Multiply and Divide Mixed Numbers and Complex Fractions

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1 OpenStax Staging module: m00 Multiply and Divide Mixed Numbers and Complex Fractions Words Numbers This work is produced by OpenStax Staging and licensed under the Creative Commons Attribution License.0 By the end of this section, you will be able to: Abstract Multiply and divide mixed numbers Translate phrases to expressions with fractions Simplify complex fractions Simplify expressions written with a fraction bar Before you get started, take this readiness quiz. )Divide and reduce, if possible: ( + ) (0 ). If you missed this problem, review. )Multiply and write the answer in simplied form: If you missed this problem, review. )Convert into an improper fraction. If you missed this problem, review.. Multiply and Divide Mixed Numbers In the previous section, you learned how to multiply and divide fractions. All of the examples there used either proper or improper fractions. What happens when you are asked to multiply or divide mixed numbers? Remember that we can convert a mixed number to an improper fraction. And you learned how to do that in. Multiply or divide mixed numbers. Step. Convert the mixed numbers to improper fractions. Step. Follow the rules for fraction multiplication or division. Step. Simplify if possible. Version.0: Sep 9, 0 :0 am

2 OpenStax Staging module: m00 Example Multiply: Convert to an improper fraction. 0 0 Multiply. Look for common factors. Remove common factors. Simplify. Notice that we left the answer as an improper fraction,, and did not convert it to a mixed number. In algebra, it is preferable to write answers as improper fractions instead of mixed numbers. This avoids any possible confusion between and. Exercise (Solution on p..) Multiply, and write your answer in simplied form:. Exercise (Solution on p..) Multiply, and write your answer in simplied form:. Example Multiply, and write your answer in simplied form: ( ). ( ) ( ) Convert mixed numbers to improper fractions. Multiply. Look for common factors. Remove common factors. Simplify. Exercise (Solution on p..) ( Multiply, and write your answer in simplied form. ).

3 OpenStax Staging module: m00 Exercise (Solution on p..) Multiply, and write your answer in simplied form.. Example Divide, and write your answer in simplied form:. Convert mixed numbers to improper fractions. Multiply the rst fraction by the reciprocal of the second. Multiply. Look for common factors. Remove common factors. Simplify. Exercise (Solution on p..) Divide, and write your answer in simplied form:. Exercise 9 (Solution on p..) Divide, and write your answer in simplied form:. Example Divide:. Convert mixed numbers to improper fractions. Multiply the rst fraction by the reciprocal of the second. Multiply. Look for common factors. Remove common factors. Simplify.

4 OpenStax Staging module: m00 Exercise (Solution on p..) Divide, and write your answer in simplied form:. Exercise (Solution on p..) Divide, and write your answer in simplied form:. Translate Phrases to Expressions with Fractions The words quotient and ratio are often used to describe fractions. In, we dened quotient as the result of division. The quotient of a and b is the result you get from dividing a by b, or a b. Let's practice translating some phrases into algebraic expressions using these terms. Example Translate the phrase into an algebraic expression: the quotient of x and. The keyword is quotient; it tells us that the operation is division. Look for the words of to nd the numbers to divide. and and The quotient of x and () This tells us that we need to divide x by. Algebraically, we can express this as x () Exercise (Solution on p..) Translate the phrase into an algebraic expression: the quotient of 9s and. Exercise (Solution on p..) Translate the phrase into an algebraic expression: the quotient of y and. Example Translate the phrase into an algebraic expression: the quotient of the dierence of m and n, and p. We are looking for the quotient of the dierence of m and, and p. This means we want to divide the dierence of m and n by p. m n p

5 OpenStax Staging module: m00 Exercise (Solution on p..) Translate the phrase into an algebraic expression: the quotient of the dierence of a and b, and cd. Exercise (Solution on p..) Translate the phrase into an algebraic expression: the quotient of the sum of p and q, and r. Simplify Complex Fractions Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called complex fraction, which is a fraction that has a fraction in either the numerator, the denominator, or both. Some examples of complex fractions are: x () To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction can be written as. Simplify a complex fraction. Step. Rewrite the complex fraction as a division problem. Step. Follow the rules for dividing fractions. Step. Simplify if possible. Example. Rewrite as division. Multiply the rst fraction by the reciprocal of the second. Multiply. Look for common factors. Remove common factors and simplify.

