Adding and Subtracting Fractions. 1. The denominator of a fraction names the fraction. It tells you how many equal parts something is divided into.


 Dina Manning
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1 Tallahassee Community College Adding and Subtracting Fractions Important Ideas:. The denominator of a fraction names the fraction. It tells you how many equal parts something is divided into.. The numerator of the fraction tells you how many of the equal parts you have.. Fractions cannot be added or subtracted unless they have the same denominators.. If the denominators are not the same, the fractions must be written in an equivalent form with the same denominators.. The Least Common Denominator (LCD) is the Least Common Multiple (LCM) of the denominators. To add or subtract fractions. If the denominators are different find the Least Common Denominator (LCD).. Build equivalent fractions.. Add or subtract the numerators.. Reduce the answer to lowest terms if necessary. We will now work through several examples following these steps.
2 Example Add 8 The denominators are different so we must find the Least Common Denominator (LCD). The LCD will be the Least Common Multiple of 8 and. See worksheet # if you do not remember how to find the LCM of two or more numbers The LCM of 8 and is so the LCD of these fractions is. This means that is the smallest number that both 8 and will divide into. We must now write two new fractions which have a denominator of and are equivalent in value to and. 8 Divide each denominator into the LCD and then multiply the original numerator by the number of times the ()() ()() original denominator divides into the LCD. 8 and This gives us a new addition problem equal to the original problem. 0 Now we add the numerators and place the sum over the LCD: 0 This fraction is already in lowest terms so it does not have to be reduced. It could be changed to a mixed number but in algebra we generally do not do that unless the fraction is the answer to a word problem. By lowest terms we mean that there is no number besides that will divide evenly into both the numerator and the denominator. In other words the Greatest Common Factor (GCF) of the numerator and denominator is.
3 Example : Add 6 0 The denominators are different so we must find the LCD. The LCD will be the LCM of 6 and 0. The LCM of 6 and 0 is so the LCD of these fractions is. This means that is the smallest number that both 6 and 0 will divide into. We will now write two fractions with the same denominator that are equal to the original fractions and 6 0. Divide by each of the original denominators and multiply the original numerators by the number of times each ()() ()() 6 and of the original denominators divides into the LCD. 0 That will give us a new addition problem equivalent to the original problem. When we add the numerators in this problem and place them over the common denominator we get a fraction that is not in lowest terms. We reduce the fraction by finding the prime factorization of both the numerator and the denominator and then canceling common factors. Note that both and were divided by. This means that is the Greatest Common Factor (GCF) of and.
4 Example : Add Note that denominators are the same. We already have a common denominator so we can add the numerators and place them over the common denominator. 6 6 Is in lowest terms? Let s find the prime factorization of the numerator and the denominator and see if there are any common factors which can be cancelled. 6 Note that the product of the cancelled factors is the GCF of 6 and. and is the GCF of 6 and or the biggest number which will divide evenly into 6 and. Example : Add 0 The denominators are different so we must find the LCD. The LCD will be the LCM of, and 0. The LCM of, and 0 is so the LCD is. This means that is the smallest number that, and 0 will all divide into. We will now rewrite the three fractions in equivalent forms with a denominator of. Divide each of the original denominators into the LCD and then multiply each of the original ()() ()() ()() numerators by the number of times the original, and denominator divides into the LCD. 0 This gives us a new addition problem equal to the original problem.
5 Is the fraction in lowest terms? Let s find the prime factorization of the numerator and the denominator and see if there are any common factors which can be cancelled. Note that and do not have any prime factors in common. This means that the biggest number that would divide evenly into and is the number. The GCF of and is. Example : Subtract 6 The denominators are different so we must find the LCD. The LCD will be the LCM the denominators and 6. The LCM of and 6 is so the LCD of these fractions will be. We will now rewrite the two fractions that have a denominator of and are equivalent to the two original fractions. Divide each of the original denominators into the LCD and then multiply each of the original ()() ()() numerators by the number of times each of the and 6 original denominator divides into the LCD. We now have a new subtraction problem which is equal to the original problem. We will subtract the numerators and place them over the common denominator. Is the fraction in lowest terms? We will find the prime factorization of the and and see if there are any common factors which can be cancelled.
6 0 Note that and were divided by. This means that the GCF of and is. Example 6: Subtract 0 8 The denominators are different so we must find the LCD. The LCD will be the LCM the denominators 0 and 8. The LCM of 0 and 8 is so the LCD will be. We will now write two fractions that have a denominator of and are equal to the original fractions. Divide the original denominators into the LCD and then multiply the original ()() ()() numerators by the number of times the 0 and 8 original denominator divides into the LCD. We now have a new subtraction problem that is equal to the original problem. 6 6 Is the fraction in lowest terms? We could find the prime factorization of the numerator and the denominator and see if there are any prime factors which could be cancelled. However, is a prime number and does not divide evenly into so we can safely say that the fraction is in lowest terms. 6
7 Practice Problems
8 Answers to Practice Problems or 0. or
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