Math 02, Intermediate Algebra Name Author: Debra Griffin Circle one: Red Bluff Main Campus Burney Weaverville Chapter 2 Rational Epressions 2. Rational Epressions and Allowed X-Values (Domain) 2 Determining the domain of a rational epression, simplifying rational epression, writing equivalent rational epressions 2.2 Multiplication and Division of Rational Epressions 5 2.3 Addition and Subtraction of Rational Epressions 8 Adding and subtracting rational epressions with like denominators, finding LCM, adding and subtracting rational epressions with unlike denominators 2.4 Comple Fractions 5 2.5 Equation Solving 8 2.6 Applications 23 Lecture Note-Taking Guide Scoring Rubric Completion of these lecture notes for etra credit points is optional. Because it is an etra credit option, the epectation of work quality is very high. There is no partial etra credit for this assignment. Any of the following criteria that are not met will result in no etra credit. Etra Credit Points 5 complete all worksteps shown correct answers neatly done in pencil correctly ordered Condition 0 Any of the required criteria have not been met hole punched fastened in folders with fasteners turned in on time labeled with name and site Page of 30
Math 02, Intermediate Algebra Section 2. 2. Domain (Allowed X-Values) and Simplifying Rational Epressions Recall that a rational number is a ratio of two integers, the second of which is non-zero. Formally, the set of rational numbers is { a b a and b are integers and b 0}. Informally, rational numbers are fractions. In algebra, a rational epression is a ratio of two polynomials, the second of which must be non-zero. Domain (Allowed X-Values) Eample (a) Find the set of real numbers for which each of the following is defined (the domain of the epression). (i) Since 0 is undefined, then cannot be 0. The domain is { 0 } (ii) 3 4 3 4 cannot be 0. What value of would make that happen? 3 4 0 +4 +4 3 + 0 4 3 3 4 3 4 3 The domain is { 4 3 } (iii) + 2 2 + 2 + cannot be 0. What value of would make that happen? 2 + 0 2 There is no real number for which 2. The domain is { } (iv) 5 2 4 2 4 cannot be 0. What value(s) of would make that happen? 2 4 0 ( + 2)( 2) 0 +20 or 20 2 2 +2 +2 +0 2 +02 2 or 2 The domain is { ±2 } Find the set of real numbers for which each of the following is defined.. (a) 5. (b) 2 2. (a) 4 5 2. (b) 0 2 3. (a) 5 3 8 3. (b) 5 2 3 00 Answers:. (b) { 0}; 2. (b) { }; 3. (b) { 0, ±0} Page 2 of 30
Math 02, Intermediate Algebra Section 2. Simplifying Rational Epressions Recall that to reduce a fraction to lowest terms, we remove factors of one, such as 0 25 2 5 5 5 2 5 We can reduce a rational epression similarly by removing factors of one. Eample (b) Simplify completely, that is, reduce to lowest terms. (i) 5 2 5 2 5 5 (ii) 2 4 5 3 2 4 5 2 4 5 ( +)( 5) 3 2 4 5 (3 +)( 5) + 3 + (iii) 2 2 2 2 2( ) ( ) 2 Simplify completely. 4. (a) 6 6 24 4. (b) 8 4 24 5. (a) 3( +) ( +) ( +) 5. (b) ( 2) 5( 2) ( 2)( + 3) 6. (a) 2 2 4 2 6. (b) 3 9 9 2 Answer: 4. (b) 2 ; 5. (b) 6 5 ; 6. (b) 3 ; + 3 3 + Page 3 of 30
