Fractions and Factors

12 Key excerpts on "Fractions and Factors"

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  eBook - PDF

  Technical Mathematics with Calculus

  • Paul A. Calter, Michael A. Calter(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  Also list at least four other expressions that are not one of the given seven, and state why each is different from those we have studied. 11–5 Simplifying Fractions From factoring, we move on to our study of fractions and fractional equations. As usual we start with a reminder of some definitions. A fraction has a numerator, a denominator, and a fraction line, or bar. This fraction can also be written on a single line as a/b. A fraction is a way of indicating a quotient of two quantities. Thus the fraction a/b can be read as “a divided by b.” Another way of indicating this same division is a  b. The quotient of two quantities is also spoken of as the ratio of those quantities. Thus a/b is the ratio of a to b. Recall that the bar or fraction line is a symbol of grouping. The quantities in the numerator must be treated as a whole, and the quantities in the denominator must be treated as a whole. ◆◆◆ Example 29: In the fraction the numerator x  4 must be treated as a whole. The 4 in the numerator, for example, cannot be divided by the 2 in the denominator, without also dividing the x by 2. Division by Zero Since division by zero is not permitted, it should be understood in our work with fractions that the denominator cannot be zero. ◆◆◆ Example 30: What values of x are not permitted in the following fraction Solution: Factoring the denominator, we get We see that an x equal to 2 or to will make or equal to zero. This will result in division by zero, so these values are not permitted. ◆◆◆ (x  3) (x  2) 3 3x x 2  x  6  3x (x  2)(x  3) 3x x 2  x  6 x  4 2 fraction line S a b dS 3  ds 3 4 3 pr 2 3  4 3 pr 1 3 . r 2 r 1 numerator denominator Î Î 336 Chapter 11 ◆ Factoring and Fractions Common Fractions and Algebraic Fractions A common fraction is one whose numerator and denominator are both integers. An algebraic fraction is one whose numerator and/or denominator contain literal quantities. ◆◆◆ Example 31: (a) The following are common fractions.

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  Technical Mathematics with Calculus

  • Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  9–1 Simplification of Fractions Parts of a Fraction A fraction has a numerator, a denominator, and a fraction line. fraction line a b numerator denominator Quotient A fraction is a way of indicating a quotient of two quantities. The fraction a/b can be read “a divided by b.” The two ways of writing a fraction, a b and a/b, are equally valid. Ratio We also speak of the quotient of two numbers or quantities as the ratio of those quantities. Thus the ratio of x to y is x y . 9 ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to: • Simplify fractional expressions. • Multiply and divide fractional expressions. • Add and subtract fractional expressions. • Simplify complex fractional expressions. • Solve fractional equations. • Solve word problems using fractional equations. • Manipulate and work with literal equations and formulas. You already know about fractions with numbers. In algebra, however, the numbers are replaced with letters, coefficients, and even entire expressions. Many equations and formulas in science and technology are in the form of a fraction. Since the rules of working with the numerators and denominators of fractions must be applied to entire algebraic expressions, we’ll need to make much use of the factoring techniques of Chapter 8 in order to simplify them. As we work with formulas that include fractions, we must be careful: it’s very easy to make mistakes when we cross multiply. Remember that any- thing you do must be done to each term on both sides of the equation. Also, don’t be intimidated by complex fractions where a numerator or denominator might contain a fraction; use your skills and take it one step at a time. Not all of this material is new to us. Some was covered in Chapter 2, and we solved simple fractional equations in Chapter 3. Fractions and Fractional Equations

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  Contemporary Mathematics for Business & Consumers

