Skill Check Lesson Presentation Lesson Quiz

Skill Check Write the first five multiples of the number. 1. 7 7, 14, 21, 28, 35 2. 9 9, 18, 27, 36, 45 11, 22, 33, 44, 55 3. 11 4. 15 15, 30, 45, 60, 75

Agriculture Crop rotation is a system in which farmers vary the crops they plant in their fields each year. Suppose a farmer grows alfalfa in a certain field every 6 years. In another field, the farmer grows alfalfa every 10 years. This year, the farmer is growing alfalfa in both fields. In how many years will the farmer grow alfalfa in both fields again?

A multiple of a whole number is the product of the number and any nonzero whole number. A multiple that is shared by two or more numbers is a common multiple. Some of the common multiples of 8 and 12 are shown in blue below. Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, . . . Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, . . . The least of the common multiples of two or more numbers is the least common multiple (LCM). The LCM of 8 and 12 is 24.

1 Finding the Least Common Multiple EXAMPLE 1 Finding the Least Common Multiple For the crop rotation system previously described, the number of years until the farmer grows alfalfa in both fields again is given by the LCM of 6 and 10. You can use one of two methods to find the LCM. Method 1 List the multiples of each number. Identify the least number that is on both lists. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 The LCM of 6 and 10 is 30. Multiples of 10: 10, 20, 30, 40, 50, 60

1 Finding the Least Common Multiple EXAMPLE 1 Finding the Least Common Multiple For the crop rotation system previously described, the number of years until the farmer grows alfalfa in both fields again is given by the LCM of 6 and 10. You can use one of two methods to find the LCM. Method 2 Find the common factors of the numbers. 6 = 2 • 3 The common factor is 2. 10 = 2 • 5 Multiply all of the factors, using each common factor only once. LCM = 2 • 3 • 5 = 30 ANSWER The farmer will grow alfalfa in both fields again in 30 years.

2 Finding the Least Common Multiple of Monomials EXAMPLE 2 Finding the Least Common Multiple of Monomials Find the least common multiple of 9xy 2 and 15x 2y. 9xy 2 = 3 • 3 • x • y • y Common factors are circled and used only once in the LCM. 15x 2y = 3 • 5 • x • x • y LCM = 3 • x • y • 3 • 5 • x • y = 45x 2y 2 ANSWER The least common multiple of 9xy 2 and 15x 2y is 45x 2y 2.

Least Common Denominator The least common denominator (LCD) of two or more fractions is the least common multiple of the denominators. You can use the LCD to compare and order fractions.

3 Comparing Fractions Using the LCD EXAMPLE 3 Comparing Fractions Using the LCD Winter Sports Last year, a winter resort had 144,000 visitors, including 45,000 snowboarders. This year, the resort had 160,000 visitors, including 56,000 snowboarders. In which year was the fraction of snowboarders greater? SOLUTION 1 Write the fractions and simplify. Number of snowboarders Total number of visitors 45,000 144,000 5 16 Last year: = = Number of snowboarders Total number of visitors 56,000 160,000 7 20 This year: = = Find the LCD of and . 5 16 7 20 2 The LCM of 16 and 20 is 80. So, the LCD of the fractions is 80.

3 . Comparing Fractions Using the LCD EXAMPLE 3 Comparing Fractions Using the LCD Winter Sports Last year, a winter resort had 144,000 visitors, including 45,000 snowboarders. This year, the resort had 160,000 visitors, including 56,000 snowboarders. In which year was the fraction of snowboarders greater? SOLUTION 3 Write equivalent fractions using the LCD. 5 16 5 • 5 16 • 5 25 80 7 20 7 • 4 20 • 4 28 80 Last year: = = This year: = = 25 80 28 < , so 5 16 7 20 . 4 Compare the numerators: ANSWER The fraction of snowboarders was greater this year.

4 Ordering Fractions and Mixed Numbers Order the numbers 3 , and EXAMPLE 4 Ordering Fractions and Mixed Numbers Order the numbers 3 , and from least to greatest. 4 15 33 10 19 6 , 1 Write the mixed number as an improper fraction. 3 4 15 3 • 15 + 4 15 49 15 = = 33 10 19 6 49 15 Find the LCD of , , and . 2 The LCM of 15, 10, and 6 is 30. So, the LCD is 30.

4 Ordering Fractions and Mixed Numbers , Order the numbers 3 , and EXAMPLE 4 Ordering Fractions and Mixed Numbers 4 15 , 33 10 19 6 Order the numbers 3 , and from least to greatest. The LCM of 15, 10, and 6 is 30. So, the LCD is 30. 3 Write equivalent fractions using the LCD. 49 15 49 • 2 15 • 2 98 30 33 10 33 • 3 10 • 3 99 30 19 6 19 • 5 6 • 5 95 30 = = = = = = 4 Compare the numerators. 95 30 98 < 99 19 6 49 15 33 10 and , so . ANSWER From least to greatest, the numbers are , and 19 6 4 15 33 10 , 3 .

Lesson Quiz 1. Find the least common multiple of 8 and 56. 56 2. Find the least common multiple of 12x2y4 and 15xy2. 60x2y4 3. Use the LCD to determine which fraction is greater, or . 4. Challenge Rewrite the variable expressions and with a common denominator.