Math 02, Intermediate Algebra Section 2. Equivalent Rational Epressions In arithmetic, we sometimes want to build up a fraction to an equivalent fraction with a new denominator in order to perform certain operations on the fraction. For eample, 4 5 4 3 5 3 2 5 Similarly, we can build up a rational epression to an equivalent one with a new denominator. 2 Eample (c) Write an epression equivalent to + with the denominator 2 +. 2 + 2 ( +) 22 2 + 7. (a) Write an epression equivalent to 5 with the denominator + 2. 7. (b) Write an epression equivalent to 3 2 with the denominator 5. 8. (a) Write an epression equivalent to 3 + 2 with the denominator 52 + 0. 8. (b) Write an epression equivalent to 5 3 with the denominator 62 8. Answers: 7. (b) 3 2 ( 5) 5 ; 8. (b) 30 2 6 2 8 Page 4 of 30
Math 02, Intermediate Algebra Section 2.2 2.2 Multiplication and Division of Rational Epressions Multiplication of Rational Epressions To multiply two fractions, we use the following property: For any real numbers a, b, c, and d (b 0 and d 0) a b c d a c b d For eample, 2 3 2 2 3 2 2 6 and /2 6 3 3 or 2 3 2 3 Multiplication of rational epressions is similar. Eample (a) Simplify 52 y y2 25. 5 2 y y2 25 52 y 2 25y 5 25 2 y2 y 5 y y 5 or 5 2 y y y 2 y 25 5 5 Eample (b) Multiply and simplify 2 4 2 2 3 + 3 2 + 6 + 8 2 4 2 2 3 + 3 ( + 2)( 2) 2 + 6 + 8 ( +)( 2) 3( +) ( + 4)( + 2) 3 + 4 Multiply and simplify.. (a) 5 5 20y 2. (b) 2 y 2 y 3 20 5 2. (a) 2 + 6 + 9 3 2 6 + 5 2 3 0 2 + 5 + 6 2. (b) 2 + 2 + 5 2 + 2 2 + 3 0 2 + 6 + 5 Answers:. (b) 3y ; 2. (b) 5 5 + 5 Page 5 of 30
Math 02, Intermediate Algebra Section 2.2 Dividing Rational Epressions To divide two fractions, we use the following property: For any real numbers a, b, c, and d (b 0, c 0, and d 0) a b c d a b d c For eample, 7 8 3 4 7 /8 2 /4 3 7 6 Division of two rational epressions is similar. Eample (c) Divide and simplify Eample (d) Divide and simplify 3 2 4y 93 3 7y 3 2 4y 3 93 7y 32 4y 3 7y 9 3 3 2 2 4 y 2 y3 5 2 6 5 3 +2 7y 3 9 3 6y 2 5 2 6 5 3 +2 5 3 +2 2 6 5 5 3( + 4) ( + 4)( 4) ( 5) 3 4 Divide and simplify. 3. (a) 2 2 y 3. (b) 8y 6y2 4. (a) 2 36 2 8 +6 8 3 2 2 4. (b) 5 2 49 2 7 +0 2 9 +4 Answers: 3. (b) 4 ; 4. (b) 3y + 7 Page 6 of 30
Math 02, Intermediate Algebra Section 2.2 Mied Practice Simplify. 5. (a) 6 2 + 6 2 + 7 + 6 20 5. (b) 8 + 8 42 + 32 6 3 6. (a) 2 y 3 6. (b) 6y 2 3y5 7. (a) + 3 4 + + 3 4 7. (b) 2 + 2 + 2 2 + + 8. (a) ( 2) 2 + 2 + 5 2 6 + 8 8. (b) ( 3) 2 ( + 3) + 3 2 2 4 + 3 Answers: 5. (b) 2 ; 6. (b) 2 y 3 ; 7. (b) ; 8. (b) 3 ( + 3)( ) Page 7 of 30
Math 02, Intermediate Algebra Section 2.3 2.3 Addition and Subtraction of Rational Epressions Addition and Subtraction of Rational Epressions with Like Denominators To add or subtract two fractions with like denominators, we use the following properties: a For any real numbers a, b, and c (c 0) c + b c a + b c and a c b c a b c For eample, 5 2 + 2 5 + 2 6 2 6 2 6 2 and 7 9 9 7 9 6 9 2 3 3 3 2 3. Addition and subtraction of rational epressions with like denominators is similar. Eample (a) Add 52 4 + 5y2 4. 5 2 4 + 5y2 4 52 + 5y 2 4 Eample (b) Subtract and simplify 5 6 2 3 2 3. 5 6 2 3 2 3 5 6 2 3 4 6 2(2 3) 2 3 2 3 2 Eample (c) Subtract and simplify 5 6 2 + 3 2 2 + 3. 5 6 2 + 3 2 5 6 ( 2) 5 6 +2 4 + 6 2(2 + 3) 2 + 3 2 + 3 2 + 3 2 + 3 2 + 3 2 Eample (d) Subtract and simplify 3 6 2 3 3 2. 3 6 2 3 3 2 3 6 2 3 (2 3) 3 6 2 3 + Eample (e) Add and simplify 22 +0 + 4 2 2 + + 5 + 2 + 7 + 6 2 2 + + 5. 2 3 4 6 2(2 3) 2 3 2 3 2 2 2 +0 + 4 2 2 + + 5 + 2 + 7 + 6 2 2 + + 5 32 +7 +0 (3 + 2)( + 5) 2 2 + + 5 (2 +)( + 5) 3 + 2 2 + Page 8 of 30
