1. Adding and Subtracting Fractions
Chapter 3
Adding and Subtracting Fractions
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2. 3.1 Adding and Subtracting Like Fractions
Objectives
1. Define like and unlike fractions.
2. Add like fractions
3. Subtract like fractions.
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3. Copyright © 2010 Pearson Education, Inc. All rights reserved.
Fractions with the same denominators are like fractions. Fractions with different denominators are unlike fractions.
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4. Copyright © 2010 Pearson Education, Inc. All rights reserved.
Parallel
Example 1
Identifying Like and Unlike Fractions
a. b.
Like fractions
All denominators are the same.
Unlike fractions
Denominators are different.
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Parallel
Example 2
Adding Like Fractions
Add and write the sum in lowest terms. a.
Add numerators
Same denominator
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7. Copyright © 2010 Pearson Education, Inc. All rights reserved.
Parallel
Example 2
Adding Like Fractions
Add and write the sum in lowest terms. b.
Add numerators
Step 1 12 16
Step 2
Sum of numerators
Same denominator
Step 3
In lowest terms
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Parallel
Example 3
Subtracting Like Fractions
Find the difference and simplify the answer. a.
Subtract numerators
Step 1
Step 2
Step 3
In lowest terms
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Parallel
Example 3
Subtracting Like Fractions
Find the difference and simplify the answer. b.
Subtract numerators
To simplify the answer, write as a mixed number.
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11. 3.2 Least Common Multiples
Objectives
1. Find the least common multiple.
2. Find the least common multiple using multiples
of the largest number.
3. Find the least common multiple using prime factorization.
4. Find the least common multiple using an alternative method.
5. Write a fraction with an indicated denominator.
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Parallel
Example 1
Finding the Least Common Multiple
Find the least common multiple of 4 and 6. List the multiples of 4. List the multiples of 6
4 • 1
4 • 2
4 • 3
4 • 4
4 • 5
4 • 6
4 • 7
4 • 8 4, 8,
12,
16,
20,
24,
28,
32,…
6 • 1
6 • 2
6 • 3
6 • 4
6 • 5
6 • 6
6 • 7
6 • 8 6,
12,
18,
24,
30,
36,
42,
48,…
The smallest number found in both lists is 12, so 12 is the least common multiple of 4 and 6.
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Parallel
Example 2
Finding the Least Common Multiple
Use multiples of the larger number to find the least common multiple of 6 and 10. List the first few multiples of , 20, 30, 40, 50, 60, 70, … Now, we check each multiple of 10 to see if it is divisible by 6.
The first multiple of 10 that is divisible by 6 is 30.
The least common multiple of 6 and 10 is 30.
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Parallel
Example 3
Applying Prime Factorization Knowledge
Use prime factorization to find the least common multiple of 12 and 45.
Find the prime factorization of each number.
12 = 2  2  3
45 = 3  3  5
LCM = 2  2  3  3  5 = 180
Check to see that 180 is divisible by 12 (yes) and (45) yes. The smallest whole number divisible by both 12 and 45 is 180.
Factors of 12
Factors of 45
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Parallel
Example 4
Using Prime Factorization
Find the least common multiple of 20, 25, and 30.
Find the prime factorization of each number.
20 = 2  2  5
25 = 5  5
30 = 2  3  5
LCM = 2  2  3  5  5 = 300
Check to see that 300 is divisible by 20 (yes), 25 (yes) and (30) yes. The smallest whole number divisible by 20, 25, and 30 is 300.
Factors of 20
Factors of 25
Factors of 30
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Parallel
Example 5
Finding the Least Common Multiple
Find the least common multiple for 7, 15, and 21.
Find the prime factorization of each number.
7 = 7
15 = 3  5
21 = 3  7
LCM = 3  5  7 = 105
The least common multiple of 7, 15, and 21 is 105.
Factors of 15
Factors of 7
Factors of 21
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Parallel
Example 5
Finding the Least Common Multiple
Find the least common multiple for 18, 20, and 30.
Find the prime factorization of each number.
18 = 2  3  3
20 = 2  2  5
30 = 2  3  5
LCM = 2  2  3  3  5 = 180
The least common multiple of 18, 20, and 30 is 180.
Factors of 18
Factors of 20
Factors of 30
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19. Some people like the following alternative method for finding the least common multiple for larger numbers.
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20. Alternative Method for Finding the Least Common Multiple
Parallel
Example 6
Alternative Method for Finding the Least Common Multiple
Find the least common multiple of 18 and 21. Start by trying to divide 18 and 21 by the first prime number in the list of primes: 2, 3, 5, 7, 11. 2
Multiply these numbers to get the LCM. 3 3 7
All quotients are 1.
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Parallel
Example 6
Alternative Method for Finding the Least Common Multiple
Find the least common multiple of 12, 30, and 50. 2 2 3 5 5
All quotients are 1.
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Parallel
Example 7
Writing a Fraction with a New Denominator
Write the fraction with a denominator of 18.
To find the new numerator, first divide 18 by 6.
Multiply both numerator and denominator of the fraction by 3.
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Parallel
Example 8
Writing Fractions with a New Denominator
Rewrite each fraction with the indicated denominator. a. b.
Divide 32 by 8, getting 4. Now multiply both the numerator and denominator by 4.
Multiply numerator and denominator by 4.
Multiply numerator and denominator by 5.
45 ÷ 9 = 5
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24. 3.3 Adding and Subtracting Unlike Fractions
Objectives
1. Add unlike fractions.
2. Add unlike fractions vertically.
3. Subtract unlike fractions.
4. Subtract unlike fractions vertically.
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To add unlike fractions, we must first change them to like fractions (fractions with the same denominator.)