  • Robert Brechner, Geroge Bergeman(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  This concept is used quite commonly in business. We may look at sales for 1 2 the year or reduce prices by 1 4 for a sale. A new production machine in your company may be 1 3 4 times faster than the old one, or you might want to cut 5 3 4 yards of fabric from a roll of material. Just like whole numbers, fractions can be added, subtracted, multiplied, divided, and even combined with whole numbers. This chapter introduces you to the various types of fractions and shows you how they are used in the business world. D ISTINGUISHING AMONG THE V ARIOUS T YPES OF F RACTIONS Technically, fractions express the relationship between two numbers set up as division. The numerator is the number on the top of the fraction. It represents the dividend in the division. The denominator is the bottom number of the fraction. It represents the divisor. The numera-tor and the denominator are separated by a horizontal or slanted line, known as the division line . This line means “divided by.” For example, the fraction 2/3 or 2 3 , read as “two-thirds,” means 2 divided by 3 , or 2 ÷ 3 . Numerator Denominator 2 3 Remember, fractions express parts of a whole unit. The unit may be dollars, feet, ounces, or anything else. The denominator describes how many total parts are in the unit. The numer-ator represents how many of the total parts we are describing or referring to. For example, an apple pie (the whole unit) is divided into eight slices (total equal parts, denominator). As a fraction, the whole pie would be represented as 8 8 . If five of the slices were eaten (parts referred to, numerator), what fraction represents the part that was eaten? The answer would be the fraction 5 8 , read “five-eighths.” Because five slices were eaten out of a total of eight, three slices, or 3 8 , of the pie is left. fractions A mathematical way of expressing a part of a whole thing. For example, 1 4 is a fraction expressing one part out of a total of four parts.

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  Teaching Fractions and Ratios for Understanding

  Essential Content Knowledge and Instructional Strategies for Teachers

  • Susan J. Lamon(Author)
  • 2020(Publication Date)
  • Routledge

    (Publisher)

  Although elementary textbooks have traditionally addressed decimals and percentages as separate (and later) topics in the math curriculum, decimals and percentages are really just special kinds of fractions with their own notation. There are several good reasons for arguing that right from the start of fraction instruction, children should be encouraged to express themselves in any or all of these forms. Children see decimals and percentages more in everyday life than they see fractions. As we know, learning in contexts is a complex and lengthy process, so “saving” children from decimals and percentages until much later is a poor excuse that only cuts short their experiences. Most compelling is the fact that research has shown that by third grade, some children have already developed preferences for expressing quantities as decimals or percentages. (In this book you will see some of their work.)

  What Are Fractions?

  Today, the word “fraction” is used in two different ways. First, it is a numeral. Second, in a more abstract sense, it is a number.

  First, fractions are bipartite symbols, a certain form for writing numbers:

  a b

  . This sense of the word fraction refers to a form for writing numbers, a notational system, a symbol, a numeral, two integers written with a bar between them.

  Second, fractions are non-negative rational numbers. Traditionally, because students begin to study fractions long before they are introduced to the integers, a and b are restricted to the set of whole numbers. This is only a subset of the rational numbers.

  The top number of a fraction is called the numerator and the bottom number is called the denominator. The order of the numbers is important. Thus, fractions are ordered pairs of numbers, so

  3 4

  is not the same as the fraction

  4 3

  . Zero may appear in the numerator, but not in the denominator.

  All of these are fractions in the sense that they are written in the form

  a b

  :

  − 3

  4

  ,

  π 2

  ,

  4 2

  ,

  − 12.2

  14.4

  ,

  1 2

  1 4

  However, they are not all fractions in the second sense of the word. Therefore, I will say fraction form when I mean the notation, and fraction when I mean non-negative rational numbers.

  Rational Numbers

  Although many people mistakenly use the terms fractions and rational numbers

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  The Everything Guide to Pre-Algebra

  A Helpful Practice Guide Through the Pre-Algebra Basics - in Plain English!

  • Jane Cassie(Author)
  • 2013(Publication Date)
  • Everything

    (Publisher)

  Chapter 5 Fractions

  By clarifying the rules and process for how you write, alter, and manipulate fractions, you’ll be getting ready to start adding variables into your math problems. The most important part of pre-algebra is clarifying the rules that you already know how to follow, so that you can still follow the rules when the numbers start getting replaced with variables. Fractions won’t just show up by themselves anymore—they’ll be mixed into math problems that test all different kinds of material.

