Download Lesson 28: Fractions and Variables

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Lesson 28  page 1
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.
1
14a
2•7•a
7•a•a
7a 2
15a
2b
14b
5a
2
1
2• 7• a
2
7• a•a
a
15a
1
1
•
5a
14b
2b
3•5•a
5•a
•
2•b •b 2•7•b
2
15a 14b
•
2b 2 5a
75a
2
28b
3
3• 5 • a
2•b • b
30xy
15x
30xy
z
yz
z
•
•
2 •7• b
5•a
yz
15x
2• 3 • 5 •x • y
z
2xy
1
2
21
2b
•
y• z
3•5•x
2xy 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
25a
x
y
5•5 a •a
2
3
x x x
• •
y y y
1
5a
© 2010 Cheryl Wilcox
4
5a
1 1
•
5a 5a
2x
5y
x3
y3
2
1
25a
4 4
•
5a 5a
2
3
2x
2x
2x
•
•
5y 5y 5y
2
2x
5y
8x 3
125 y 3
16
25a 2
4
2x 2x 2x 2x
•
•
•
5y 5y 5y 5y
16x 4
625 y 4
Free Pre-Algebra
Lesson 28  page 2
Fractions with Fractions
Since a fraction bar is also a division symbol, fractions stacked in fractions are really just (intimidating) division problems.
Example: Simplify.
a
7
5
7
5
2
2
2
7 1
•
5 2
2
a
4x
5
a a
a2
10x
3
1 2
2
a
a
7
10
•
4x
5
10x
3
2
4 x
3
•
5
10 x
6
25
5
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
3
•
2•3 3
21
18
8
9
8 2
•
3•3 2
16
18
To add
7
8
6
9
7
6a
7
3
•
2•3•a 3
21
18a
7
6a
8
9
8 2•a
•
3•3 2•a
16a
18a
8
9a 2
7
3•a
•
2•3•a 3•a
18a 2
8
2
•
3•3•a •a 2
, we convert to equivalent fractions with a common denominator and add the numerators:
I can add 21 and 16 because both are just numbers. But if I want to add the fractions in the second box,
21
21a
16
18a 2
21
16
37
18
18
18
7
8
6a
9
,
16a
means that I have to add 21 16a . Since 21 and 16a are not like
18a 18a
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.)
converting to a common denominator
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.
© 2010 Cheryl Wilcox
.
Free Pre-Algebra
Lesson 28  page 3
Example: Find equivalent fractions with a common denominator, then add.
2
5x
2
2
•
5•x 2
4
10x
3
10
3
x
•
2•5 x
3x
10x
2
5x
3
10
3x
10x
4
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
2y
x
2
3• y 2
•
x
3
•
2• y 3
x
3y
2x
6y
x
2y
3x
6y
2x
6y
3x
6y
x
2y
x
3y
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
x
2y
2x
6y
3x
6y
x
3y
x
6y
3x
6y
2x
6y
3x
6y
x
6y
2x
6y
x
6y
Mixed Operations
The order of operations is of course the same.
Example: Simplify.
5
1
•6
2
5
1
•6
2
5 3
2
5
1
2
5
1
2
4
© 2010 Cheryl Wilcox
2
3 1
•
2 5
2
5
4
4
1
4
1
4
3
4
4
3 1
•
2 5
60
10
7
•9
10
7
•9
10
6
3
10
63
10
Free Pre-Algebra
Lesson 28  page 4
Simplifying Algebraic Expressions
It’s the same, the same, the same, the same, the same,
thesamesamesame…as it ever was.
One of These Things is Not Like the Others
Example: Simplify.
1
x
2
x
7
x
4
7x
4
2
Since
Common denominator:
x
3x
4
3
x
4
x 2
•
2 2 2
7x
7x
4
2•2
2x
4
But
2x
4
7x
4
3
3
x means • x
4
4
3
3 x
x
•
4
4 1
3
4x
7x
4
5x
4
3x
4
3 1
•
4 x
3x
4
3
x
4
Add:
3
4x
3
4x
Example: Simplify.
10
3
x
5
1
2
20
2
10
6x
© 2010 Cheryl Wilcox
3
5
5
5
x
10
1
2
x
5
3
4
4
20
4x
x
5
15
5
20
3
4
Free Pre-Algebra
Lesson 28  page 5
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
16t 2 72t
63
Find the height after 51/4 seconds.
5
Find the height after 1/4 second.
h
1
4
16
2
1
4
72
16 1
1 16
72 1
1 4
1 18 63
1
4
21
4
2
21
16
4
h
63
72
16 441
1
16
63
21
4
72 21
1 4
441 378 63
It hits the ground after 5
80 feet above ground
x
1
3
2
4
3
4•1
1
6
2
1
3
3
4
3
2
3
4 1
2
4
1
6
2
3
2
1
6
2 1
•
3 2
1
6
2
6

© 2010 Cheryl Wilcox
63
0 feet.
Example: Evaluate the expression when x = –2/3.
4x
63
1
6
1
6
1
seconds.
4
Free Pre-Algebra
Lesson 27  page 6
Lesson 28: Fractions and Variables
Worksheet
1. Simplify
3. Divide
60xy
144 x
90m
77
5. Simplify
2
.
63m
.
121
9a
3
2
7. Find equivalent fractions with a common denominator.
7
4a
Name __________________________________________
2. Multiply
45a 56b
•
8b
60
4a
4. Change to a multiplication and simplify
5
72 9
•
6. Simplify
9
7
8. Add
7
4a
2
5
.
6
5
6
9. Find equivalent fractions with a common denominator.
7x
9
5x
6
© 2010 Cheryl Wilcox
10. Subtract
7x
9
5x
.
