Fractions and Variables - Diablo Valley Collegevoyager.dvc.edu/~lmonth/PreAlg/lesson28student.pdf · · 2011-02-25Lesson 28 Fractions and Variables ... learned so far applies. Multiplying
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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.
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.
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