  Introduction to Fractions

  There are three big ways that mathematicians represent non-integer numbers: decimals, fractions, and percentages. Let’s say you have one orange, and you put a knife right in the middle and slice. You take a piece, and your friend takes a piece. You could say, “I have half of the orange.” Or, your friend could say, “I have 50 percent of the orange.” Or, you could say, “I have .5 oranges.” All of these sentences mean the same thing.

  Fractions are pretty much everyone’s least favorite part of math, but they don’t have to be. The key is to take the time to understand what they mean instead of just trying to memorize the rules you are supposed to follow. Once you understand why the rules are what they are, they are much easier to remember.

  Numerator and Denominator

  A fraction is just one number on top of another number. The top number is called the numerator , and the bottom number is called the denominator . These words are specific to fractions—they don’t mean anything except “top number” and “bottom number.”

  The bottom number tells you how many pieces of something make up the whole. For example, when you cut that orange in half, there are two pieces, so the denominator would be 2. The top number tells you how many pieces you have. In the same example, the numerator would be 1, because you only have one piece of the orange.

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  Basic Math & Pre-Algebra All-in-One For Dummies (+ Chapter Quizzes Online)

  • Mark Zegarelli(Author)
  • 2022(Publication Date)
  • For Dummies

    (Publisher)

  (See the nearby sidebar for why this works.) » When the numerator of one fraction and the denominator of the other are divisible by the same number, factor this number out of both. In other words, divide the numerator and denominator by that common factor. (For more on how to find factors, see Chapter 9.) For example, suppose you want to multiply the following two numbers: 5 13 13 20 × . You can make this problem easier by canceling out the number 13, as follows: 5 13 13 20 5 20 13 13 1 1    You can make it even easier by noticing that 20 and 5 are both divisible by 5, so you can also factor out the number 5 before multiplying: CHAPTER 11 Fractions and the Big Four Operations 193   5 13 13 20 1 1 4 1 Now, multiply across to complete the problem: = 1 4 ONE IS THE EASIEST NUMBER With fractions, the relationship between the numbers, not the actual numbers themselves, is most important. Understanding how to multiply and divide fractions can give you a deeper understand- ing of why you can increase or decrease the numbers within a fraction without changing the value of the whole fraction. When you multiply or divide any number by 1, the answer is the same number. This rule also goes for fractions, so 3 8 1 3 8 1 3 8 5 13 1 5 13 1 5 13 67 70 1 67 70 1            and 3 8 and 5 13 and 67 70  67 70 And as I discuss in Chapter 10, when a fraction has the same number in both the numerator and the denominator, its value is 1. In other words, the fractions 2 2 3 3 , , and 4 4 are all equal to 1. Look what happens when you multiply the fraction 3 4 by 2 2 : 3 4 2 2 3 2 4 2 6 8      The net effect is that you’ve increased the terms of the original fraction by 2. But all you’ve done is multiply the fraction by 1, so the value of the fraction hasn’t changed.

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  Basic Mathematics with Early Integers

  • Charles P. McKeague(Author)
  • 2011(Publication Date)
  • XYZ Textbooks

    (Publisher)

