Download 4.4 Adding and Subtracting Like Fractions, Least Common

Math 100 Class Notes
Evaluating with Fractions
Examples: Given the following replacement values, evaluate for (a) xy and (b) x ¸ y
10.
(a)
= ;
=
11.
(a)
= ;
=
(b)
(b)
4.4 Adding and Subtracting Like Fractions, Least Common Denominator, and
Equivalent Fractions
Fractions that have the same or common
denominator are called like fractions.
Fractions that have different denominators are
called unlike fractions.
Like Fractions
Unlike Fractions
2
4
and
5
5
2
3
and
3
4
5
−3
and
7
7
5
5
and
6
12
Adding and Subtracting Fractions
If a, b, and c, are numbers and b is not 0, then
+
=
+
and also
−
− =
Helpful Hint: To add or subtract fractions with the same denominator, add or subtract their
numerators and write the sum or difference over the common denominator.
21
22
R. Getso
Examples: Add.
1.
+
2.
+
3.
+
Do not forget to write the answer in simplest form. If it is not in simplest form, divide out all
common factors larger than 1.
To add or subtract fractions that have unlike, or different, denominators, we write the fractions as
equivalent fractions with a common denominator.
Least Common Denominator
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.)
Method 1: Finding the LCD of a List of Fractions Using Multiples of the Largest Number
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 the list of fractions using Method 1.
4.
and
Method 2: Finding the LCD of a List of Denominators Using Prime Factorization
Step1: Write the prime factorization of each denominator.
Step 2: For each different prime factor in Step 1, circle the greatest number of times that factor
occurs in any one factorization.
Step 3:
The LCD is the product of the circled factors.
Example: Find the LCD of the list of fractions using Method 2.
5.
and
Math 100 Class Notes
23
Example: Find the LCD.
6.
and
7.
and
8.
and
9.
,
,
Equivalent Fractions
To add or subtract unlike fractions, first write equivalent fractions with the LCD as the
denominator.
To write an equivalent fraction,
∙
= ∙ =
∙
where , and are nonzero numbers.
Examples:
10. Write
5
as an equivalent fraction with a
12
denominator of 24.
11. Write
3
as an equivalent fraction with a
5
denominator of 35.
13. Write
12. Write
7
as an equivalent fraction with a
8
denominator of 64.
2x
as an equivalent fraction with a
3
denominator of 12.
24
R. Getso
4.5 Adding and Subtracting UNLike Fractions
Adding and Subtracting Fractions
Step 1: Find the least common denominator (LCD) of the fractions.
Step 2: Write each fraction as an equivalent fraction whose denominator is the LCD.
Step 3: Add or subtract the like fractions.
Step 4: Write the sum or difference in simplest form.
Examples: Add or subtract as indicated.
1.
3.
+
+
2.
−
4.
+ −
Math 100 Class Notes
Writing Fractions in Order
One important application of the least common denominator is to use the LCD to help order or
compare fractions.
?
Insert < or > to form a true sentence.
The LCD for these fractions is 35.
Write each fraction as an equivalent fraction with a denominator of 35.
=
∙
∙
=
Compare the numerators of the equivalent fractions.
Since 21 > 20, then
>
Examples: Insert <, > or =
=
∙
∙
=
. Thus, > .
5.
6.
6.1 Ratio and Proportion
Writing Ratio as Fractions
A ratio is the quotient of two quantities.
For example, a percent can be thought of as a ratio, since it is the quotient of a number and 100.
53% =
or the ratio of 53 to 100.
The ratio of a number a to a number b is their quotient. Ways of writing ratios are:
a to b,
a : b, and
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