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2.5 Least Common Multiple and Comparing Fractions 1 Multiples A multiple of a natural number is a product of that number and any natural number. In other words, take the number and multiply it by 1, 2, 3, … Example: Multiples of 6: 6, 12, 18, 24, 30, 36, … (6 1=6, 6 2=12, 6 3=18, 6 4=24, 6 5=30, etc.) The least common multiple, or LCM, of two natural numbers is the smallest number that is a multiple of both numbers. Least Common Multiple Example: Observe the list of multiples for the following numbers. 6: 6, 12, 18, 24, 30, 36, 42, 48, 54 … 8: 8, 16, 24, 32, 40, 48, 56, 64, … The LCM is the smallest number that appears in both lists. 24 is the smallest number that appears in both lists, therefore it is the LCM. There are other common multiples such as 48, but 24 is the lowest common multiple. Next Slide

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2.5 Least Common Multiple and Comparing Fractions 2 Finding a LCM is part of the process to adding and subtracting fractions with unlike denominators so it is very important. Writing down a list of multiples is an ok method as long as the numbers are relatively small. What if the the numbers are 84 and 90. Then, making a list of multiples is not too cool. Fortunately, we have a better method. Procedure: Finding the LCM using the Prime Factorization Method Step 1. Find the prime factorization of each number. Step 2. Create a product of factors, using each factor the greatest number of times that it occurs in any one factorization. Prime Factorization Method for Finding the LCM (Least Common Multiple) Example 1: Find the LCM of 15 and 24. Step 1. Find the prime factorization of each number. 15 35 24 46 22 2 3 Step 2. Write LCM= below the prime factorization of each number. Consider each prime number. Write it the greatest number of times it occurs in any one factorization. 2: it occurs three times in the 24. 3: it occurs once. 5: it occurs once. Now multiply. This is the LCM. LCM = 120 Answer: 630 Your Turn Problem #1 Find the LCM of 35 and 90

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2.5 Least Common Multiple and Comparing Fractions 3 Example 2: Find the LCM of 12, 30, and 70. Step 1. Find the prime factorization of each number. Step 2. Write LCM= below the prime factorization of each number. Consider each prime number. Write it the greatest number of times it occurs in any one factorization. 2: it occurs twice. 3: it occurs once. 5: it occurs once. Now multiply. This is the LCM. LCM = 420 Answer: 1200 Your Turn Problem #2 Find the LCM of 24, 150 and 240 12 26 23 30 56 23 70 710 25 7: it occurs once.

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2.5 Least Common Multiple and Comparing Fractions 4 Equivalent Fractions Recall the Multiplicative Identity (also called the multiplication property of 1): a 1 = a. When we multiply a number by 1, we get the same number. Example: 5 1 = 5 Also, any fraction with the same numerator and denominator is equal to 1.,, If we take a fraction, such as ½, and multiply it by 1, we get the same number. As long as we multiply the numerator and denominator by the same number, we will obtain an equivalent fraction. Next Slide

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2.5 Least Common Multiple and Comparing Fractions 5 Example 3. Write the following fractions as an equivalent fraction with the given denominator. Solution: We want an equivalent fraction with a denominator of 40. What number multiplied by 8 will give 40? (or divide 40 by 8) Answer: 5 Therefore, if we multiply the denominator by 5 to get 40, multiply the numerator by 5 to obtain the desired equivalent fraction. We want an equivalent fraction with a denominator of 27. What number multiplied by 9 will give 27? (or divide 40 by 8) Answer: 3 Therefore, if we multiply the denominator by 3 to get 27, multiply the numerator by 3 to obtain the desired equivalent fraction. Your Turn Problem #3 Write the following fractions as an equivalent fraction with the given denominator.

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2.5 Least Common Multiple and Comparing Fractions 6 Example 4: Write an inequality symbol between the following numbers. Your Turn Problem #4 Write an inequality symbol between the following numbers. Lowest Common Denominator (LCD) The procedure for finding a LCD is exactly the same as finding a LCM. We only use the numbers from the denominators. Step 1. Find the prime factorization of each number. Step 2. Create a product of factors, using each factor the greatest number of times that it occurs in any one factorization. Next Slide

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2.5 Least Common Multiple and Comparing Fractions 7 12 26 23 15 35Example 5: Find the LCD for the following fractions: a) Find the LCM of 12 and 15.b) Find the LCM of 24 and 30. 24 46 22 2 3 30 56 23 Your Turn Problem #5 Find the LCD for the following fractions: Answer: 1260

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2.5 Least Common Multiple and Comparing Fractions 8 Example 6: Convert each fraction to an equivalent fraction using LCD for the denominator. Then determine the direction for an inequality symbol between the two fractions. Solution: From example 3, we determined the LCD is 60. Now write each as equivalent fractions with a denominator =60. Multiply the 1 st fraction by 5 on the numerator and denominator. Multiply the 2 nd fraction by 4 on the numerator and denominator. Now that the denominators are identical, we can determine the larger fraction. The inequality symbol opens towards the second fraction. Your Turn Problem #6 Convert each fraction to an equivalent fraction using LCD for the denominator. Then determine the direction for an inequality symbol between the two fractions.

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2.5 Least Common Multiple and Comparing Fractions 9 Ordering Fractions If the denominators of two or more fractions are identical, then we can arrange the fractions from by simply looking at the numerators. Example: Arrange from smallest to largest. Solution: Find the LCD. Write each as an equivalent fraction using the LCD. Now that the denominators are identical, we can determine the larger fraction. The 3 rd fraction is the smallest and the middle fraction is the largest. If the denominators are not the same, write each as an equivalent fraction using the LCD. Example 7: Arrange from smallest to largest: Your Turn Problem #7 Arrange from smallest to largest. The End. B.R. 5-30-08

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