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Chapter 5: Rational Expressions

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Chapter 14
Rational
Expressions
Chapter Sections
14.1 – Simplifying Rational Expressions
14.2 – Multiplying and Dividing Rational Expressions
14.3 – Adding and Subtracting Rational Expressions with the
Same Denominator and Least Common Denominators
14.4 – Adding and Subtracting Rational Expressions with
Different Denominators
14.5 – Solving Equations Containing Rational Expressions
14.6 – Problem Solving with Rational Expressions
14.7 – Simplifying Complex Fractions
Martin-Gay, Developmental Mathematics
2
В§ 14.1
Simplifying Rational
Expressions
Rational Expressions
Rational expressions can be written in the form
where P and Q are both polynomials and Q п‚№ 0.
P
Q
Examples of Rational Expressions
3x пЂ« 2 x пЂ­ 4
4x пЂ« 3y
4x пЂ­ 5
2 x пЂ­ 3 xy пЂ« 4 y
2
2
Martin-Gay, Developmental Mathematics
3x
2
2
4
4
Evaluating Rational Expressions
To evaluate a rational expression for a particular
value(s), substitute the replacement value(s) into the
rational expression and simplify the result.
Example
Evaluate the following expression for y = пЂ­2.
yпЂ­2
пЂ­4
4
пЂ­2 пЂ­ 2
пЂЅ
пЂЅ
пЂЅ
пЂ­ 5 пЂ« (пЂ­ 2)
пЂ­7
7
пЂ­5пЂ« y
Martin-Gay, Developmental Mathematics
5
Evaluating Rational Expressions
In the previous example, what would happen if we
tried to evaluate the rational expression for y = 5?
yпЂ­2
3
5пЂ­2
пЂЅ
пЂЅ
пЂ­5 пЂ« 5
пЂ­5пЂ« y
0
This expression is undefined!
Martin-Gay, Developmental Mathematics
6
Undefined Rational Expressions
We have to be able to determine when a
rational expression is undefined.
A rational expression is undefined when the
denominator is equal to zero.
The numerator being equal to zero is okay
(the rational expression simply equals zero).
Martin-Gay, Developmental Mathematics
7
Undefined Rational Expressions
Find any real numbers that make the following rational
expression undefined.
Example
9x пЂ« 4x
3
15 x пЂ« 45
The expression is undefined when 15x + 45 = 0.
So the expression is undefined when x = пЂ­3.
Martin-Gay, Developmental Mathematics
8
Simplifying Rational Expressions
Simplifying a rational expression means writing it in
lowest terms or simplest form.
To do this, we need to use the
Fundamental Principle of Rational Expressions
If P, Q, and R are polynomials, and Q and R are not 0,
PR
QR
пЂЅ
P
Q
Martin-Gay, Developmental Mathematics
9
Simplifying Rational Expressions
Simplifying a Rational Expression
1) Completely factor the numerator and
denominator.
2) Apply the Fundamental Principle of Rational
Expressions to eliminate common factors in the
numerator and denominator.
Warning!
Only common FACTORS can be eliminated from
the numerator and denominator. Make sure any
expression you eliminate is a factor.
Martin-Gay, Developmental Mathematics
10
Simplifying Rational Expressions
Example
Simplify the following expression.
7 x пЂ« 35
x пЂ« 5x
2
пЂЅ
7 ( x пЂ« 5)
x ( x пЂ« 5)
пЂЅ
Martin-Gay, Developmental Mathematics
7
x
11
Simplifying Rational Expressions
Example
Simplify the following expression.
x пЂ« 3x пЂ­ 4
2
x пЂ­ x пЂ­ 20
2
пЂЅ
( x пЂ« 4 )( x пЂ­ 1)
( x пЂ­ 5 )( x пЂ« 4 )
Martin-Gay, Developmental Mathematics
пЂЅ
x пЂ­1
xпЂ­5
12
Simplifying Rational Expressions
Example
Simplify the following expression.
7пЂ­ y
yпЂ­7
пЂЅ
пЂ­ 1( y пЂ­ 7 )
yпЂ­7
пЂЅ пЂ­1
Martin-Gay, Developmental Mathematics
13
В§ 14.2
Multiplying and Dividing
Rational Expressions
Multiplying Rational Expressions
Multiplying rational expressions when P,
Q, R, and S are polynomials with Q п‚№ 0
and S п‚№ 0.
P R
PR
пѓ— пЂЅ
Q S
QS
Martin-Gay, Developmental Mathematics
15
Multiplying Rational Expressions
Note that after multiplying such expressions, our result
may not be in simplified form, so we use the following
techniques.
