Mathematics
Lowest Common Denominator
The lowest common denominator is the smallest multiple that two or more denominators can be divided evenly into. In mathematics, it is used when adding or subtracting fractions to find a common denominator. By finding the lowest common denominator, fractions can be added or subtracted without changing their values.
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10 Key excerpts on "Lowest Common Denominator"
- Robert Brechner, Geroge Bergeman(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
To add or subtract fractions, the denominators must be the same. If they are not, we must find a common multiple, or common denominator , of all the denominators in the problem. The most efficient common denominator to use is the least common denominator, or LCD. By using the LCD, you avoid raising fractions to terms higher than necessary. D ETERMINING THE L EAST C OMMON D ENOMINATOR (LCD) OF T WO OR M ORE F RACTIONS The least common denominator (LCD) is the smallest number that is a multiple of each of the given denominators. We can often find the LCD by inspection (i.e., mentally) just by using the definition. For example, if we want to find the LCD of 1 4 and 1 6 , we think (or write out, if we wish): Multiples of 4 are 4, 8, 12, 16, 20, 24 , and so on Multiples of 6 are 6, 12, 18, 24, 30 , and so on By looking at these two lists, we see that 12 is the smallest multiple of both 4 and 6 . Thus, 12 is the LCD. Sometimes, especially when we have several denominators or the denominators are relatively large numbers, it is easier to use prime numbers to find the LCD. A prime number is a whole number greater than 1 that is evenly divisible only by itself and 1 . Following are prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and so on common denominator A com-mon multiple of all the denominators in an addition or subtraction of frac-tions problem. A common denomina-tor of the fractions 1 4 + 3 5 is 40 . 2-6 least common denominator (LCD) The smallest and, therefore, most efficient common denominator in addition or subtraction of frac-tions. The least common denomina-tor of the fractions 1 4 + 3 5 is 20 . prime number A whole number greater than 1 that is divisible only by itself and 1 . For example, 2, 3, 5, 7 , and 11 are prime numbers. 2 SECTION II iStock.com/ MarsBars Copyright 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
eBook - PDF- Michael Kennamer(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
100 Unit 2: Basic Principles Find least common multiple (LCM). 4, 8, 12 , 16, 20, 24, and so on. 3, 6, 9, 12 , 15, 18, 21, and so on. 6, 12 , 18, 24, 30, and so on. Solution: The least common multiple is 12; therefore, the LCD is 12. Skill Sharpener Find the least common denominator for each of the following sets of fractions. 1. 7 8 1 4 2 3 2. 8 10 12 20 3. 1 2 6 8 3 4 4. 2 3 3 4 4 5 5. 1 2 3 4 6. 4 5 3 10 7. 2 3 5 6 8. 3 8 7 16 9. 3 4 4 5 10. 7 8 7 16 LCM Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Chapter 6: Fractions 101 Objective 6.21 Add fractions with uncommon denominators. To add fractions with uncommon denominators , first find the least common denominator. Next, convert each fraction to a fraction using the least common denominator. Then add the numerators and reduce as necessary. The problem: Add the following fractions with uncommon denominators: 3 4 1 1 3 5 _____ Find the LCM and the LCD. 4, 8, 12, 16, 20, and so on. 3, 6, 9, 12, 15, and so on. The LCM is 12, so the LCD will be 12. Now change the denominators in both fractions to the LCD. 3 4 1 1 3 5 _____ 12 1 12 5 _____ Multiply each numerator by the number of times the original denominators go into the LCD. 3 4 1 1 3 3 3 3 4 12 1 12 Now multiply the original numerators by the shaded factors to determine new numerators. 9 12 1 4 12 Add. 9 12 1 4 12 5 13 12 Solution: 3 4 1 1 3 5 13 12 or 1 1 12 Skill Sharpener Add the following fractions with uncommon denominators and reduce to lowest terms. 1. 1 4 1 1 16 5 2. 3 9 1 2 3 5 3 3 4 5 12 3 3 3 5 9 1 3 4 5 4 Copyright 2017 Cengage Learning.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
After that, we simply add or subtract as we did in our first three examples. We provide a more detailed outline of these steps in the following box. Step 1: Factor each denominator completely. Step 2: Find the least common denominator (LCD) of the fractions. The LCD will be a product containing each factor the most number of times it appears in any single denominator. Step 3: Rewrite each fraction as an equivalent fraction with the common denominator by multiplying both numerator and denominator by the factors from the LCD that are missing in the denominator. Step 4: Add or subtract the numerators and keep the common denominator. Step 5: Reduce to lowest terms, if possible. HOW TO Add or Subtract Fractions with Different Denominators Example 4 reviews this step-by-step procedure in adding two rational numbers. Add: 3 __ 14 + 7 __ 30 . SOLUTION Step 1: Factor each denominator. We factor both denominators into prime factors. Factor 14: 14 = 2 ⋅ 7 Factor 30: 30 = 2 ⋅ 3 ⋅ 5 Step 2: Find the LCD. Because the LCD must be divisible by 14, it must have factors of 2 and 7. It must also be divisible by 30 and, therefore, have factors of 2, 3, and 5. We do not need to repeat the 2 that appears in both the factors of 14 and those of 30. Therefore, LCD = 2 ⋅ 3 ⋅ 5 ⋅ 7 = 210 The least common denominator for a set of denominators is the smallest expression that is divisible by each of the denominators. least common denominator DEFINITION EXAMPLE 4 436 CHAPTER 6 Rational Expressions and Rational Functions Step 3: Change to equivalent fractions. Because we want each fraction to have a denominator of 210 and at the same time keep its original value, we multiply each by 1 in the appropriate form. Change 3 __ 14 to a fraction with denominator 210. 