Math 02, Intermediate Algebra Section 2.3 Add or subtract and then and simplify if possible.. (a) 5 3 + 4 4. (b) 7 9 6 + 3 + 5 6 2. (a) 5 2 + 2 2 2. (b) 3 + 2 + 3. (a) 5 5 3. (b) 8 + 8 8 4. (a) 2 2 2 2 + 3 4 2 2 + 3 4 4. (b) 2 2 50 2 + 4 5 2 25 2 + 4 5 Answers:. (b) 5 2 ; 2. (b) 2; 3. (b) ; 4. (b) 3 5 Page 9 of 30
Math 02, Intermediate Algebra Section 2.3 Least Common Multiples A multiple of a number is the product of that number and a natural number. Consider the multiples of 5 and 4 as shown below: 5 4 5 5 4 4 5 2 0 4 2 8 5 3 5 4 3 2 5 4 20 4 4 6 5 5 25 4 5 20 5 6 30 4 6 24 5 7 35 4 7 28 5 8 40 4 8 32 5 9 45 4 9 36 5 0 50 4 0 40 Common multiples of 4 and 5 are 20, 40, 60, 80,... The Least Common Multiple of 4 and 5 is 20. The smallest number in any list of common multiples is the Least Common Multiple. We can denote the Least Common Multiple of 4 and 5 as LCM (4, 5) 20. Additionally, we can describe the Least Common Multiple of a set of numbers as the product of the distinct prime factors of the numbers, each raised only to the highest degree with which it occurs in any of the numbers of the set. If we write the prime factorization of the numbers, we can then build the LCM as follows. LCM (20, 36) 20 2 3 3 5 36 2 2 3 2 2 3 3 2 5 360 This LCM includes all of the factors of 20 and only the factors of 36 that are not present in the factors of 20. LCM (20, 36) 360 Finding the LCM of rational epressions is similar. LCM (3 4 y, 4 3 yz) 3 4 y 3 4 y 4 3 y 4 3 y z 3 4 y 4 z 2 4 yz LCM (3 4 y, 4 3 y) 2 4 yz Eample (f ) Find LCM (5 2 y 4 z, 0yz 3 ) 5 2 y 4 z 3 5 2 y 4 z 0yz 3 2 5 yz 3 LCM (5 2 y 4 z, 0yz 3 ) 3 2 5 2 y 4 z 3 30 2 y 4 z 3 Page 0 of 30
Math 02, Intermediate Algebra Section 2.3 5. (a) LCM (5 2 y, 50y 5 ) 5. (b) LCM (3 4 y 5, 2y 3 ) 6. (a) LCM (2 + 3, 2) 6. (b) LCM (3, 2 5) 7. (a) LCM ( + 2, 5) 7. (b) LCM (, 4) 8. (a) LCM (5 20, 2 8 + 6) 8. (b) LCM (6 2, 2 0 + 6) Answer: 5. (b) 2 4 y 5 ; 6. (b) 3(2 5); 7. (b) ( )( 4); 8. (b) 6( 2)( 8); Page of 30
Math 02, Intermediate Algebra Section 2.3 Addition and Subtraction of Rational Epressions with Unlike Denominators To add or subtract fractions with unlike denominators, we build up the fractions so that the denominators are the same and then use one of the previous properties. For eample, since the least common multiple of 3 and 4 is 2, we have 2 3 + 3 4 2 3 4 4 + 3 4 3 3 8 2 + 9 2 7 2 To add or subtract rational epressions, we first find the LCM of the denominators, build up the epressions so that the denominators are the same, then combine the fractions as shown in eamples a e. Eample (g) Add 52 3 + 5y2 4. LCM(3, 4) 3 4 2 5 2 3 + 5y2 4 52 3 4 4 + 5y2 4 3 3 202 2 + 5y2 2 202 +5y 2 2 Eample (h) Subtract and simplify 5 6 2 3. LCM(2 3, ) (2 3) 2 3. 5 6 5 6 2 3 2 3 2 3 2 3 5 6 2 3 2 3 2 3 5 6 (2 3) 2 3 5 6 2 + 3 2 3 3 3 2 3 Page 2 of 30
Math 02, Intermediate Algebra Section 2.3 Eample (i) Subtract and simplify + 3 + 4. LCM( + 3, + 4) ( + 3)( + 4) + 3 + 4 + 3 + 4 + 4 + 4 + 3 + 3 + 4 ( + 3)( + 4) + 3 ( + 3)( + 4) + 4 ( + 3) ( + 3)( + 4) + 4 3 ( + 3)( + 4) ( + 3)( + 4) Eample (j) Subtract and simplify 5 3 2 6 + 9. LCM( 3, 2 6 + 9) LCM( 3, ( 3)( 3)) ( 3)( 3) 5 3 2 6 + 9 5 3 3 3 2 6 + 9 5 5 2 6 + 9 5 5 2 6 + 9 4 5 2 6 + 9 2 6 + 9 Eample (k) Add and simplify + 2 + 2 5 + + 2 2 + 3 0. LCM( 2 + 2 5, 2 + 3 0) LCM (( + 5)( 3), ( + 5)( 2)) ( + 5)( 3)( 2) + 2 + 2 5 + + 2 2 + 3 0 + ( + 5)( 3) + + 2 ( + 5)( 2) + ( + 5)( 3) 2 2 + + 2 ( + 5)( 2) 3 3 ( +)( 2) ( + 5)( 3)( 2) + ( + 2)( 3) ( + 5)( 2)( 3) 2 2 ( + 5)( 3)( 2) + 2 6 ( + 5)( 2)( 2) 2 2 2 8 ( + 5)( 3)( 2) Page 3 of 30