Multiplying rational expressions
1) Factor the numerators and denominators.
2) Multiply the numerators and multiply the
denominators.
3) Simplify or write the product in lowest terms
by applying the fundamental principle to all
common factors.
Martin-Gay, Developmental Mathematics
16
Multiplying Rational Expressions
Example
Multiply the following rational expressions.
6x
2
10 x
3
пѓ—
5x
12
пЂЅ
2 пѓ—3пѓ— x пѓ— x пѓ—5пѓ— x
2 пѓ—5пѓ— x пѓ— x пѓ— x пѓ—2 пѓ—2 пѓ—3
Martin-Gay, Developmental Mathematics
пЂЅ
1
4
17
Multiplying Rational Expressions
Example
Multiply the following rational expressions.
(m пЂ­ n)
mпЂ«n
2
пѓ—
m
m пЂ­ mn
2
пЂЅ
( m пЂ­ n )( m пЂ­ n ) пѓ— m
(m пЂ« n) пѓ— m (m пЂ­ n)
пЂЅ
mпЂ­n
mпЂ«n
Martin-Gay, Developmental Mathematics
18
Dividing Rational Expressions
Dividing rational expressions when P, Q, R,
and S are polynomials with Q п‚№ 0, S п‚№ 0 and
R п‚№ 0.
P
R
P S
PS
п‚ё
пЂЅ
пѓ— пЂЅ
Q S
Q R QR
Martin-Gay, Developmental Mathematics
19
Dividing Rational Expressions
When dividing rational expressions, first
change the division into a multiplication
problem, where you use the reciprocal of the
divisor as the second factor.
Then treat it as a multiplication problem
(factor, multiply, simplify).
Martin-Gay, Developmental Mathematics
20
Dividing Rational Expressions
Example
Divide the following rational expression.
( x пЂ« 3)
5
2
п‚ё
5 x пЂ« 15
пЂЅ
25
( x пЂ« 3 )( x пЂ« 3 ) пѓ— 5 пѓ— 5
5 пѓ— 5 ( x пЂ« 3)
пЂЅ
( x пЂ« 3)
2
пѓ—
5
25
5 x пЂ« 15
пЂЅ
xпЂ«3
Martin-Gay, Developmental Mathematics
21
Units of Measure
Converting Between Units of Measure
Use unit fractions (equivalent to 1), but with
different measurements in the numerator and
denominator.
Multiply the unit fractions like rational
expressions, canceling common units in the
numerators and denominators.
Martin-Gay, Developmental Mathematics
22
Units of Measure
Example
Convert 1008 square inches into square feet.
(1008 sq
пѓ¦ 1 ft пѓ¶ пѓ¦ 1 ft пѓ¶
in) пѓ— пѓ§
пѓ· пѓ—пѓ§
пѓ·пЂЅ
пѓЁ 12 in пѓё пѓЁ 12 in пѓё
(2В·2В·2В·2В·3В·3В·7 in В·
in)
1 ft
1 ft
пѓ¦
пѓ¶ пѓ¦
пѓ¶
пѓ—пѓ§
пѓ· пѓ—пѓ§
пѓ·пЂЅ
пѓЁ 2 пѓ— 2 пѓ— 3 in пѓё пѓЁ 2 пѓ— 2 пѓ— 3 in пѓё
7 sq. ft.
Martin-Gay, Developmental Mathematics
23
В§ 14.3
Adding and Subtracting Rational
Expressions with the Same
Denominator and Least Common
Denominators
Rational Expressions
If P, Q and R are polynomials and Q п‚№ 0,
P
пЂ«
Q
пЂЅ
PпЂ«Q
R
R
R
P
Q
PпЂ­Q
R
пЂ­
R
пЂЅ
R
Martin-Gay, Developmental Mathematics
25
Adding Rational Expressions
Example
Add the following rational expressions.
4pпЂ­3
2pпЂ«7
пЂ«
3p пЂ«8
2pпЂ«7
пЂЅ
4p пЂ­3пЂ«3p пЂ«8
2pпЂ«7
Martin-Gay, Developmental Mathematics
пЂЅ
7pпЂ«5
2pпЂ«7
26
Subtracting Rational Expressions
Example
Subtract the following rational expressions.
8y
yпЂ­2
пЂ­
16
yпЂ­2
пЂЅ
8 y пЂ­ 16
yпЂ­2
пЂЅ
8( y пЂ­ 2)
yпЂ­2
Martin-Gay, Developmental Mathematics
пЂЅ 8
27
Subtracting Rational Expressions
Example
Subtract the following rational expressions.