3 __ 14 ⋅ 15 __ 15 = 45 ___ 210 14 contains a factor of 2 and 7, but is missing the factors 3 and 5.
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
In order to add fractions that do not have common denominators, such as 3 8 1 1 4 1 7 16 , it is necessary to change to equivalent fractions with common denominators. Multiplying the denominators does give a common denominator, but it could be a very large number. We often find it easier to determine the Lowest Common Denominator. The Lowest Common Denominator is the smallest denominator that is evenly divisible by each of the denominators of the fractions being added. Stated in another way, the Lowest Common Denominator is the smallest denominator into which each denominator can be divided without leaving a remainder. c PROCEDURE To find the Lowest Common Denominator ● ● Determine the smallest number into which all denominators can be divided without leaving a remainder. ● ● Use this number as a common denominator. Example 1 Find the Lowest Common Denominator of 3 8 , 1 4 , and 7 16 . The smallest number into which 8, 4, and 16 can be divided without leaving a remainder is 16. Write 16 as the Lowest Common Denominator. UNIT 2 Addition of Common Fractions and Mixed Numbers OBJECTIVES After studying this unit you should be able to ● ● Determine Lowest Common Denominators. ● ● Express fractions as equivalent fractions having Lowest Common Denominators. ● ● Add fractions and mixed numbers. A machinist must be able to add fractions and mixed numbers in order to determine the length of stock required for a job, the distances between various parts of a machined piece, and the depth of holes and cutouts in a workpiece. UNIT 2 ADDITION OF COMMON FRACTIONS AND MIXED NUMBERS 9 Example 2 Find the Lowest Common Denominator of 3 4 , 1 3 , 7 8 , and 5 12 . The smallest number into which 4, 3, 8, and 12 can be divided is 24. The Lowest Common Denominator is 24. Note: In this example, denominators such as 48, 72, and 96 are common denominators because 4, 3, 8, and 12 divide evenly into these numbers, but they are not the Lowest Common Denominators.
eBook - ePub- Dave Fox(Author)
- 2012(Publication Date)
- Routledge(Publisher)
Suppose that you have been given one piece of a cake that has been divided equally into four parts and another piece from a second cake that has been cut into 16 parts. How much cake do you have?It is not possible to add together two fractions which have different denominators. The denominator is not a value but an indicator of how many parts a whole has been divided into. To carry out this addition it is necessary to find a common denominator:If the smallest denominator will divide exactly into the largest denominator then the larger of the two denominators is the common denominator since this is the smallest number that both will divide into. Four will divide into sixteen, four times. This tells us that is four times larger than ; alternatively it can be suggested that to change 4 into 16 it was multiplied by four. To keep its true value a fraction must have both its denominator and its numerator increased or decreased in the same way and by the same factor:The common denominator of and is 16. It is not the only common denominator. Other examples could be 32, 48, 64. Sixteen is the smallest number that both 4 and 16 will divide into and is called the Lowest Common Denominator – LCD.Find the LCDs of the following (answers at the end of the chapter):In all the cases the smaller denominator will divide directly and equally into the larger. Consider the following fractions, and . The smallest number that both 6 and 9 will divide equally into is 18. How have we achieved this? Basically from remembered experience, but without experience this is not possible. Since 6 will not divide equally into 9 an alternative is to multiply the two denominators together:6 × 9 = 54Both numbers will now divide exactly into 54 and our addition can continue but some extra work may be required later on. This will be to get the fraction in its lowest terms, but this will be dealt with later. Returning to the problem of adding two fractions, it is necessary to ensure the denominators are the same.ExamplesEach fraction has to have the same denominator. Since 4 will divide into 8 the common denominator can be 8. We now have to change a quarter into an equivalent value with 8 as its denominator. To change 4 into 8 it has been multiplied by 2. The numerator 1 must be multiplied by 2.
