Download (Adding or Subtracting Like Fractions).

Professor Busken – 4.4 Adding & Subtracting Like Fractions, and Least Common Denominators (LCDs)
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Learning Objectives:
☞ Add or Subtract Like Fractions
☞ Add or Subtract Given Fractional Replacement Values
☞ Solve Problems by Adding & Subtracting Like Fractions
☞ Find the Least Common Denominator (LCD) of a List of Fractions
☞ Writing Equivalent Fractions
Definition 1. Fractions with the same denominator are called like fractions.
Theorem 1 (Adding or Subtracting Like Fractions). Suppose a, b and c numbers and b is not
0, then
a b a−b
a b a+b
+ =
and
− =
c c
c
c c
c
✎ Add or subtract the given fractions.
1.
−
99
−9
99
−9 + 99
90
9 · 10
9 10
9
9
9
+
=
+
=
=
=
=
·
=
·1 =
100 100 100 100
100
100 10 · 10 10 10 10
10
2.
23 1
−
11 11
2.
3.
11
7
+
30 30
3.
4.
3
7
−
10 10
4.
5.
−
13x 13x
−
28
28
5.
6.
1
9x
+
11 11
6.
7.
7x 15x
−
16 16
7.
Professor Busken – 4.4 Adding & Subtracting Like Fractions, and Least Common Denominators (LCDs)
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Recall that the Natural numbers are 1, 2, 3, 4, . . ..
Definition 2. A multiple of a number is the product of that number and any Natural number.
For example, the multiples of 3 are
3 · 1 = 3, 3 · 2 = 6, 3 · 3 = 9, 3 · 4 = 12, and so on.
☞ Least Common Denominators (LCDs)
To add or subtract fractions that have unlike, or different, denominators, we write the fractions
as equivalent fractions with a common denominator. The smallest common denominator is
called the least common denominator (LCD).
Definition 3. The least common denominator (LCD) of a list of fractions is the smallest
positive number divisible by all the denominators in the list. (The least common denominator
is also the least common multiple (LCM) of the denominators.)
1
17
For example, the LCD of and
is 24 since 24 is the smallest positive number divisible by
6
8
both 6 and 8.
Finding the LCD: Method 1
Step 1: Write the multiples of the largest denominator (starting with the number itself) until
a multiple common to all denominators in the list is found.
Step 2: The multiple found in Step 1 is the LCD.
Example: Find the LCD of
1
9
and .
8
16
Solution: Write multiples of 16 until one that is also a multiple of 8 is found.
16 · 1 = 16 a multiple of 8
The LCD is 16.
Example: Find the LCD of
17
29
and .
30
25
Solution: Write multiples of 30 until one that is also a multiple of 25 is found.
The LCD is 150.
30 · 1 = 30
30 · 2 = 60
30 · 3 = 90
30 · 4 = 120
30 · 5 = 150
not a multiple of 25
not a multiple of 25
not a multiple of 25
not a multiple of 25
is a multiple of 25
Professor Busken – 4.4 Adding & Subtracting Like Fractions, and Least Common Denominators (LCDs)
✎ Find the LCD of each list of fractions.
8.
8
17
and
9
12
8.
9.
7
13
and
12
20
9.
10.
1
12 ,
−
17
7
and
24
36
10.
11.
4
25 ,
−
7
17
and
6
15
11.
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Professor Busken – 4.4 Adding & Subtracting Like Fractions, and Least Common Denominators (LCDs)
4
17
29
and
must be a multiple of both 30 and 25 . Examine the prime
Recall that the LCD of
30
25
factorization of both numbers:
25 = 5 · 5
30 = 2 · 3 · 5
The LCD is 5 · 5 · 2 · 3 = 150. We circle (and then multiply) the greatest number of factors
for each different prime number in the set of prime factorizations to find the LCD. Remember,
150 is the smallest number that both 25 and 30 divide into evenly (remainder of 0).
Finding the LCD: Method 2
Step 1. Write the prime factorization of each denominator.
Step 2. For each different prime factor, circle the greatest number of times that factor occurs
in any one factorization.
Step 3. The LCD is the product of the circled factors.
✎ Find the LCD of each list of fractions.
12.
13 1
108 , 40
12.
13.
1 1 1
18 , 24 , 42
13.
Professor Busken – 4.4 Adding & Subtracting Like Fractions, and Least Common Denominators (LCDs)
✎ Find the LCD of each list of fractions.
14.
3 1 13
4 , 14 , 20
14.
15.
−
1 1
36 , 24
15.
16.
11 7 17
20 , 24 , 45
16.
17.
2 3 4
9 , x2 , x4
17.
5
Professor Busken – 4.4 Adding & Subtracting Like Fractions, and Least Common Denominators (LCDs)
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✎ Writing Equivalent Fractions
Recall that equivalent fractions are fractions that represent the same part of a whole. We will
need to know how to write equivalent fractions to add and subtract fractions with different, or
unlike denominators.
To write an equivalent fraction,
a a c a·c
= · =
b b c b·c
where a, b and c are nonzero numbers.
Example: Write
7
as an equivalent fraction with a denominator of 24.
8
7
=
8 24
Solution: Since 24 = 8 · 3, we multiply
3
7
by 1 in the form of .
8
3
7 7
7 3 7·3
21
= ·1 = · =
=
8 8
8 3 8·3
24
21
X To check that this result is correct, write
in simplest form, or compute the cross products
24
7
21
between and
are check if they are equal.
8
24
✎ Write an equivalent fraction with the given denominator.
18.
3=
19.
3
=
4 48
19.
20.
7x
=
9
27
20.
21.
7
=
11 132x
21.
5
18.