3y
y пЂ« 3 y пЂ­ 10
2
6
пЂ­
y пЂ« 3 y пЂ­ 10
2
3( y пЂ­ 2 )
( y пЂ« 5 )( y пЂ­ 2 )
пЂЅ
пЂЅ
3y пЂ­ 6
y пЂ« 3 y пЂ­ 10
2
пЂЅ
3
yпЂ«5
Martin-Gay, Developmental Mathematics
28
Least Common Denominators
To add or subtract rational expressions with
unlike denominators, you have to change
them to equivalent forms that have the same
denominator (a common denominator).
This involves finding the least common
denominator of the two original rational
expressions.
Martin-Gay, Developmental Mathematics
29
Least Common Denominators
To find a Least Common Denominator:
1) Factor the given denominators.
2) Take the product of all the unique factors.
Each factor should be raised to a power equal
to the greatest number of times that factor
appears in any one of the factored
denominators.
Martin-Gay, Developmental Mathematics
30
Least Common Denominators
Example
Find the LCD of the following rational expressions.
1
,
3x
6 y 4 y пЂ« 12
6 y пЂЅ 2 пѓ—3y
4 y пЂ« 12 пЂЅ 4 ( y пЂ« 3 ) пЂЅ 2 ( y пЂ« 3 )
2
So the LCD is 2 пѓ— 3 y ( y пЂ« 3 ) пЂЅ 12 y ( y пЂ« 3 )
2
Martin-Gay, Developmental Mathematics
31
Least Common Denominators
Example
Find the LCD of the following rational expressions.
4
,
4x пЂ­ 2
x пЂ« 4 x пЂ« 3 x пЂ« 10 x пЂ« 21
2
2
x пЂ« 4 x пЂ« 3 пЂЅ ( x пЂ« 3 )( x пЂ« 1)
2
x пЂ« 10 x пЂ« 21 пЂЅ ( x пЂ« 3 )( x пЂ« 7 )
2
So the LCD is (x пЂ« 3)(x пЂ« 1)(x пЂ« 7)
Martin-Gay, Developmental Mathematics
32
Least Common Denominators
Example
Find the LCD of the following rational expressions.
3x
,
4x
2
5x пЂ­ 5 x пЂ­ 2x пЂ« 1
2
2
5 x пЂ­ 5 пЂЅ 5 ( x пЂ­ 1) пЂЅ 5 ( x пЂ« 1)( x пЂ­ 1)
2
2
x пЂ­ 2 x пЂ« 1 пЂЅ ( x пЂ­ 1)
2
2
So the LCD is 5(x пЂ« 1)(x - 1)
Martin-Gay, Developmental Mathematics
2
33
Least Common Denominators
Example
Find the LCD of the following rational expressions.
1
,
2
xпЂ­3 3пЂ­ x
Both of the denominators are already factored.
Since each is the opposite of the other, you can
use either x – 3 or 3 – x as the LCD.
Martin-Gay, Developmental Mathematics
34
Multiplying by 1
To change rational expressions into equivalent
forms, we use the principal that multiplying
by 1 (or any form of 1), will give you an
equivalent expression.
P R
Pпѓ—R
пЂЅ
пѓ—1 пЂЅ
пѓ— пЂЅ
Q
Q
Q R Qпѓ—R
P
P
Martin-Gay, Developmental Mathematics
35
Equivalent Expressions
Example
Rewrite the rational expression as an equivalent
rational expression with the given denominator.
3
9y
3
9y
5
пЂЅ
5
3
9y
5
пЂЅ
72 y
пѓ—
8y
4
8y
4
9
пЂЅ
24 y
4
72 y
9
Martin-Gay, Developmental Mathematics
36
В§ 14.4
Adding and Subtracting
Rational Expressions with
Different Denominators
Unlike Denominators
As stated in the previous section, to add or
subtract rational expressions with different
denominators, we have to change them to
equivalent forms first.
Martin-Gay, Developmental Mathematics
38
Unlike Denominators
Adding or Subtracting Rational Expressions with
Unlike Denominators
1) Find the LCD of all the rational
expressions.
2) Rewrite each rational expression as an
equivalent one with the LCD as the
denominator.
3) Add or subtract numerators and write result
over the LCD.
4) Simplify rational expression, if possible.
Martin-Gay, Developmental Mathematics
39
Adding with Unlike Denominators
Example
Add the following rational expressions.