6 OpenStax Staging module: m00 Exercise 0 (Solution on p..). Exercise (Solution on p..). Example. Rewrite as division. Multiply the rst fraction by the reciprocal of the second. Multiply; the product will be negative. Look for common factors. Remove common factors and simplify. Exercise (Solution on p..). Exercise (Solution on p..) 9. 0 Example 9 x xy. Rewrite as division. Multiply the rst fraction by the reciprocal of the second. x xy x xy x xy Multiply. x Look for common factors. xy x x y Remove common factors and simplify. y

7 OpenStax Staging module: m00 Exercise (Solution on p..) a ab. Exercise (Solution on p..) p pq. Example 0. Rewrite as division. Change the mixed number to an improper fraction. Multiply the rst fraction by the reciprocal of the second. Multiply. Look for common factors. Remove common factors and simplify. Exercise 9 (Solution on p..). Exercise 0 (Solution on p..).

8 OpenStax Staging module: m00 Simplify Expressions with a Fraction Bar Where does the negative sign go in a fraction? Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. Remember that fractions represent division. The fraction could be the result of dividing, a negative by a positive, or of dividing, a positive by a negative. When the numerator and denominator have dierent signs, the quotient is negative. If both the numerator and denominator are negative, then the fraction itself is positive because we are dividing a negative by a negative. = negative = positive () negative For any positive numbers a and b, a b = a b = a b Example Which of the following fractions are equivalent to?,,, The quotient of a positive and a negative is a negative, so and are also negative. () is negative. Of the fractions listed, Exercise (Solution on p..) Which of the following fractions are equivalent to?,,, Exercise (Solution on p..) Which of the following fractions are equivalent to?,,, Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses. For example, + means ( + ) ( ). The order of operations tells us to simplify the numerator and the denominator rstas if there were parenthesesbefore we divide. We'll add fraction bars to our set of grouping symbols from to have a more complete set here.

9 OpenStax Staging module: m00 9 Simplify an expression with a fraction bar. Step. Simplify the numerator. Step. Simplify the denominator. Step. Simplify the fraction. Example +. + Simplify the expression in the numerator. Simplify the expression in the denominator. Simplify the fraction. Exercise (Solution on p..) +. Exercise (Solution on p..) +. Example () +.

10 OpenStax Staging module: m00 0 Use the order of operations. Multiply in the numerator and use the exponent in the denominator. Simplify the numerator and the denominator. Simplify the fraction. () + + Exercise (Solution on p..) () +. Exercise 9 (Solution on p..) Example ( ). () +. Use the order of operations (parentheses rst, then exponents). ( ) () Simplify the numerator and denominator. Simplify the fraction. Exercise (Solution on p..) ( ). Exercise (Solution on p..) (+) +.

11 OpenStax Staging module: m00 Example ( )+( ) (). ( )+( ) Multiply. () +( ) Simplify. Divide. Exercise (Solution on p..) ( )+( ) ()+. Exercise (Solution on p..) ( )+9( ) (). We encourage you to go to Appendix B to take the Self Check for this section. Access these online resources for additional instruction and practice with mixed numbers and complex fractions. Division Involving Mixed Numbers Evaluate a Complex Fraction Key Concepts ˆ To multiply or divide mixed numbers, rst convert the mixed numbers to improper fractions. See Example. ˆ A fraction is another way to represent a quotient or ratio of two values. See Example. ˆ To simplify a complex fraction, rst rewrite it as a division problem. See Example. ˆ For any positive numbers a and b, a b = a b = a b. See Example 0. ˆ The fraction bar is a grouping symbol; simplify the numerator and denominator separately before dividing. See Example

12 OpenStax Staging module: m00. Practice Makes Perfect Multiply and Divide Mixed Numbers In the following exercises, multiply and write the answer in simplied form. Exercise 0 Exercise (Solution on p..) 9 Exercise Exercise 9 (Solution on p..) 0 Exercise ( 0 ) Exercise (Solution on p..) ( ) 9 Exercise 9 Exercise (Solution on p..) 0 In the following exercises, divide, and write your answer in simplied form. Exercise Exercise (Solution on p..) 9 Exercise Exercise (Solution on p..) Exercise Exercise 9 (Solution on p..) 0 Exercise 0 9 ( ) Exercise (Solution on p..) ( ) Translate Phrases to Expressions with Fractions In the following exercises, translate each English phrase into an algebraic expression. Exercise the quotient of u and Exercise (Solution on p..) the quotient of v and

13 OpenStax Staging module: m00 Exercise the quotient of p and q Exercise (Solution on p..) the quotient of a and b Exercise the quotient of r and the sum of s and 0 Exercise (Solution on p..) the quotient of A and the dierence of and B Simplify Complex Fractions In the following exercises, simplify the complex fraction. Exercise 9 Exercise 9 (Solution on p..) Exercise 0 Exercise (Solution on p..) 9 0 Exercise Exercise (Solution on p..) 9 0 Exercise Exercise (Solution on p..) 0 Exercise m n Exercise (Solution on p..) r s Exercise x 9 Exercise 9 (Solution on p..) y Exercise 0 0 Exercise (Solution on p..) Exercise 9 Exercise (Solution on p..) Simplify Expressions with a Fraction Bar In the following exercises, simplify.

