COMPLEX FRACTIONS 7.5. section. Complex Fractions

Size: px

Start display at page:

Download "COMPLEX FRACTIONS 7.5. section. Complex Fractions"

Della Wilson
7 years ago
Views:

$Write a rational epression for the fraction of the barn that they complete in one day by working together. Evaluate the epression for 5. GETTING MORE INVOLVED 65. Writing.$

1 5 (7 ) Chapter 7 Rational Epressions 6. Barn painting. Melanie can paint a certain barn by herself in days. Her helper Melissa can paint the same barn by herself in days. Write a rational epression for the fraction of the barn that they complete in one day by working together. Evaluate the epression for 5. GETTING MORE INVOLVED 65. Writing. Write a step-by-step procedure for adding rational epressions. 66. Writing. Eplain why fractions must have the same denominator to be added. Use real-life eamples. FIGURE FOR EXERCISE COMPLEX FRACTIONS In this Comple Fractions section Using the LCD to Simplify Comple Fractions Applications In this section we will use the idea of least common denominator to simplify comple fractions. Also we will see how comple fractions can arise in applications. Comple Fractions A comple fraction is a fraction having rational epressions in the numerator, denominator, or both. Consider the following comple fraction: Numerator of comple fraction 5 8 Denominator of comple fraction To simplify it, we can combine the fractions in the numerator as follows: We can combine the fractions in the denominator as follows: Now divide the numerator by the denominator:

2 Using the LCD to Simplify Comple Fractions 7.5 Comple Fractions (7 ) 55 A comple fraction can be simplified by writing the numerator and denominator as single fractions and then dividing, as we just did. However, there is a better method. The net eample shows how to simplify a comple fraction by using the LCD of all of the single fractions in the comple fraction. E X A M P L E calculator close-up You can check Eample with a calculator as shown here. Using the LCD to simplify a comple fraction Use the LCD to simplify. 5 8 The LCD of,,, and 8 is. Now multiply the numerator and denominator of the comple fraction by the LCD: Multiply the numerator and denominator by the LCD Simplify CAUTION We simplify a comple fraction by multiplying the numerator and denominator of the comple fraction by the LCD. Do not multiply the numerator and denominator of each fraction in the comple fraction by the LCD. In the net eample we simplify a comple fraction involving variables. E X A M P L E A comple fraction with variables Simplify.

3 56 (7 ) Chapter 7 Rational Epressions helpful hint When students see addition or subtraction in a comple fraction, they often convert all fractions to the same denominator. This is not wrong, but it is not necessary. Simply multiplying every fraction by the LCD eliminates the denominators of the original fractions. The LCD of the denominators,, and is : ( ) Multiply the numerator and denominator by. ( ) Simplify. The numerator of this answer can be factored, but the rational epression cannot be reduced. The general strategy for simplifying a comple fraction is stated as follows. Strategy for Simplifying a Comple Fraction. Find the LCD for all the denominators in the comple fraction.. Multiply both the numerator and the denominator of the comple fraction by the LCD. Use the distributive property if necessary.. Combine like terms if possible.. Reduce to lowest terms when possible. E X A M P L E study tip Studying in an environment similar to the one in which you will be tested can increase your chances of recalling information. When possible, review for a test in the classroom in which you will take the test. Simplifying a comple fraction Simplify. Because and are opposites, we can use ( )( ) as the LCD. Multiply the numerator and denominator by ( )( ): ( )( ) ( )( ) ( )( ) ( )( ) ( ) ()( ) ( ) Combine like terms.

$Applications 7.5 Comple Fractions (7 5) 57 As their name suggests, comple fractions arise in some fairly comple situations.$

4 Applications 7.5 Comple Fractions (7 5) 57 As their name suggests, comple fractions arise in some fairly comple situations. E X A M P L E Fast-food workers A survey of college students found that of the female students had jobs and of the male students had jobs. It was also found that of the female students worked in fast-food restaurants and 6 of the male students worked in fast-food restaurants. If equal numbers of male and female students were surveyed, then what fraction of the working students worked in fast-food restaurants? Let represent the number of males surveyed. The number of females surveyed is also. The total number of students working in fast-food restaurants is. 6 The total number of working students in the survey is. So the fraction of working students who work in fast-food restaurants is 6. The LCD of the denominators,,, and 6 is. Multiply the numerator and denominator by to eliminate the fractions as follows: 6 6 Multiply numerator and denominator by Combine like terms. 5 Reduce. 5 So (or about 6%) of the working students work in fast-food restaurants. WARM-UPS True or false? Eplain your answer.. The LCD for the denominators,, 6, and is.. The LCD for the denominators a b, b a, and 6 is 6a 6b. 7. The fraction 7 is a comple fraction. 98. The LCD for the denominators a and a is a The largest common denominator for the fractions,, and is.