eBook - PDF- Alan Tussy, R. Gustafson(Authors)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
In the numerator, combine like terms: . In an attempt to simplify, we can factor as . However, the numerator and denominator have no common factors. The result is in simplest form. 2 x ( x 7) 2 x 2 14 x x 2 x 2 2 x 2 2 x 2 14 x ( x 7)( x 8) ( x 2 14 x ) x 2 14 x Self Check 4 Subtract: a. b. 3 y 2 ( y 3)( y 3) 3 y 2 y ( y 3)( y 3) x 2 3 x x 1 5 x 1 x 1 Problems 33 and 39 Now Try Find the Least Common Denominator. We will now discuss two skills that are needed for adding and subtracting rational expressions that have unlike denominators. To begin, let’s consider To add these expressions, we must express them as equivalent expressions with a common denominator. The least common denominator (LCD) is usually the easiest one to use. 11 8 x 7 18 x 2 To find the LCD of a set of rational expressions: 1. Factor each denominator completely. 2. The LCD is a product that uses each different factor obtained in step 1 the greatest number of times it appears in any one factorization. Finding the LCD Don’t make this common error by forgetting to write within parentheses. x 2 10 x 4 x 9 x 3 4 x 9 Copyright 201 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 7.3 Adding and Subtracting with Like Denominators; Least Common Denominators 535 EXAMPLE 5 Find the LCD of each pair of rational expressions: a. and b. and Strategy We will begin by factoring completely the denominator of each rational expression. Then we will form a product using each factor the greatest number of times it appears in any one factorization.
- James Deitz, James Southam(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 10 Part 1 Fundamental Applications a. Subtract 5 6 from 7 8 . b. Subtract 2 9 10 from 6 5 6 . The least common denominator is 24. The least common denominator is 30. 7 8 5 21 24 6 5 6 5 6 25 30 5 5 55 30 2 5 6 5 2 20 24 2 2 9 10 5 2 2 27 30 5 2 2 27 30 1 24 3 28 30 5 3 14 15 COMPLETE ASSIGNMENT 1.1. ✔ C O N C E P T C H E C K 1 . 2 In fractions, multiplication and division do not require common denominators. This means that multiplication and division are simpler than addition and subtraction. Recall that any mixed number can be changed to an improper fraction. Also, a whole number can be writ-ten as an improper fraction by writing the whole number in the numerator with a denomi-nator of 1. For example, the whole number 5 can be written as the improper fraction 5 1 . Multiplying Fractions and Mixed Numbers to Multiply Fractions and Mixed Numbers 1. Change any mixed (or whole) numbers to improper fractions. 2. Multiply all the numerators to get the numerator of the product. 3. Multiply all the denominators to get the denominator of the product. 4. Change the product to a proper fraction or mixed number in lowest terms. S T E P S EXAMPLE L EXAMPLE M STEP 1 STEPS 2 & 3 STEP 4 STEPS 2 & 3 STEP 4 1 2 3 3 4 5 5 5 3 3 4 5 5 5 3 4 3 3 5 5 20 15 5 1 5 15 5 1 1 3 2 3 3 4 5 3 5 6 5 2 3 4 3 5 3 3 5 3 6 5 40 90 5 4 9 Note: The word of often means multiply when it is used with fractions. For example, you know that “ 1 2 of 6 bottles” is 3 bottles. And 1 2 of 6 5 1 2 3 6 1 5 6 2 5 3 . For this reason, multiplication may even be the most important arithmetic operation with fractions.