15
,
8
7a 6a
15
8
пЂ«
7a
90
42 a
пЂЅ
6a
пЂ«
56
42 a
6 пѓ— 15
6 пѓ— 7a
пЂЅ
пЂ«
146
42 a
7 пѓ—8
7 пѓ— 6a
пЂЅ
пЂЅ
73
21 a
Martin-Gay, Developmental Mathematics
40
Subtracting with Unlike Denominators
Example
Subtract the following rational expressions.
5
,
3
2x пЂ­ 6 6 пЂ­ 2x
5
2x пЂ­ 6
пЂ­
3
6 пЂ­ 2x
8
2x пЂ­ 6
пЂЅ
пЂЅ
5
2x пЂ­ 6
2пѓ—2пѓ—2
2 ( x пЂ­ 3)
пЂЅ
пЂ«
3
2x пЂ­ 6
пЂЅ
4
xпЂ­3
Martin-Gay, Developmental Mathematics
41
Subtracting with Unlike Denominators
Example
Subtract the following rational expressions.
7
2x пЂ­ 3
7
2x пЂ­ 3
7
2x пЂ­ 3
пЂ­
7
пЂ­3пЂЅ
6x пЂ­ 9
2x пЂ­ 3
and 3
2x пЂ­ 3
пЂЅ
пЂ­
3( 2 x пЂ­ 3)
2x пЂ­ 3
7 пЂ­ 6x пЂ« 9
2x пЂ­ 3
пЂЅ
Martin-Gay, Developmental Mathematics
пЂЅ
16 пЂ­ 6 x
2x пЂ­ 3
42
Adding with Unlike Denominators
Example
Add the following rational expressions.
4
x
,
x пЂ­ x пЂ­ 6 x пЂ« 5x пЂ« 6
4
x
4
x
пЂ«
пЂЅ
пЂ«
пЂЅ
2
2
x пЂ­ x пЂ­ 6 x пЂ« 5x пЂ« 6
( x пЂ­ 3 )( x пЂ« 2 ) ( x пЂ« 3 )( x пЂ« 2 )
2
4 ( x пЂ« 3)
( x пЂ­ 3 )( x пЂ« 2 )( x пЂ« 3 )
4 x пЂ« 12 пЂ« x пЂ­ 3 x
2
пЂ«
( x пЂ« 3 )( x пЂ« 2 )( x пЂ­ 3 )
пЂЅ
x пЂ« x пЂ« 12
2
2
( x пЂ« 2 )( x пЂ­ 3 )( x пЂ« 3 )
x ( x пЂ­ 3)
пЂЅ
( x пЂ« 2 )( x пЂ­ 3 )( x пЂ« 3 )
Martin-Gay, Developmental Mathematics
43
В§ 14.5
Solving Equations
Containing Rational
Expressions
Solving Equations
First note that an equation contains an equal sign
and an expression does not.
To solve EQUATIONS containing rational
expressions, clear the fractions by multiplying
both sides of the equation by the LCD of all the
fractions.
Then solve as in previous sections.
Note: this works for equations only, not
simplifying expressions.
Martin-Gay, Developmental Mathematics
45
Solving Equations
Example
Solve the following rational equation.
5
3x
пЂ«1 пЂЅ
Check in the original equation.
7
6
пѓ¦ 5
пѓ¶ пѓ¦7пѓ¶
6 xпѓ§
пЂ« 1пѓ· пЂЅ пѓ§ пѓ·6 x
пѓЁ 3x
пѓё пѓЁ6пѓё
10 пЂ« 6 x пЂЅ 7 x
10 пЂЅ x
5
7
пЂ«1 пЂЅ
3 пѓ— 10
6
5
пЂ«1 пЂЅ
7
30
6
1
7
пЂ«1 пЂЅ
6
Martin-Gay, Developmental Mathematics
6
true
46
Solving Equations
Example
Solve the following rational equation.
1
пЂ­
1
пЂЅ
2x x пЂ«1
1 пѓ¶
пѓ¦ 1
6 x пЂЁ x пЂ« 1пЂ©пѓ§
пЂ­
пѓ·пЂЅ
пѓЁ 2x x пЂ«1пѓё
3 пЂЁ x пЂ« 1пЂ© пЂ­ 6 x пЂЅ
1
3x пЂ« 3x
пѓ¦
пѓ¶
1
пѓ§пѓ§
пѓ·пѓ· 6 x пЂЁ x пЂ« 1пЂ©
пѓЁ 3 x ( x пЂ« 1) пѓё
2
2
3x пЂ« 3 пЂ­ 6x пЂЅ 2
3 пЂ­ 3x пЂЅ 2
пЂ­ 3x пЂЅ пЂ­1
xпЂЅ 1
3
Martin-Gay, Developmental Mathematics
Continued.