14 OpenStax Staging module: m00 Exercise Which of the following fractions are equivalent to?,,, Exercise (Solution on p..) Which of the following fractions are equivalent to 9? 9, 9, 9, 9 Exercise Which of the following fractions are equivalent to?,,, Exercise (Solution on p. 9.) Which of the following fractions are equivalent to?,,, Exercise + Exercise 9 (Solution on p. 9.) 9+ Exercise Exercise 9 (Solution on p. 9.) 9 Exercise 9 Exercise 9 (Solution on p. 9.) + Exercise Exercise 9 (Solution on p. 9.) + Exercise 9 9 Exercise 9 (Solution on p. 9.) +9 + Exercise 9. 0 Exercise 99 (Solution on p. 9.). Exercise 00.. Exercise 0 (Solution on p. 9.). 9. Exercise 0 Exercise 0 (Solution on p. 9.) + 0 Exercise

15 OpenStax Staging module: m00 Exercise 0 (Solution on p. 9.) Exercise 0..0 Exercise 0 (Solution on p. 9.).9. Exercise Exercise 09 (Solution on p. 9.) Exercise 0 Exercise (Solution on p. 9.) Exercise +() Exercise (Solution on p. 9.) +() Exercise. ( ) 9.. Exercise (Solution on p. 9.) 9. ( ).. Exercise 9( ) ( ) ( ) ( 9) Exercise (Solution on p. 9.) (9 ) ( 9) ( ) ( 9). Everyday Math Exercise Baking A recipe for chocolate chip cookies calls for cups of our. Graciela wants to double the recipe. (a) How much our will Graciela need? Show your calculation. Write your result as an improper fraction and as a mixed number. (b) Measuring cups usually come in sets with cups for,,,, and cup. Draw a diagram to show two dierent ways that Graciela could measure out the our needed to double the recipe. Exercise 9 (Solution on p. 9.) Baking A booth at the county fair sells fudge by the pound. Their award winning Chocolate Overdose fudge contains cups of chocolate chips per pound. (a) How many cups of chocolate chips are in a half-pound of the fudge? (b) The owners of the booth make the fudge in 0-pound batches. How many chocolate chips do they need to make a 0-pound batch? Write your results as improper fractions and as a mixed numbers.

16 OpenStax Staging module: m00. Writing Exercises Exercise 0 Explain how to nd the reciprocal of a mixed number. Exercise (Solution on p. 9.) Explain how to multiply mixed numbers. Exercise Randy thinks that is. Explain what is wrong with Randy's thinking. Exercise (Solution on p. 9.) Explain why,, and are equivalent.

17 OpenStax Staging module: m00 Solutions to Exercises in this Module Solution to Exercise (p. ) Solution to Exercise (p. ) 9 Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) 9s Solution to Exercise (p. ) y Solution to Exercise (p. ) a b cd Solution to Exercise (p. ) p+q r Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) 0 Solution to Exercise (p. ) b Solution to Exercise (p. ) q Solution to Exercise (p. ) 9. Solution to Exercise (p. ) Solution to Exercise (p. ), Solution to Exercise (p. ), Solution to Exercise (p. 9) 0 9 Solution to Exercise (p. 9)

18 OpenStax Staging module: m00 Solution to Exercise (p. 0) Solution to Exercise (p. 0) Solution to Exercise (p. 0) 9 Solution to Exercise (p. 0) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) v Solution to Exercise (p. ) a b Solution to Exercise (p. ) A B Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) 0 Solution to Exercise (p. ) Solution to Exercise (p. ) r s Solution to Exercise (p. ) 9 y Solution to Exercise (p. ) Solution to Exercise (p. )

19 OpenStax Staging module: m00 9 Solution to Exercise (p. ) 9, 9 Solution to Exercise (p. ), Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) 0 Solution to Exercise (p. ) Solution to Exercise (p. ) 0 Solution to Exercise (p. ) Solution to Exercise (p. ) (a) = Solution to Exercise (p. ) Answers will vary. Solution to Exercise (p. ) Answers will vary. Glossary cups; (b) 0 = cups Denition : complex fraction fraction in which the numerator or the denominator contains a fraction

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