5 58 (7 6) Chapter 7 Rational Epressions WARM-UPS (continued) Questions 6 0 refer to the following comple fractions: b a) b) 5 5 a c) d) 6. To simplify (a), we multiply the numerator and denominator by To simplify (b), we multiply the numerator and denominator by a b ab. 8. The comple fraction (c) is equivalent to. 9. If, then (c) represents a real number. 0. The comple fraction (d) can be written as EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is a comple fraction?. What are the two ways to simplify a comple fraction? Simplify each comple fraction. See Eample Simplify each comple fraction. See Eample. a b.. b a 5 a.. a y y a 5 a a w b b w w 5 b 9w Simplify each comple fraction. See Eample. y a y a

7.5 Comple Fractions (7 7) 59 6.. 5 a a a 7. a a 8. ( a ) y y 8 y y 5y 6 y 5.. 5 5 9. 0. a a b a b a b 5 a 5. a 6. m m 7. m m 8. w w 9. 5 w w 0. a b a b. b a b a. 9 6 9 y y y y Solve each problem.

6 7.5 Comple Fractions (7 7) a a a 7. a a 8. ( a ) y y 8 y y 5y 6 y a a b a b a b 5 a 5. a 6. m m 7. m m 8. w w 9. 5 w w 0. a b a b. b a b a y y y y Solve each problem. See Eample.. Sophomore math. A survey of college sophomores showed that 5 6 of the males were taking a mathematics class and of the females were taking a mathematics class. One-third of the males were enrolled in calculus, and 5 of the females were enrolled in calculus. If just as many males as females were surveyed, then what fraction of the surveyed students taking mathematics were enrolled in calculus? Rework this problem assuming that the number of females in the survey was twice the number of males.. Commuting students. At a well-known university, of the undergraduate students commute, and of the graduate students commute. One-tenth of the undergraduate students drive more than 0 miles daily, and 6 of the graduate students drive more than 0 miles daily. If there are twice as many undergraduate students as there are graduate students, then what fraction of the commuters drive more than 0 miles daily? Simplify each comple fraction a 5 a 5 y y y y ab b ab 5 a b 6a b FIGURE FOR EXERCISE

7 60 (7 8) Chapter 7 Rational Epressions GETTING MORE INVOLVED. Eploration. Simplify,, and. a) Are these fractions getting larger or smaller as the fractions become more comple? b) Continuing the pattern, find the net two comple fractions and simplify them. c) Now what can you say about the values of all five comple fractions?. Discussion. A comple fraction can be simplified by writing the numerator and denominator as single fractions and then dividing them or by multiplying the numerator and denominator by the LCD. Simplify the comple fraction y 6 y y by using each of these methods. Compare the number of steps used in each method, and determine which method requires fewer steps. In this section Equations with Rational Epressions Etraneous s E X A M P L E helpful hint Note that it is not necessary to convert each fraction into an equivalent fraction with a common denominator here. Since we can multiply both sides of an equation by any epression we choose, we choose to multiply by the LCD. This tactic eliminates the fractions in one step. 7.6 SOLVING EQUATIONS WITH RATIONAL EXPRESSIONS Many problems in algebra can be solved by using equations involving rational epressions. In this section you will learn how to solve equations that involve rational epressions, and in Sections 7.7 and 7.8 you will solve problems using these equations. Equations with Rational Epressions We solved some equations involving fractions in Section.. In that section the equations had only integers in the denominators. Our first step in solving those equations was to multiply by the LCD to eliminate all of the denominators. Integers in the denominators Solve 6. The LCD for,, and 6 is 6. Multiply each side of the equation by 6: Original equation Multiply each side by 6. ( ) Simplify. Enclose in parentheses. 6 Subtract 7 from each side. Divide each side by.

6.3. section. Building Up the Denominator. To convert the fraction 2 3 factor 21 as 21 3 7. Because 2 3

$6.3. section. Building Up the Denominator. To convert the fraction 2 3 factor 21 as 21 3 7. Because 2 3$ 0 (6-18) Chapter 6 Rational Epressions GETTING MORE INVOLVED 7. Discussion. Evaluate each epression. a) One-half of 1 b) One-third of c) One-half of d) One-half of 1 a) b) c) d) 8 7. Eploration. Let R