47
Solving Equations
Example Continued
Substitute the value for x into the original
equation, to check the solution.
1
2 1
пЂ­
1
пЂЁ 3пЂ© пЂЁ 3пЂ©пЂ«1
1
3
пЂ­
3
2
4
6
3
4
пЂ­
4
пЂЅ
1
пЂЁ 3пЂ©
3 1
пЂЅ
пЂЅ
2
пЂЁ 3пЂ©
пЂ«3 1
1
1 пЂ«1
3
3
4
true
So the solution is
xпЂЅ 1
3
Martin-Gay, Developmental Mathematics
48
Solving Equations
Example
Solve the following rational equation.
xпЂ«2
x пЂ« 7 x пЂ« 10
2
пЂЅ
1
3x пЂ« 6
пЂ­
1
xпЂ«5
xпЂ«2
1 пѓ¶
пѓ¦
пѓ¶ пѓ¦ 1
3 пЂЁ x пЂ« 2 пЂ©пЂЁ x пЂ« 5 пЂ©пѓ§ 2
пЂ­
пѓ·пЂЅпѓ§
пѓ· 3 пЂЁ x пЂ« 2 пЂ©пЂЁ x пЂ« 5 пЂ©
пѓЁ x пЂ« 7 x пЂ« 10 пѓё пѓЁ 3 x пЂ« 6 x пЂ« 5 пѓё
3пЂЁ x пЂ« 2 пЂ© пЂЅ пЂЁ x пЂ« 5 пЂ© пЂ­ 3пЂЁ x пЂ« 2 пЂ©
3x пЂ« 6 пЂЅ x пЂ« 5 пЂ­ 3x пЂ­ 6
3x пЂ­ x пЂ« 3x пЂЅ 5 пЂ­ 6 пЂ­ 6
5 x пЂЅ пЂ­7
xпЂЅпЂ­7
5
Martin-Gay, Developmental Mathematics
Continued.
49
Solving Equations
Example Continued
Substitute the value for x into the original
equation, to check the solution.
пЂ­7
пЂЁпЂ­ 75 пЂ©
2
5
пЂ«2
пЂЁ
пЂ«7 пЂ­7
5
пЂ© пЂ« 10
пЂЅ
1
1
пЂ­
3 пЂ­7
пЂ«6
пЂ­7
пЂ«5
5
5
пЂЁ
3
49
25
5
пЂ­ 49
5
пЂ« 10
5
18
пЂЅ
пЂЅ
пЂ©
1
пЂ­ 21
5
9
пЂ­
5
5
18
So the solution is
пЂ«6
пЂЁ
пЂ­
пЂ©
1
18
5
true
x пЂЅ пЂ­7
Martin-Gay, Developmental Mathematics
5
50
Solving Equations
Example
Solve the following rational equation.
1
x пЂ­1
пЂЅ
2
x пЂ«1
пѓ¦ 1 пѓ¶ пѓ¦ 2 пѓ¶
пЂЁ x пЂ­ 1пЂ©пЂЁ x пЂ« 1пЂ©пѓ§
пѓ·пЂЅпѓ§
пѓ· пЂЁ x пЂ­ 1 пЂ©пЂЁ x пЂ« 1 пЂ©
пѓЁ x пЂ­1пѓё пѓЁ x пЂ«1пѓё
x пЂ« 1 пЂЅ 2 пЂЁ x пЂ­ 1пЂ©
x пЂ«1 пЂЅ 2x пЂ­ 2
3пЂЅ x
Continued.
Martin-Gay, Developmental Mathematics
51
Solving Equations
Example Continued
Substitute the value for x into the original
equation, to check the solution.
1
2
пЂЅ
3 пЂ­1 3 пЂ«1
1
2
пЂЅ
2
4
true
So the solution is x = 3.
Martin-Gay, Developmental Mathematics
52
Solving Equations
Example
Solve the following rational equation.
