2-6
Adding and Subtracting with Unlike Denominators
Preview
Evaluating Algebraic Expressions
Warm Up
California Standards
Lesson Presentation
2-6
Adding and Subtracting with Unlike Denominators
Warm Up
Evaluating Algebraic Expressions
A proper fraction has an odd number as
the numerator and 10 as the
denominator. This fraction is divided by a
proper fraction with denominator 5. Will
the denominator of the quotient be odd
or even? Explain and show examples.
2-6
Adding and Subtracting with Unlike Denominators
California
Evaluating Algebraic Expressions
Standards
NS1.2 Add, subtract, multiply, and divide
rational numbers (integers, fractions, and
terminating decimals) and take positive rational
numbers to whole-number powers.
NS2.2 Add and subtract fractions by
using factoring to find a common denominator.
2-6
Adding and Subtracting with Unlike Denominators
Evaluating
Algebraic
To add
and subtract
fractions Expressions
with unlike
denominators, first find a common
denominator using one of these methods:
Method 1 Find a common denominator by
multiplying the denominators.
Method 2 Find the least common
denominator (LCD).
2-6
Adding and Subtracting with Unlike Denominators
Additional Example 1A: Adding and Subtracting
Fractions with Unlike Denominators
Evaluating
Add
or subtract.
Method 1:
Algebraic Expressions
2
1
+
7
8
2 8
1 7
=
+
7 8
8 7
7
16
+
=
56
56
Find a common
denominator: 8(7)=56.
Multiply by fractions
equal to 1.
Rewrite with a common
denominator.
7 + 16 = 23
=
56
56
Add numerators.
Keep the denominator.
2-6
Adding and Subtracting with Unlike Denominators
Additional Example 1B: Adding and Subtracting
Fractions with Unlike Denominators
Add
or subtract. Algebraic Expressions
Evaluating
Method 2:
Write as improper
1
5
7
13
1 6 – 18 = –
fractions.
6
8
Multiples of 6: 6, 12, 18, 24, 30 List the multiples of
each denominator
Multiples of 8: 8, 16, 24, 32
and find the LCD.
= 7 4 – 13 3
8 3
6 4
28
39
=
–
24
24
28 – 39
11
=
=
–
24
24
Multiply by fractions equal
to 1.
Rewrite with the LCD.
Subtract numerators.
Keep the denominator.
2-6
Adding and Subtracting with Unlike Denominators
Partner Share! Example 1A
Add or subtract.
Evaluating
Algebraic Expressions
Method
1:
=
=
=
5
1
+
8
3
1 8
5 3
+
8
3
8 3
8
15
+
24 24
Find a common
denominator: 3(8)=24.
Multiply by fractions
equal to 1.
23
8 + 15
=
24
24
Add numerators.
Keep the denominator.
Rewrite with the LCD.
2-6
Adding and Subtracting with Unlike Denominators
Partner Share! Example 1B
Add or subtract.
Method
2:
Evaluating
Algebraic Expressions
Write as an
1
3
3
13
=
26 + 4
6 + 4
improper fraction.
Multiples of 6: 6, 12, 18, 24, 30 List the multiples of
each denominator
Multiples of 4: 4, 8, 12, 16
and find the LCD
=
13
6
=
26
12
=
2
2
+
+
9
12
3
4
3
3
Multiply by fractions
equal to 1.
Rewrite with the LCD.
26 + 9 = 35 = 2 11 Add numerators.
12 Keep the denominator.
12
12
2-6
Adding and Subtracting with Unlike Denominators
Additional Example 2: Using Factoring to Find the
LCD
25
37
Evaluating
Find
+
. WriteAlgebraic
the answer Expressions
in simplest form.
56
84
Factors of 56: 2 ∙ 2 ∙ 2 ∙ 7 Write the prime
factorization of each
Factors of 84: 2 ∙ 2 ∙ 3 ∙ 7 denominator. Circle the
common factors.
List all the prime factors of
the denominators, using
2, 2, 2, 7, 3
the circled factors only
once.
2 ∙ 2 ∙ 2 ∙ 7 ∙ 3 = 168
The LCD is 168.
Multiply.
