2

3

7

2

9

5

13. Combine like terms.

5x

6

2x

3

x

2

14. Use the distributive property to simplify

452x

5

8

9.

15. The height of an object t seconds after being tossed in

the air is given by the equation h 16t2 64t 36 .

Find the height after 2

1

2 seconds.

16. Evaluate the expression

x

3

3

x when

x

4

9.




Homework 28A Name _________________________________________

1. A rectangle has length 21/2 inches and width 11/4 inches.

a. Find the area of the rectangle.

b. Find the perimeter of the rectangle.

2. Find the equivalent temperature using the formulas given. Write the answers using mixed numbers.

a. Find C

5(F 32)

9 when F = (–2/5)ºF.

b. Find F

9C

532 when C = 37ºC.

3. A triangle has base 7 inches and height 5 inches. Find the area of the triangle.

4. A box has length 341/2 inches, width 8 inches, and height 221/2 inches. Find the volume.

5. A runner ran 7 mph for 3/4 of an hour. What distance did she run?

6. There are 24 potstickers in each box, and Anton has 41/2 boxes. How many potstickers does Anton have?




13

15

7

12

8. Add 7

13

156

7

12.


1

5x

8

25

10. Subtract

1

5x

8

25.

11. How many 51/2 foot lengths can be cut from a 50-foot roll of tape?

12. Divide

9y

25x

15x

27.

13. Simplify

6n

7

3n

14.

14. Simplify

602

15x

3

4.

15. Evaluate

5

2xwhen x 0 .

16. Evaluate

x3

4

5when

x

3

4.




Homework 28A Answers



A lw 21

21

1

4

5

2•

5

4

25

83

1

8 square inches


P 2L 2W

2 •5

22 •

5

42

55

25 2

1

27

1

2 inches


a. Find C

5(F 32)

9 when F = (–2/5)ºF.

C

5(2

532)

9

5 322

5

9

5162

5

9

162

918ºC

b. Find F

9C

532 when C = 37ºC.

F9(37)

532

333

532

663

532 98

3

5ºF


A1

2bh

1

2• 7 • 5

35

217

1

2 square inches


V lwh 341

28 22

1

2

69

2•

8

2

1•

45

26210 cubic inches

5. A runner ran 7 mph for 3/4 of an hour. What distance did she run?

d rt 73

4

21

4

51

4 miles

6. There are 24 potstickers in each box, and Anton has 41/2 boxes. How many potstickers does Anton have?

24 potstickers

1 box• 4

1

2 boxes

24

12

•9

2 potstickers 108 potstickers




13

15

13

3 • 5•

2 • 2

2 • 2

52

60

7

12

7

2 • 2 • 3•

5

5

35

60

8. Add 7

13

156

7

12.

752

606

35

60(7 6) (

52

60

35

60)

1387

6013 1

27

6014

9

20


1

5x

1

5 • x•

5

5

5

25x

8

25

8

5 • 5•

x

x

8x

25x

10. Subtract

1

5x

8

25.

5

25x

8x

25x

8x 5

25x


50 51

250

11

2

50

1•

2

11

100

119

1

11

12. Divide

9y

25x

15x

27.

9

3

y

25x•

27

155

x

81y

125x 2

13. Simplify

6n

7

3n

14.

12n

14

3n

14

15n

14

14. Simplify

602

15x

3

4.

60

42

15x 60

153

4

8x 45

15. Evaluate

5

2xwhen x 0 .

5

2(0)

5

0 undefined

16. Evaluate

x3

4

5when

x

3

4.

3

4

3

4

5

0

50




Homework 28B Name __________________________________________





a. Find C

5(F 32)

9 when F = (–111/5)ºF.

b. Find F

9C

532 when C = 0ºC.



5. A runner ran 71/2 mph for 1/2 of an hour. What distance did she run?

6. There are 36 buffalo wings in each bag, and Abbie has 71/2 bags. How many wings does Abbie have?




11

12

13

20

8. Add 5

11

123

13

20.


8

x 2

7

2x

10. Subtract

7

2x

8

x 2.


12. Divide

21

20x

7x

60.

13. Simplify

6

5y

1

2y .

14. Simplify

368a

9

1

4.

15. Evaluate

5x

x1

3

when x

2

3.

16. Evaluate

2x

3when

x

3

4.

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