12
9пЂ­a
2
пЂ«
3
3пЂ« a
пЂЅ
2
3пЂ­a
3 пѓ¶ пѓ¦ 2 пѓ¶
пѓ¦ 12
пЂЁ3 пЂ­ a пЂ©пЂЁ3 пЂ« a пЂ©пѓ§
пЂ«
пѓ·пЂЅпѓ§
пѓ· пЂЁ3 пЂ­ a пЂ©пЂЁ3 пЂ« a пЂ©
2
3пЂ« a пѓё пѓЁ3пЂ­a пѓё
пѓЁ9пЂ­a
12 пЂ« 3 пЂЁ3 пЂ­ a пЂ© пЂЅ 2 пЂЁ3 пЂ« a пЂ©
12 пЂ« 9 пЂ­ 3 a пЂЅ 6 пЂ« 2 a
21 пЂ­ 3 a пЂЅ 6 пЂ« 2 a
15 пЂЅ 5 a
3пЂЅ a
Martin-Gay, Developmental Mathematics
Continued.
53
Solving Equations
Example Continued
Substitute the value for x into the original
equation, to check the solution.
12
3
2
пЂ«
пЂЅ
2
3пЂ«3 3пЂ­3
9пЂ­3
12
0
пЂ«
3
5
пЂЅ
2
0
Since substituting the suggested value of a into the
equation produced undefined expressions, the
solution is пѓ†.
Martin-Gay, Developmental Mathematics
54
Solving Equations with Multiple Variables
Solving an Equation With Multiple Variables for
One of the Variables
1) Multiply to clear fractions.
2) Use distributive property to remove
grouping symbols.
3) Combine like terms to simplify each side.
4) Get all terms containing the specified
variable on the same side of the equation,
other terms on the opposite side.
5) Isolate the specified variable.
Martin-Gay, Developmental Mathematics
55
Solving Equations with Multiple Variables
Example
Solve the following equation for R1
1
пЂЅ
1
пЂ«
1
R
R1 R 2
1 пѓ¶
пѓ¦ 1 пѓ¶ пѓ¦ 1
пѓ· RR 1 R 2
RR 1 R 2 пѓ§ пѓ· пЂЅ пѓ§пѓ§
пЂ«
пѓ·
пѓЁ R пѓё пѓЁ R1 R 2 пѓё
R1 R 2 пЂЅ RR 2 пЂ« RR 1
R1 R 2 пЂ­ RR 1 пЂЅ RR 2
R 1 пЂЁ R 2 пЂ­ R пЂ© пЂЅ RR 2
R1 пЂЅ
RR 2
R2 пЂ­ R
Martin-Gay, Developmental Mathematics
56
В§ 14.6
Problem Solving with
Rational Equations
Ratios and Rates
Ratio is the quotient of two numbers or two
quantities.
The ratio of the numbers a and b can also be
a
written as a:b, or .
b
The units associated with the ratio are important.
The units should match.
If the units do not match, it is called a rate, rather
than a ratio.
Martin-Gay, Developmental Mathematics
58
Proportions
Proportion is two ratios (or rates) that are
equal to each other.
a
b
пЂЅ
c
d
We can rewrite the proportion by multiplying
by the LCD, bd.
This simplifies the proportion to ad = bc.
This is commonly referred to as the cross product.
Martin-Gay, Developmental Mathematics
59
Solving Proportions
Example
Solve the proportion for x.
x пЂ«1
xпЂ«2
пЂЅ
5
3
3 пЂЁ x пЂ« 1пЂ© пЂЅ 5 пЂЁ x пЂ« 2 пЂ©
3 x пЂ« 3 пЂЅ 5 x пЂ« 10
пЂ­ 2x пЂЅ 7
x пЂЅ пЂ­7
2
Martin-Gay, Developmental Mathematics
Continued.
60
Solving Proportions
Example Continued
Substitute the value for x into the original
equation, to check the solution.
пЂ­7
пЂ­7
пЂ«1
2
2
пЂ«2
пЂЅ
5
3
пЂ­5
2 пЂЅ 5
3
пЂ­3
2
true
So the solution is
x пЂЅ пЂ­7
Martin-Gay, Developmental Mathematics
2
61
Solving Proportions
Example
If a 170-pound person weighs approximately 65 pounds
on Mars, how much does a 9000-pound satellite weigh?
170 - pound person on Earth
9000 - pound satellite
on Earth
пЂЅ
65 - pound person on Mars
x - pound satellite
on Mars
170 x пЂЅ 9000 пѓ— 65 пЂЅ 585 , 000
x пЂЅ 585000 / 170 п‚» 3441 pounds
Martin-Gay, Developmental Mathematics
62
Solving Proportions
Example
Given the following prices charged for
various sizes of picante sauce, find the best
buy.
• 10 ounces for $0.99
• 16 ounces for $1.69
• 30 ounces for $3.29
Continued.