  2.1 The Meaning and Properties of Fractions 147 The Number 1 and Fractions There are two situations involving fractions and the number 1 that occur fre- quently in mathematics. The first is when the denominator of a fraction is 1. In this case, if we let a represent any number, then } 1 a } 5 a for any number a The second situation occurs when the numerator and the denominator of a fraction are the same nonzero number: } a a } 5 1 for any nonzero number a EXAMPLE 5 Simplify each expression. a. } 2 1 4 } b. } 2 2 4 4 } c. } 4 2 8 4 } d. } 7 2 2 4 } SOLUTION In each case we divide the numerator by the denominator: a. } 2 1 4 } 5 24 b. } 2 2 4 4 } 5 1 c. } 4 2 8 4 } 5 2 d. } 7 2 2 4 } 5 3 Comparing Fractions We can compare fractions to see which is larger or smaller when they have the same denominator. EXAMPLE 6 Write each fraction as an equivalent fraction with denomi- nator 24. Then write them in order from smallest to largest. } 5 8 } } 5 6 } } 3 4 } } 2 3 } SOLUTION We begin by writing each fraction as an equivalent fraction with denominator 24. } 5 8 } 5 } 1 2 5 4 } } 5 6 } 5 } 2 2 0 4 } } 3 4 } 5 } 1 2 8 4 } } 2 3 } 5 } 1 2 6 4 } Now that they all have the same denominator, the smallest fraction is the one with the smallest numerator and the largest fraction is the one with the largest numerator. Writing them in order from smallest to largest we have: } 1 2 5 4 } , } 1 2 6 4 } , } 1 2 8 4 } , } 2 2 0 4 } or } 5 8 } , } 2 3 } , } 3 4 } , } 5 6 } 148 Chapter 2 Fractions 1: Multiplication and Division DESCRIPTIVE STATISTICS Scatter Diagrams and Line Graphs The table and bar chart give the daily gain in the price of eCollege.com stock for one week in the year 2000, when stock prices were given in terms of frac- tions instead of decimals. Figure 6 below shows another way to visualize the information in the table.

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  N1 Mathematics

  • J Daniels, M Kropman, J Daniels, M Kropman(Authors)
  • 2014(Publication Date)
  • Future Managers

    (Publisher)

  92 Module 3 • Factorisation and fractions 3.2 Algebraic fractions In this section monomial denominators will be covered. Fractions can be simplified by dividing the numerator and the denominator by the same factor. Example 4 8 ← ← numerator denominator = 4 1 4 2 × × • Cancel 4. = 1 2 or 4 8 = 4 8 2 1 • 4 divides into 8 twice. = 1 2 • The denominator of a fraction can never be equal to 0. a 0 is undefined, for example, 2 0 is undefined. • 0 a = 0, for example, 0 2 = 0 Pre-knowledge • You need to know the following when working with algebraic fractions. The laws of exponents. How to factorise expressions: common factor and grouping How to find the LCM. • Forms of fractions Fraction Examples Explanation Proper fraction 3 4 ; 2 7 ; – 7 9 ; 3 6 Numerator is smaller than denominator Improper fraction 6 5 ; – 11 4 ; 9 7 Numerator is bigger than denominator Mixed fraction 1 2 3 ; 5 3 7 ; 10 1 2 Whole number and a fraction 1 1 + 2 3 = 1 2 3 2 Equivalent fraction 5 10 = 1 2 ; 19 95 = 1 5 Multiplying the numerator and the denominator by the same number (except 0) results in a fraction equivalent to the given fraction. • Splitting up a fraction: a b c + = a c + b c • a b c + ≠ a b + a c • ( a + b ) 2 ≠ a 2 + b 2 • ( a + b ) 2 = ( a + b )( a + b ) = a 2 + 2 ab + b 2 93 N1 Mathematics| Hands-On 3.2.1 Simplifying fractions The numerators in the examples below are not equal to 0. Example 1 Simplify the following. Fraction Solution Explanation 1. 5 xy x ( x ≠ 0) 5 / / xy x = 5 y • Monomial numerator and denominator; x x = 1 2. 3 6 xy xz ( x ≠ 0; z ≠ 0) / / / / 3 2 3 xy xz . = y z 2 • 3 6 = / × / × 3 1 3 2 = 1 2 or 3 6 2 1 = 1 2 3. 16 4 3 2 2 2 ab c a bc − ( x ≠ 0; b ≠ 0; c ≠ 0) 4 4 4 3 2 2 2 . / -/ ab c a bc = – 4 2 b a • – 16 4 = –4; a a 2 = a 1–2 = a –1 = 1 a • b b 3 = b 3–1 = b 2 ; c c 2 2 = c 2–2 = c 0 = 1 • 16 4 − = – 16 4 • − − 3 3 = +1 = 1 • − 3 3 = –1 • − a b = a b − = – a b 4.

