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Fractions and Factors<br />In order to work with fractions efficiently, it is important to understand some concepts about factors, prime numbers and composite numbers<br />We will use these concepts to develop equivalent fractions when we need to reduce a fraction to lowest terms or change fractions to one with a common denominator.<br />
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Factors<br />Factors are the numbers that are multiplied together to get a product.<br />We can write 12 as the product of two factors in any of the following ways:<br /> 3 • 4 = 12 2 • 6 = 12 1 • 12 = 12<br />If asked for all of the factors of 12, the answer would be:<br />1, 2, 3, 4, 6, and 12<br />
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Finding All Factors<br />Try using the Rainbow Method to find all factors of a given number. For example, to find all of the factors of 36:<br />Start with 1 which is a factor of every number. Since 1 X 36 = 36, we place 1 at one end and 36 at the other. <br />36<br />1<br />
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Finding All Factors<br />Try using the Rainbow Method to find all factors of a given number. For example, to find all of the factors of 36:<br />Since 2 is a factor of 36, and 2 X 18 = 36, we place the factors 2 and 18 inside the first set of factors.<br /> 36<br />1 2<br />
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Finding All Factors<br />Try using the Rainbow Method to find all factors of a given number. For example, to find all of the factors of 36:<br />Since 3 is a factor of 36, and 3 X 12 = 36, we place the factors 3 and 12 inside the next set of factors<br />1 2 3<br />12 18 36<br />
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Finding All Factors<br />Try using the Rainbow Method to find all factors of a given number. For example, to find all of the factors of 36:<br />Since 4 is a factor of 36, and 4 X 9 = 36, we place the factors 4 and 9 inside the other sets of factors.<br />1 2 3 4 <br />9 12 18 36<br />
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Finding All Factors<br />Try using the Rainbow Method to find all factors of a given number. For example, to find all of the factors of 36:<br />Now we try 5. But that is not a factor of 36, so we go on to 6. 6 X 6 = 36, so we include the factor 6 in our rainbow. We have already captured all of the factors greater than 6, so we are done.<br />1 2 3 4 <br />6<br />9 12 18 36<br />Solution: All of the factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36<br />
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Prime and Composite Numbers<br />A Prime number can only be divided by 1 and itself.<br /> The first 10 prime numbers are:<br /> 2, 3, 5, 7, 11, 13, 17, 19, 23, 29<br /> <br />A Composite number is composed of more than one prime factor and can be divided by other factors, as well as 1 and itself.<br /> The first 10 composite numbers are:<br /> 4, 6, 8, 9, 10, 12, 14, 15, 16, 18<br /> <br />NOTE: The number 1is considered neitherprime nor composite.<br />
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Prime Factorization<br />It is sometimes necessary to be able to break a number down into its prime factors. This process is called Prime Factorization. We can use a factor tree to determine the prime factorization of a number.<br />To determine the prime factorization of 12, we first choose any set of factors for 12, such as 3 X 4<br /> 12<br /> / <br /> 4<br /> / <br /> 2 2<br />3 is already a prime number, but 4 is not,<br />So we break it down into its factors<br />Solution: the prime factorization of 12 is 3 •2 • 2<br />
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Prime Factorization<br />Another example: Let’s define 120 as a product of its prime factors (find the prime factorization of 120)<br /><ul><li> First, we find any two factors of 120.
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If not prime, we factor again and circle</li></ul> the primes<br /><ul><li>Continue until we only have primes.</li></ul>Solution: the prime factorization of 120is 2 • 2 •2 •3 • 5<br />
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Reduce a Fraction to Lowest Terms<br />Write fraction in lowest terms using the Prime Factoring method.<br />First, write the numerator and denominator<br />as the product of their primes.<br />Divide out any common factors. <br />Since 2 and 3 have no more common factors<br />The fraction is in lowest terms.<br />
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Finding an Equivalent Fraction<br />3<br />Let’s say we need to rewrite — with a denominator of 15.<br />5<br />Remember that if we multiply numerator<br />and denominator by the same number, we get <br />an equivalent fraction.<br />Since 5 •3 = 15, we need to multiply the<br />numerator by 3 as well.<br />= 1 -- So when we multiply both numerator and denominator by 3<br /> we are multiplying the original fraction by 1.<br />
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Lowest Common Denominator<br />To find the LCD (Lowest Common Denominator) for two fractions, determine the prime factorization for each denominator <br />The LCD will include each different factor, and those factors will be used the maximum number of times it appears in any factorization<br /> To find the LCD for and , first list the prime factorization for each denominator.<br />3<br />—<br />20<br />5<br />—<br />24<br />20 = 2 • 2 • 5 and 24 = 2 • 2 • 2 • 3<br />5 appears once in the factorizations, 3 also appears once, but 2 appears at most three times, so the LCD will be 2 • 2 • 2 • 3 • 5 or 120.<br />
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Change to Common Denominator<br />Once we have found the LCD for two fractions we can change them to equivalent fractions with a common denominator.<br />Since 20 • 6 = 120 we multiply the numerator by 6 as well<br />Since 24 • 5 = 120, we multiply the numerator by 5 as well<br />
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