Martin-Gay, Developmental Mathematics
63
Solving Proportions
Example Continued
Size
Price
Unit Price
10 ounces
$0.99
$0.99/10 = $0.099
16 ounces
$1.69
$1.69/16 = $0.105625
30 ounces
$3.29
$3.29/30 п‚» $0.10967
The 10 ounce size has the lower unit price, so it is the
best buy.
Martin-Gay, Developmental Mathematics
64
Similar Triangles
In similar triangles, the measures of
corresponding angles are equal, and
corresponding sides are in proportion.
Given information about two similar triangles,
you can often set up a proportion that will
allow you to solve for the missing lengths of
sides.
Martin-Gay, Developmental Mathematics
65
Similar Triangles
Example
Given the following triangles, find the unknown
length y.
12 m
10 m
5m
y
Continued
Martin-Gay, Developmental Mathematics
66
Similar Triangles
Example
1.) Understand
Read and reread the problem. We look for the corresponding
sides in the 2 triangles. Then set up a proportion that relates
the unknown side, as well.
2.) Translate
By setting up a proportion relating lengths of corresponding
sides of the two triangles, we get
12
5
пЂЅ
10
y
Martin-Gay, Developmental Mathematics
Continued
67
Similar Triangles
Example continued
3.) Solve
12
пЂЅ
5
10
y
12 y пЂЅ 5 пѓ— 10 пЂЅ 50
y пЂЅ 50
12
пЂЅ 25
6
meters
Continued
Martin-Gay, Developmental Mathematics
68
Similar Triangles
Example continued
4.) Interpret
Check: We substitute the value we found from
the proportion calculation back into the problem.
12
5
пЂЅ
10
25
6
пЂЅ
60
25
true
State: The missing length of the triangle is 25 6 meters
Martin-Gay, Developmental Mathematics
69
Finding an Unknown Number
Example
The quotient of a number and 9 times its reciprocal
is 1. Find the number.
1.) Understand
Read and reread the problem. If we let
n = the number, then
1
n
= the reciprocal of the number
Continued
Martin-Gay, Developmental Mathematics
70
Finding an Unknown Number
Example continued
2.) Translate
The quotient of
a number
n
is
1
and 9 times its reciprocal
п‚ё
пѓ¦1пѓ¶
9пѓ§ пѓ·
пѓЁnпѓё
=
1
Continued
Martin-Gay, Developmental Mathematics
71
Finding an Unknown Number
Example continued
3.) Solve
пѓ¦ 1пѓ¶
n п‚ё пѓ§9пѓ— пѓ· пЂЅ 1
пѓЁ nпѓё
пѓ¦9пѓ¶
nп‚ёпѓ§ пѓ· пЂЅ1
пѓЁnпѓё
nпѓ—
n
пЂЅ1
9
n пЂЅ9
2
n пЂЅ 3, пЂ­ 3
Martin-Gay, Developmental Mathematics
Continued
72
Finding an Unknown Number
Example continued
4.) Interpret
Check: We substitute the values we found from the
equation back into the problem. Note that nothing in
the problem indicates that we are restricted to positive
values.
1 пѓ¶
пѓ¦
пЂ­ 3 п‚ё пѓ§9 пѓ—
пѓ· пЂЅ1
пѓЁ пЂ­3пѓё
пѓ¦ 1пѓ¶
3 п‚ё пѓ§9пѓ— пѓ· пЂЅ 1
пѓЁ 3пѓё
3п‚ё3пЂЅ1
true
пЂ­ 3 п‚ё пЂ­3 пЂЅ 1
true
State: The missing number is 3 or –3.
Martin-Gay, Developmental Mathematics
73
Solving a Work Problem
Example
An experienced roofer can roof a house in 26 hours. A
beginner needs 39 hours to do the same job. How long will it
take if the two roofers work together?
1.) Understand
Read and reread the problem. By using the times for each
roofer to complete the job alone, we can figure out their
corresponding work rates in portion of the job done per hour.
Time in hrs
Experienced roofer 26
Beginner roofer
39
Together
t
Portion job/hr
1/26
/39
1/t
Martin-Gay, Developmental Mathematics
Continued
74
Solving a Work Problem
Example continued
2.) Translate
Since the rate of the two roofers working together
would be equal to the sum of the rates of the two
roofers working independently,
1
26
пЂ«
1
39
пЂЅ
1
t
Continued
Martin-Gay, Developmental Mathematics
75
Solving a Work Problem
Example continued
3.) Solve
1
26
пЂ«
1
39
пЂЅ
1
t
1 пѓ¶ пѓ¦1пѓ¶
пѓ¦ 1
78 t пѓ§
пЂ«
пѓ· пЂЅ пѓ§ пѓ· 78 t
пѓЁ 26 39 пѓё пѓЁ t пѓё
3 t пЂ« 2 t пЂЅ 78
5 t пЂЅ 78
t пЂЅ 78 / 5 or 15.6 hours
Continued
Martin-Gay, Developmental Mathematics
76
Solving a Work Problem
Example continued
4.) Interpret
Check: We substitute the value we found from the
proportion calculation back into the problem.
