4: Fractions
- Page ID
- 46120
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Often in life, whole amounts are not exactly what we need. A baker must use a little more than a cup of milk or part of a teaspoon of sugar. Similarly a carpenter might need less than a foot of wood and a painter might use part of a gallon of paint. In this chapter, we will learn about numbers that describe parts of a whole. These numbers, called fractions, are very useful both in algebra and in everyday life. You will discover that you are already familiar with many examples of fractions!
- 4.1: Visualize Fractions (Part 1)
- A fraction is a way to represent parts of a whole. The denominator b represents the number of equal parts the whole has been divided into, and the numerator a represents how many parts are included. The denominator, b, cannot equal zero because division by zero is undefined. A mixed number consists of a whole number and a fraction. When a fraction has a numerator that is smaller than the denominator, it is called a proper fraction, and its value is less than one.
- 4.2: Visualize Fractions (Part 2)
- Equivalent fractions are fractions that have the same value. When working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. We can use the inequality symbols to order fractions. Remember that a > b means that a is to the right of b on the number line. As we move from left to right on a number line, the values increase.
- 4.4: Multiply and Divide Fractions (Part 1)
- A fraction is considered simplified if there are no common factors, other than 1, in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.
- 4.5: Multiply and Divide Fractions (Part 2)
- The reciprocal of the fraction a/b is b/a, where a ≠ 0 and b ≠ 0. A number and its reciprocal have a product of 1. To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator. To divide fractions, multiply the first fraction by the reciprocal of the second.
- 4.6: Find Multiples and Factors (Part 1)
- A number is a multiple of n if it is the product of a counting number and n. If a number m is a multiple of n, then we say that m is divisible by n. If a • b = m, then a and b are factors of m, and m is the product of a and b. To find all the factors of a counting number, divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor. Then, list all the factor pairs and write all the factors in order from smallest to largest.
- 4.7: Find Multiples and Factors (Part 2)
- A prime number is a counting number greater than 1 whose only factors are 1 and itself. A composite number is a counting number that is not prime. To determine if a number is prime, divide it by each of the primes, in order, to see if it is a factor of the number. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.
- 4.8: Prime Factorization and the Least Common Multiple (Part 1)
- The prime factorization of a number is the product of prime numbers that equals the number. This can be found using either the tree method or the ladder method. The tree method involves writing the factors below the number and connecting them to the number with small line segments. The ladder method involves dividing the given number by its smallest prime factor. The composite number is the product of all the primes used in either method, which should give the same result.
- 4.9: Prime Factorization and the Least Common Multiple (Part 2)
- The least common multiple (LCM) is the smallest number that is a multiple of two numbers. The LCM of two numbers can be found by listing their multiples or using the prime factors method. The listing method involves writing out the multiples of each number until the first multiple common to both lists is found. The prime factors method involves writing each number as a product of primes, matching primes vertically when possible, and then multiplying the factors together to obtain the LCM.
- 4.10: Multiply and Divide Mixed Numbers and Complex Fractions (Part 1)
- To multiply or divide mixed numbers, convert the mixed numbers to improper fractions. Then follow the rules for fraction multiplication or division and then simplify if possible. A complex fraction is a fraction in which the number and/or denominator contains a fraction. To simplify a complex fraction, rewrite the complex fraction as a division problem. Then follow the rules for dividing fractions and then simplify if possible.
- 4.11: Multiply and Divide Mixed Numbers and Complex Fractions (Part 2)
- Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. When the numerator and denominator have different signs, the quotient is negative. If both the numerator and denominator are negative, then the fraction is positive because we are dividing a negative by a negative. Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses.
- 4.12: Add and Subtract Fractions with Common Denominators
- To add fractions, add the numerators and place the sum over the common denominator. To subtract fractions, subtract the numerators and place the difference over the common denominator.
- 4.13: Add and Subtract Fractions with Different Denominators (Part 1)
- The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators. To find the LCD of two fractions, factor each denominator into its primes. Then list the primes, matching primes in columns when possible, and bring down the columns. Finally, multiply the factors together, the product is the LCM of the denominators which is also the LCD of the fractions.
- 4.14: Add and Subtract Fractions with Different Denominators (Part 2)
- In fraction multiplication, you multiply the numerators and denominators together, respectively. To divide fractions, you multiply the first fraction by the reciprocal of the second. For fraction addition, add the numerators together and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD. Likewise, for fraction subtraction, subtract the numerators and place the difference over the common denominator.
- 4.15: Add and Subtract Mixed Numbers (Part 1)
- To add mixed numbers with a common denominator, first rewrite the problem in vertical form. Then, add the whole numbers and the fractions together. Finally, simplify the sum if possible. An alternate method for adding mixed numbers is to convert the mixed numbers to improper fractions and then add the improper fractions. This method is usually written horizontally.
- 4.16: Add and Subtract Mixed Numbers (Part 2)
- To subtract mixed numbers with common denominators, first rewrite the problem in vertical form and compare the two fractions. If the top fraction is larger than the bottom fraction, subtract the fractions and then the whole numbers. If the top fraction is not larger than the bottom fraction, in the top mixed number, take one whole and add it to the fraction part, making a mixed number with an improper fraction. Then subtract the fractions and then the whole numbers. Lastly, simplify if possible.
- 4.18: Solve Equations with Fractions (Part 1)
- The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, or a fraction. To determine whether a number is a solution to an equation, first substitute the number for the variable in the equation. Then simplify the expressions on both sides of the equation and determine if the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.
- 4.19: Solve Equations with Fractions (Part 2)
- To solve real-world problems, we first need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we translate math notation into a sentence to answer the question.
Figure 4.1 - Bakers combine ingredients to make delicious breads and pastries. (credit: Agustín Ruiz, Flickr)
Contributors
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."