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Parallel
Example 1
Adding Unlike Fractions
Add
The least common multiple of 6 and 12 is 12.
Write the fractions as like fractions with a denominator of 12. This is the least common denominator (LCD).
Step 1
Step 2
Step 3
Step 3 is not needed because the fraction is in lowest terms.
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Parallel
Example 2
Adding Fractions
Add the fractions using the three steps. Simplify all answers.
The least common multiple of 4 and 8 is 8.
Rewritten as like fractions
Step 1
Step 2
Step 3
Step 3 is not needed because the fraction is in lowest terms.
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Parallel
Example 3
Add unlike fractions vertically
Add the following fractions vertically. a.
Rewritten as like fractions
Add the numerators.
Denominator is 24, the LCD.
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29. Copyright © 2010 Pearson Education, Inc. All rights reserved.
Parallel
Example 3
Add unlike fractions vertically
Add the following fractions vertically. b.
Rewritten as like fractions
Add the numerators.
Denominator is 24, the LCD.
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Parallel
Example 4
Subtracting Unlike Fractions
Subtract. Simplify all answers.
Rewritten as like fractions
Step 1
Subtract numerators.
Step 2
Step 3
Step 3 is not needed because the fraction is in lowest terms.
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Parallel
Example 4
Subtracting Unlike Fractions
Subtract. Simplify all answers.
Rewritten as like fractions
Step 1
Subtract numerators.
Step 2
Step 3
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32. Copyright © 2010 Pearson Education, Inc. All rights reserved.
Parallel
Example 5
Vertical Subtraction of Fractions
Subtract the following fractions vertically. a.
Rewritten as like fractions
Subtract the numerators.
Denominator is 15, the LCD.
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33. Copyright © 2010 Pearson Education, Inc. All rights reserved.
Parallel
Example 5
Vertical Subtraction of Fractions
Subtract the following fractions vertically. b.
Rewritten as like fractions
Subtract the numerators.
Denominator is 15, the LCD.
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34. 3.4 Adding and Subtracting Mixed Numbers
Objectives
1. Estimate an answer, then add or subtract
mixed numbers.
2. Estimate an answer, then subtract mixed
numbers by regrouping.
3. Add or subtract mixed numbers using an
alternate method.
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Add or subtract mixed numbers by adding or subtracting the fraction parts and then the whole number parts. It is a good idea to estimate the answer first.
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Parallel
Example 1
Adding and Subtracting Mixed Numbers
First estimate the answer. Then add or subtract to find the exact answer. a.
Estimate
Exact
Rounds to
Rounds to
Sum of whole numbers.
Sum of fractions.
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Parallel
Example 1
Adding and Subtracting Mixed Numbers
First estimate the answer. Then add or subtract to find the exact answer. b.
30 is the least common denominator.
Estimate
Exact
Rounds to
Rounds to
Subtract whole numbers.
Subtract fractions.
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38. Copyright © 2010 Pearson Education, Inc. All rights reserved.
Parallel
Example 2
Simplify and Regroup When Adding Mixed Numbers
First estimate, and then add
Estimate
Exact
Rounds to
Becomes
Rounds to
Sum of whole numbers.
Sum of fractions.
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Parallel
Example 3
Regroup When Subtracting Mixed Numbers
a. First estimate, and then subtract.
Estimate
Exact
Rounds to
Rounds to
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41. Copyright © 2010 Pearson Education, Inc. All rights reserved.
Parallel
Example 3
Regroup When Subtracting Mixed Numbers
b. First estimate, and then subtract.
Estimate
Exact
Rounds to
Rounds to
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Parallel
Example 3
Regroup When Subtracting Mixed Numbers
First estimate, and then subtract.
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An alternate method for adding or subtracting mixed numbers is to first change the mixed numbers to improper fractions. Then, rewrite the unlike fractions as like fractions. Finally, add or subtract the numerators and write the answer in lowest terms.
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Parallel
Example 4
Adding or Subtracting Mixed Numbers Using an Alternate Method
Add.
6 is the least common denominator
Change to improper fractions.
Mixed number
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Parallel
Example 4
continued
Adding or Subtracting Mixed Numbers Using an Alternate Method
Subtract.
15 is the least common denominator
Change to improper fractions.
Mixed number
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46. 3.5 Order Relations and the Order of Operations
Objectives
1. Identify the greater of two fractions.
2. Use exponents with fractions.
3. Use the order of operations with fractions.
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There are times when we want to compare the size of two numbers. For example, we might want to know which is the greater amount, the larger size, or the longer distance.
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49. Using Less-Than and Greater-Than Symbols
Parallel
Example 1
Using Less-Than and Greater-Than Symbols
Rewrite the following using < and > symbols. a. b.
The number farther to the left on the number line is less.
The number farther to the right on the number line is greater.
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51. Identifying the Greater Fraction
Parallel
Example 2
Identifying the Greater Fraction
Determine which fraction in each pair is greater. a. b.
Rewrite both fractions with a common denominator of 40
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Exponents were used to write repeated multiplication in an earlier chapter.
exponent
exponent
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Parallel
Example 3
Using Exponents with Fractions
Simplify. a. b.
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Parallel
Example 3
Using Exponents with Fractions
Simplify. c. 4 4 2
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Parallel
Example 4
Using the Order of Operations with Fractions
Simplify by using the order of operations. a.
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Parallel
Example 4
Using the Order of Operations with Fractions
Simplify by using the order of operations. b.
Simplify the exponent.
Multiply.
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