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Procedure for Determining Lowest Common Denominators When it is difficult to determine the Lowest Common Denominator, a procedure using prime factors is used. A factor is a number being multiplied. A prime number is a whole number other than 0 and 1 that is divisible only by itself and 1. Some examples of prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and 23. A prime factor is a factor that is a prime number. To determine the Lowest Common Denominator, first factor each of the denomina-tors into prime factors. List each prime factor as many times as it appears in any one denominator. Multiply all of the prime factors listed. EXAMPLES 1. Find the Lowest Common Denominator of 5 6 , 1 5 , and 3 16 . Factor each of the denominators into prime factors. The prime factors 3 and 5 are each used as factors only once. List these factors. The prime factor 2 is used once for denominator 6 and four times for denominator 16. List 2 as a factor four times. 6 5 2 3 3 5 5 5 s prime d 16 5 2 3 2 3 2 3 2 3 3 5 3 2 3 2 3 2 3 2 5 240 Ans Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 60 Section I FUNDAMENTALS OF GENERAL MATHEMATICS If the same prime factor is used in 2 or more denominators, it is listed only for the denominator for which it is used the greatest number of times. Multiply all prime factors listed to obtain the Lowest Common Denominator. 2. Find the Lowest Common Denominator of 9 10 , 3 8 , 4 9 , and 7 15 . Factor each of the denominators into prime factors. Multiply all prime factors listed to obtain the Lowest Common Denominator.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
Rational What is the LCM of and The multiples of are: and the multiples of are: Therefore, their LCM is the smallest number on both lists. What is the LCM of and The multiples of are: and the multiples of are: So their LCM is Note that, by definition, if a and b are two rationals (or integers), there are integers m and n such that LCM( a , b ) = m × a = n × b . This implies that m and n are coprime; otherwise they could be divided by their common divisor, giving a common multiple less that the least common multiple, which is absurd. The above examples illustrate this fact. ________________________ WORLD TECHNOLOGIES ________________________ Applications When adding, subtracting, or comparing vulgar fractions, it is useful to find the least common multiple of the denominators, often called the Lowest Common Denominator, because each of the fractions can be expressed as a fraction with this denominator. For instance, where the denominator 42 was used because it is the least common multiple of 21 and 6. Formulas Fundamental theorem of arithmetic According to the fundamental theorem of arithmetic a positive integer is the product of prime numbers, and, except for their order, this representation is unique: where the exponents n 2 , n 3 , ... are non-negative integers; for example, 84 = 2 2 3 1 5 0 7 1 11 0 13 0 ... Given two integers and their least common multiple and greatest common divisor are given by the formulas and Since this gives ________________________ WORLD TECHNOLOGIES ________________________ In fact, any rational number can be written uniquely as the product of primes if negative exponents are allowed. When this is done, the above formulas remain valid. Using the same examples as above: Lattice-theoretic The positive integers may be partially ordered by divisibility: if a divides b (i.e. if b is an integer multiple of a ) write a ≤ b (or equivalently, b ≥ a ).
eBook - PDF- Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis(Authors)
- 2020(Publication Date)
- Openstax(Publisher)
complex fraction equivalent fractions fraction least common denominator (LCD) mixed number proper and improper fractions reciprocal simplified fraction CHAPTER 4 REVIEW KEY TERMS A complex fraction is a fraction in which the numerator or the denominator contains a fraction. Equivalent fractions are two or more fractions that have the same value. A fraction is written a b . in a fraction, a is the numerator and b is the denominator. A fraction represents parts of a whole. The denominator b is the number of equal parts the whole has been divided into, and the numerator a indicates how many parts are included. The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators. A mixed number consists of a whole number a and a fraction b c where c ≠ 0 . It is written as a b c , where c ≠ 0 . The fraction a b is proper if a < b and improper if a > b . The reciprocal of the fraction a b is b a where a ≠ 0 and b ≠ 0 . A fraction is considered simplified if there are no common factors in the numerator and denominator. KEY CONCEPTS 4.1 Visualize Fractions • Property of One ◦ Any number, except zero, divided by itself is one. a a = 1 , where a ≠ 0 . • Mixed Numbers ◦ A mixed number consists of a whole number a and a fraction b c where c ≠ 0 . ◦ It is written as follows: a b c c ≠ 0 • Proper and Improper Fractions ◦ The fraction ab is a proper fraction if a < b and an improper fraction if a ≥ b . • Convert an improper fraction to a mixed number. Divide the denominator into the numerator. Identify the quotient, remainder, and divisor. Write the mixed number as quotient remainder divisor . • Convert a mixed number to an improper fraction. Multiply the whole number by the denominator. Add the numerator to the product found in Step 1. Write the final sum over the original denominator. • Equivalent Fractions Property ◦ If a, b, and c are numbers where b ≠ 0 , c ≠ 0 , then a b = a ⋅ c b ⋅ c .
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