6
2
Free Pre-Algebra
5
11. Simplify
4
Lesson 27  page 7
1
2
2
12. Simplify
13. Combine like terms.
5x
6
2x
3
x
2
15. The height of an object t seconds after being tossed in
the air is given by the equation h
16t 2 64t 36 .
Find the height after 2
© 2010 Cheryl Wilcox
1
seconds.
2
2 7
3 2
9
5
14. Use the distributive property to simplify 45
16. Evaluate the expression
x
3
3
x
when x
2x
5
4
.
9
8
.
9
Free Pre-Algebra
Lesson 28  page 8
Lesson 28: Fractions and Variables
Homework 28A
1. A rectangle has length 21/2 inches and width 11/4 inches.
a. Find the area of the rectangle.
Name _________________________________________
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 the perimeter of the rectangle.
b. Find F
9C
5
32 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?
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 28  page 9
7. Find equivalent fractions with a common denominator.
13
15
8. Add 7
13
7
.
6
15
12
7
12
9. Find equivalent fractions with a common denominator.
1
5x
10. Subtract
1
5x
8
.
25
8
25
11. How many 51/2 foot lengths can be cut from a 50-foot roll
of tape?
13. Simplify
15. Evaluate
6n
7
3n
.
14
5
when x
2x
© 2010 Cheryl Wilcox
12. Divide
9y
25x
14. Simplify 60
15x
.
27
2
x
15
x
0.
16. Evaluate
5
3
.
4
3
4 when x
3
.
4
Free Pre-Algebra
Lesson 28  page 10
Lesson 28: Fractions and Variables
Homework 28A Answers
1. A rectangle has length 21/2 inches and width 11/4 inches.
a. Find the area of the rectangle.
2
A lw
5 5
1
2
1
25
8
•
2 4
1
4
2. Find the equivalent temperature using the formulas given.
Write the answers using mixed numbers.
1
3 square inches
8
5
2•
2
2
5
C
4
1
5 2
2
A
1
1
bh
•7•5
2
2
35
1
17 square inches
2
2
d
rt
5
© 2010 Cheryl Wilcox
7
3
4
1
miles
4
21
4
2
5
5
162
5
9
32 when C = 37ºC.
9(37)
333
32
32
5
5
3
3
66
32 98 ºF
5
5
4. A box has length 341/2 inches, width 8 inches, and height
221/2 inches. Find the volume.
V
34
lwh
1
1
8 22
2
2
2
69
2
5. A runner ran 7 mph for 3/4 of an hour. What distance did
she run?
32
18ºC
F
1
7 inches
2
3. A triangle has base 7 inches and height 5 inches. Find the
area of the triangle.
5
9
9C
5
b. Find F
2
5
5
2
32)
162
9
5
when F = (–2/5)ºF.
9
2L 2W
2•
32)
9
5(
b. Find the perimeter of the rectangle.
P
5(F
a. Find C
•
8 45
•
1 2
6210 cubic inches
6. There are 24 potstickers in each box, and Anton has 41/2
boxes. How many potstickers does Anton have?
24 potstickers
1 box
12
24 •
9
2
•4
1
2
boxes
potstickers
108 potstickers
Free Pre-Algebra
Lesson 28  page 11
7. Find equivalent fractions with a common denominator.
13
15
13 2 • 2
•
3•5 2•2
52
60
7
12
7
5
•
2•2•3 5
35
60
1 5
•
5•x 5
5
25x
8
25
8 x
•
5•5 x
8x
25x
52
35
52 35
6
(7 6) (
)
60
60
60 60
87
27
9
13
13 1
14
60
60
20
1
2
50 2
•
1 11
13. Simplify
6n
7
10. Subtract
1
5x
50
11
2
100
11
9
12. Divide
9y
25x
3
5
when x
2x
5
2(0)
15n
14
2
x
15
4
60
x
0.
16. Evaluate
5
undefined
0
125x 2
5
3
.
4
2
15
15
x
3
60
4
45
3
4 when x
3
4
3
4
5
© 2010 Cheryl Wilcox
81y
5
14. Simplify 60
3n
14
8x 5
25x
9 y 27
•
25x 15 x
8x
15. Evaluate
8x
25x
15x
.
27
1
11
3n
.
14
12n
14
8
.
25
5
25x
11. How many 51/2 foot lengths can be cut from a 50-foot roll
of tape?
50 5
13
7
.
6
15
12
7
9. Find equivalent fractions with a common denominator.
1
5x
8. Add 7
3
.
4
0
5
0
Free Pre-Algebra
Lesson 28  page 12
Lesson 28: Fractions and Variables
Homework 28B
1. A rectangle has length 23/4 inches and width 31/2 inches.
a. Find the area of the rectangle.
Name __________________________________________
2. Find the equivalent temperature using the formulas given.
Write the answers using mixed numbers.
a. Find C
b. Find the perimeter of the rectangle.
b. Find F
5(F
32)
9
9C
5
when F = (–111/5)ºF.
32 when C = 0ºC.
3. A triangle has base 17 inches and height 25 inches. Find
the area of the triangle.
4. A box has length 141/2 inches, width 12 inches, and height
23/4 inches. Find the volume.
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?
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 28  page 13
7. Find equivalent fractions with a common denominator.
11
12
8. Add 5
11
13
.
3
12
20
13
20
9. Find equivalent fractions with a common denominator.
8
10. Subtract
7
2x
8
x2
.
x2
7
2x
11. How many 31/4 foot lengths can be cut from a 60-foot roll
of tape?
13. Simplify
6
y
5
1
y.
2
5x
15. Evaluate
x
1
3
when x
© 2010 Cheryl Wilcox
12. Divide
21
20x
14. Simplify 36
2
.
3
16. Evaluate
7x
.
60
8a
9
1
.
4
2x
when x
3
3
.
4