2-6
Adding and Subtracting with Unlike Denominators
Additional Example 2 Continued
25
37
Evaluating
Expressions
Find
+
. WriteAlgebraic
the answer in
simplest form.
56
84
168 ÷ 56 = 3
168 ÷ 84 = 2
3
2
37
25
=
+
84 2
56 3
Multiply by fractions equal to 1 to
get a common denominator.
= 75 + 74
168 168
Rewrite using the LCD.
= 149
168
Add numerators. Keep the
denominator.
2-6
Adding and Subtracting with Unlike Denominators
Partner Share! Example 2
19
Algebraic
FindEvaluating
+ 9 . Write
the answerExpressions
in simplest form.
32
80
Factors of 32: 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 Write the prime
factorization of each
Factors of 80: 2 ∙ 2 ∙ 2 ∙ 2 ∙ 5 denominator. Circle
the common factors.
2, 2, 2, 2, 2, 5
List all the prime factors of
the denominators, using the
circled factors only once.
2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 5 = 160
The LCD is 160.
Multiply.
2-6
Adding and Subtracting with Unlike Denominators
Partner Share! Example 2 Continued
19
Algebraic
FindEvaluating
+ 9 . Write
the answerExpressions
in simplest form.
32
80
160 ÷ 32 = 5
160 ÷ 80 = 2
5
2 Multiply by fractions that equal to
9
19
=
+
80 2 1 to get a common denominator.
32 5
= 95 + 18
160 160
= 113
160
Rewrite using the LCD.
Add numerators.
Keep the denominator.
2-6
Adding and Subtracting with Unlike Denominators
Additional Example 3: Consumer Application
Two dancers are making necklaces from ribbon
Algebraic
Expressions
for Evaluating
their costumes.
They need pieces
measuring
13 3 inches and 12 7 inches. How much ribbon
8
4
will be left over after the pieces are cut from a
36-inch length?
Subtract both amounts
3
7
36 – 12 8 –13 4
from 36 to find the
amount of ribbon left.
=
36
1
55
– 103
–
4
8
288
103
–
8
8
75
,
or 9 3
=
8
8
=
– 110
8
Write as improper fractions.
The LCD is 8. Simplify.
There will be
3
98
inches left.
2-6
Adding and Subtracting with Unlike Denominators
Partner Share! Example 3
Fred and Jose are building a tree house. They
need to cut a 6 3 foot piece of wood and a 4 5
Evaluating
4 Algebraic Expressions
12
foot piece of wood from a 12 foot board. How
much of the board will be left?
Subtract both amounts
3
5
12 – 6 4 – 4 12
from 12 to find the
amount of board left.
53
12
27
=
– 4 – 12
Write as improper fractions.
1
= 144 – 81
12
12
=
10
,
or 5
12
6
–
53
12
The LCD is 12.
Simplify.
5
There will be 6 foot left.
2-6
Adding and Subtracting with Unlike Denominators
Additional Example 4: Evaluating Expressions with
Rational Numbers
Evaluating
Expressions
4 Algebraic
5
Evaluate t –
t–
5
for t =
4
= 5 – 4
5
5
6
4 6
5 5
–
=
5 6
6 5
25
24
=
–
30
30
1
=
30
6
.
Substitute
5
6
for t.
Multiply by fractions
equal to 1.
Rewrite with a common
denominator: 6(5) = 30.
Simplify.
2-6
Adding and Subtracting with Unlike Denominators
Partner Share! Example 4
Evaluate
5
9
7
– h for h =– 12
.
Evaluating Algebraic Expressions
7
Substitute – 12
for h.
Multiply by fractions
equal to 1.
Rewrite with the LCD.
Simplify.
2-6
Adding and Subtracting with Unlike Denominators
Lesson Review!
Add or subtract.
5
1
1
Algebraic
Expressions
1. Evaluating
+7
14
2
2.
2
83
3.
3
5
–
1
76
1
12
1
2
–2
+
3
–2 15
9
4. Evaluate 1 3 – n for n = 16
.
8
1
62
13
16
inches tall. Judy is 5
5. Robert is 5 feet
feet 3 34 inches tall. How much taller is
3
Robert than Judy? 2 4 in.