1
пЂ«
26
3
78
1
пЂЅ
39
пЂ«
2
78
1
78
5
пЂЅ
5
78
true
State: The roofers would take 15.6 hours working
together to finish the job.
Martin-Gay, Developmental Mathematics
77
Solving a Rate Problem
Example
The speed of Lazy River’s current is 5 mph. A boat travels 20
miles downstream in the same time as traveling 10 miles
upstream. Find the speed of the boat in still water.
1.) Understand
Read and reread the problem. By using the formula d=rt, we
can rewrite the formula to find that t = d/r.
We note that the rate of the boat downstream would be the rate
in still water + the water current and the rate of the boat
upstream would be the rate in still water – the water current.
Distance rate time = d/r
Down 20
r + 5 20/(r + 5)
Up
10
r–5
10/(r – 5)
Continued
Martin-Gay, Developmental Mathematics
78
Solving a Rate Problem
Example continued
2.) Translate
Since the problem states that the time to travel
downstairs was the same as the time to travel
upstairs, we get the equation
20
rпЂ«5
пЂЅ
10
rпЂ­5
Continued
Martin-Gay, Developmental Mathematics
79
Solving a Rate Problem
Example continued
3.) Solve
20
rпЂ«5
пЂЅ
10
rпЂ­5
пѓ¦ 20 пѓ¶ пѓ¦ 10 пѓ¶
пЂЁr пЂ« 5 пЂ©пЂЁr пЂ­ 5 пЂ©пѓ§
пѓ·пЂЅпѓ§
пѓ· пЂЁ r пЂ« 5 пЂ©пЂЁ r пЂ­ 5 пЂ©
пѓЁrпЂ«5пѓё пѓЁrпЂ­5пѓё
20 пЂЁ r пЂ­ 5 пЂ© пЂЅ 10 пЂЁ r пЂ« 5 пЂ©
20 r пЂ­ 100 пЂЅ 10 r пЂ« 50
10 r пЂЅ 150
r пЂЅ 15 mph
Martin-Gay, Developmental Mathematics
Continued
80
Solving a Rate Problem
Example continued
4.) Interpret
Check: We substitute the value we found from the
proportion calculation back into the problem.
20
15 пЂ« 5
20
20
пЂЅ
пЂЅ
10
15 пЂ­ 5
10
10
true
State: The speed of the boat in still water is 15 mph.
Martin-Gay, Developmental Mathematics
81
В§ 14.7
Simplifying Complex
Fractions
Complex Rational Fractions
Complex rational expressions (complex
fraction) are rational expressions whose
numerator, denominator, or both contain one or
more rational expressions.
There are two methods that can be used when
simplifying complex fractions.
Martin-Gay, Developmental Mathematics
83
Simplifying Complex Fractions
Simplifying a Complex Fraction (Method 1)
1) Simplify the numerator and denominator of
the complex fraction so that each is a single
fraction.
2) Multiply the numerator of the complex
fraction by the reciprocal of the denominator
of the complex fraction.
3) Simplify, if possible.
Martin-Gay, Developmental Mathematics
84
Simplifying Complex Fractions
Example
x
2
x
2
пЂ«2
пЂ­2
x
пЂЅ 2
x
2
пЂ«
пЂ­
4
xпЂ«4
2 пЂЅ
4
2 пЂЅ
xпЂ­4
2
xпЂ«4
2
xпЂ«4
пѓ—
пЂЅ
2
xпЂ­4
xпЂ­4
2
Martin-Gay, Developmental Mathematics
85
Simplifying Complex Fractions
Method 2 for simplifying a complex fraction
1) Find the LCD of all the fractions in both the
numerator and the denominator.
2) Multiply both the numerator and the
denominator by the LCD.
3) Simplify, if possible.
Martin-Gay, Developmental Mathematics
86
Simplifying Complex Fractions
Example
1
пЂ«
2
6пЂ« 4y
3 6y
пѓ— 2 пЂЅ
2
5 6y
6
y
пЂ­
5
y
пЂ­
y 6
2
y
1
2
2
Martin-Gay, Developmental Mathematics
87
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