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  The Origins of Mathematical Knowledge in Childhood

  • Catherine Sophian(Author)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  5 difficult to understand.

  An even more fundamental problem is that notions such as two out of five parts fail to capture constraints on the nature of the parts and their relation to the whole that are critical to understanding how the numerator and denominator of a fraction determine its magnitude. If we picture receiving two cookies from a plate containing three, or five cookies from a plate containing eight, it is hard to escape the idea that the latter quantity is greater. The plate has more on it, we get more to eat, and there is also more left over for someone else! In order to compare fraction magnitudes, we need to recognize that the focus is on the relation between the parts and the whole, not on the size of either one. Whatever the whole is, it counts as 1, and the size of each part is

  1 n

  , where n is the number of parts, no matter how big the original whole was. Therefore, increasing the number of parts in the whole along with the number of those parts that are included in the quantity being described—as in a comparison of two cookies out of three versus five cookies out of eight—does not necessarily make the fraction itself larger.

  This is a critical difference between part–whole reasoning as it applies to additive structures versus to fractions. In additive structures, the numerical size of each part is free to vary independently of the other part, and the whole, which is the aggregate of the parts, changes as the sizes of the parts change. In the case of fractions, the whole is the unit against which a quantity of interest is being measured, and fractional subunits are created by partitioning that whole into equal segments. Because fractions describe a quantity relative to a whole, the size of the whole cannot be changed without also changing the relation between the quantity of interest and that whole, and thus the fractional value of the quantity of interest.

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  Introductory Mathematics

  • J Daniels, M Kropman, J Daniels, M Kropman(Authors)
  • 2014(Publication Date)
  • Future Managers

    (Publisher)

  • The lowest common multiple ( LCM ) of two or more numbers is the smallest possible number into which all the numbers will divide exactly. • Factorising Taking out the highest common factor. Common factors are the factors shared by two or more numbers. A common factor can be a number or a variable, or a combination of numbers and variables, for example: 4 x 2 y 3 z 2 + 16 x 3 y 4 z 2 – 64 xyz 3 = 4 xyz 2 ( xy 2 + 4 x 2 y 3 – 16 z ) Grouping An expression with more than three terms can be factorised by grouping terms to find a common factor, for example: ax + ay – bx – by = ( ax + ay ) + (– bx – by ) = a ( x + y ) + b (– x – y ) = a ( x + y ) – b ( x + y ) = ( x + y )( a + b ) • Algebraic fractions Rules of fractions Rule Explanation 1 a b × c d = ac bd Also: a × c d = a 1 × c d = ac d When multiplying fractions, multiply all the numerators and then all the denominators . Restrictions: b ≠ 0; d ≠ 0 105 Introductory Mathematics| Hands-On Rule Explanation 2 a b ÷ c d = a b × d c = ad bc When dividing fractions, change the operation from division to multiplication and invert whatever is after the division sign (multiplying with the reciprocal of the fraction). This is called the tips and times rule. Restrictions: b ≠ 0; c ≠ 0; d ≠ 0 3 a b + c b = a c b + or a b + c d = ad bc bd + To add or subtract fractions, the denominator must be the same . Restrictions: b ≠ 0; d ≠ 0 • Multiplying and dividing fractions Steps for multiplying fractions Step 1 Cancel factors in the numerator with factors in the denominator. Step 2 Multiply the numerators by numerators. Step 3 Multiply the denominators by denominators. Step 4 Simplify the answer to its simplest form. Steps for dividing fractions Step 1 First change the ÷ (division) sign to a × (multiplication) sign, and invert the fraction after the division sign. Step 2 Follow the steps above for multiplying fractions.

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  Teaching Middle School Mathematics

  • Douglas K. Brumbaugh(Author)
  • 2013(Publication Date)
  • Routledge

    (Publisher)

  and the product is shown as the overlapped region.

