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Chapter 2 Fractions

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**The Least Common Denominator and Creating Equivalent Fractions**

2.6 The Least Common Denominator and Creating Equivalent Fractions

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**Least Common Multiple (LCM)**

The multiples of a number are the products of that number and the numbers 1, 2, 3, 4, 5, … The multiples of 3 are 3, 6, 9, 12, 15, … 3 1 3 2 3 3 The least common multiple, or LCM, of two natural numbers is the smallest number that is a multiple of both.

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**Least Common Multiple (LCM)**

Example: Find the LCM of 4 and 6. The multiples of 4 are 4, 8, 12, 16, 20, 24 … The multiples of 6 are 6, 12, 18, 24, 30, 36 … The first number that appears on both lists is the LCM. 12 is the least common multiple of 4 and 6.

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**Least Common Denominator (LCD)**

A least common denominator (LCD) of two or more fractions is the smallest number that can be divided evenly by each of the fractions’ denominators. Since 4 can be divided into 12, the LCD of is 12.

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**Least Common Denominator (LCD)**

Example: Find the LCD for 4 5 = 20 20 is also the smallest number that can be divided by 4 and 5 without a remainder. The LCD of is 20.

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**Finding the Least Common Denominator**

Three-Step Procedure for Finding the LCD Write each denominator as the product of prime factors. List all the prime factors that appear in either product. Form a product of those prime factors, using each factor the greatest number of times it appears in any one denominator. Example: Find the LCD for Product of primes 2 2 3 2 3 5 Prime factors in either product: 2 2 3 5 The LCD is 60.

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**Creating Equivalent Fractions**

Fractions with unlike denominators cannot be added. The LCD is 20. To change the denominators and make them the same, 1) find the LCD and 2) build up the addends into equivalent fractions that have the LCD as the denominator. The building fraction property

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**Building Fraction Property**

For whole numbers a, b, and c where b 0, c 0, Example: Build to an equivalent fraction with a LCD of 20.

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