  Traditionally students are asked to express answers for in lowest terms (often referred to as reduce, simplify, or divide out common factors). You could get , something smaller which is what reduced means. We know that is not the intention of the statement, but in the strictest sense, the fraction has been reduced. A similar discussion could be made about “simplify” from the standpoint of asking why one fraction is simpler than one of its equivalent forms. Again, we know what we mean by simplify, but the word is not overly descriptive. Divide out common factors is a good prerequisite skill for an algebraic setting like , where some students will say . With divide out common factors, clarification of the algebraic process is a lot easier. The point is that so much of what is done in middle school mathematics is critical groundwork for future study. Your care as you explain the basics makes the mathematical lives of your students much easier as they progress through the curriculum. Your colleagues will appreciate the effort too!

  The exercise raises one more issue. If we, as teachers, insist that students divide out common factors in their work, how do we justify using ? Should we not set a good example?

  Your Turn 8.29. Show someone who is not in your class how to “reduce” a fraction and describe their reaction. 8.30. Should students be expected to express fractions where the greatest common factor (GCF) between the numerator and denominator is one? Explain your reasoning.

  Having already dealt with equivalent fractions, students should be ready to handle the intricacies of by expressing it as , which gives . The traditional task now is to find an equivalent form of , where the numerator and denominator are relatively prime (GCF of the numerator and denominator is 1). Prior work with equivalent fractions should make this a manageable task. Is there a better way to get the numerator and denominator to be relatively prime? Here, . Students should be encouraged to look for patterns. From equivalent fractions, they should know that going from to involves both the numerator and denominator being divided by three. They should conclude that the division could be done before multiplying, thus the phrase “dividing out common factors.” That would generate the form you are familiar with: . It is not incorrect to multiply the numerators and denominators first and then divide out common factors, but it is not as efficient. Errors can occur because larger numbers are involved. Eventually students encounter somethin like , which is much easier to do if the common factors are divided out initially as opposed to dividing out the common factors of

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  Basic College Mathematics

  • Charles P. McKeague(Author)
  • 2015(Publication Date)
  • XYZ Textbooks

    (Publisher)

  g m8 g m8 Lowest Terms A fraction is said to be in lowest terms if the numerator and the denominator have no factors in common other than the number 1. 2.3 Prime Numbers, Factors, and Simplifying Fractions and Mixed Numbers 115 The fraction 3 _ 4 is in lowest terms, because 3 and 4 have no factors in common except the number 1. Reducing a fraction to lowest terms is simply a matter of dividing the numerator and the denominator by all the factors they have in common. We know from Property 2 of Section 2.1 that this will produce an equivalent fraction. Reduce the fraction 12 __ 15 to lowest terms by first factoring the numerator and the denominator into prime factors and then dividing both the numerator and the denominator by the factor they have in common. Solution The numerator and the denominator factor as follows: 12 = 2 ⋅ 2 ⋅ 3 and 15 = 3 ⋅ 5 The factor they have in common is 3. Property 2 tells us that we can divide both terms of a fraction by 3 to produce an equivalent fraction. So 12 ___ 15 = 2 ⋅ 2 ⋅ 3 _______ 3 ⋅ 5 Factor the numerator and the denominator completely = 2 ⋅ 2 ⋅ 3 ÷ 3 __________ 3 ⋅ 5 ÷ 3 Divide by 3 = 2 ⋅ 2 ____ 5 = 4 __ 5 The fraction 4 _ 5 is equivalent to 12 __ 15 and is in lowest terms, because the numerator and the denominator have no factors other than 1 in common. We can shorten the work involved in reducing fractions to lowest terms by using a slash to indicate division. For example, we can write the above problem this way: 12 ___ 15 = 2 ⋅ 2 ⋅ 3 _______ 3 ⋅ 5 = 4 __ 5 So long as we understand that the slashes through the 3’s indicate that we have divided both the numerator and the denominator by 3, we can use this notation. Applying the Concepts Laura is having a party. She puts 4 six-packs of diet soda in a cooler for her guests. At the end of the party she finds that only 4 sodas have been consumed. What fraction of the sodas are left? Write your answer in lowest terms. Solution She had 4 six-packs of soda, which is 4(6) = 24 sodas.

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