Math 02, Intermediate Algebra Section 2.3 Simplify. 9. (a) 3 4y + 3 2y 2 9. (b) y + 2 3y 3 0. (a) 5 2 + 5 0. (b) 2 2 2. (a) 4 3 + + 3 + 2 9. (b) 2 5 + + 5 + 4 2 25 Answer: 9. (b) 3y + ; 0. (b) 3y 3 2 26 ;. (b) 2 2 + ( 5)( + 5) Page 4 of 30
Math 02, Intermediate Algebra Section 2.4 2.4 Comple Fractions A comple fraction is a fraction in which the numerator or denominator or both contain a fraction, such as We will call the primary numerator, the primary denominator, and b and d the secondary denominators. Eample (a) Simplify 3 7. 4 We can rewrite the comple fraction as a division problem, and then write the division problem as an equivalent multiplication problem: Alternatively, we could multiply the primary numerator and denominator by the least common multiple of the secondary denominators: Eample (b) Simplify We can simplify the primary numerator and denominator + 2 y 4 +. y and then proceed as. in eample (a) Alternatively, we could multiply the primary numerator and denominator by LCM(, y, 4) 4y. Page 5 of 30
Math 02, Intermediate Algebra Section 2.4 Simplify. 2. (a) 5 3 Simplify. 3. (b) 6 5 y 2. (a) 2 5 + 3 + 2 2. (b) 2 3 + a 5 a + 2 a 3. (a) + 3. (b) + 2 a 2 + 3 a Answers:. (b) y ; 2. (b) 0 2a + 3 ; 3. (b) 2 a + 2 a + 3 Page 6 of 30
Math 02, Intermediate Algebra Section 2.4 Simplify. 4. (a) + y 2 Simplify. 4. (b) a + 2 ab a a 5. (a) + 3 3 + + 3 5. (b) 2 a + 5 a a 5 4 a + 5 6. (a) 2 + + 6. (b) 5 3a a + a a Answer: 4. (b) a + 2 ; 5. (b) ab b a 2 3a 0 ; 6. (b) a 2 4 a 5 5a 2 3a 3 Page 7 of 30
Math 02, Intermediate Algebra Section 2.5 2.5 Solving Equations containing Rational Epressions Proportions A proportion is an equation stating that two ratios are equivalent and is written in the form Let s eplore the proportion Suppose we wanted to transform this equation into one without denominators. We could use the multiplication property of equality to multiply both sides of the equation by 5. We could use the multiplication property of equality to multiply both sides of the equation by 0. Notice, by the color coding here and in the original proportion, that we could have obtained this result directly by performing what is called cross-multiplying. Means and Etremes Property of Proportions For any real numbers a, b, c, and d where b 0 and d 0, a b c ad bc Etremes d Means 20 20 The product of the means of a proportion is equal to the product of its etremes. That is, by cross multiplying, equivalency is preserved. Eample (a) Solve + 6 + 7. + 6 Check: 3 Check: 2 + 7 Apply the ( + 7) 6( +) Means and 2 + 7 6 + 6 Etremes 6 6 2 Property. + 0 + 6 2 + 6 6 6 2 + 6 0 ( + 3)( 2) 0 + 3 0 or 2 0 3 3 +2 +2 + 0 3 + 0 2 3 2 The solution set is { 3, 2}. Page 8 of 30
Math 02, Intermediate Algebra Section 2.5 Eample (b) Solve 2 2 3 6. LCM(2, 3, 6) 6 Check: 3 3 2( 2) 3 2 + 4 2 + 7 7 7 2 6 By the Multiplication Property of Equality, multiply both sides of the equation by 6 to transform the equation into one with no denominators. The solution set is {3}. 3 Etraneous Solutions Eample (c) Solve + 3 2 3. Check: 3 LCM(, 3) ( 3) 3 + ( 2) 2 3 2 2 2 2 3 2 4 + 3 + 3 0 2 4 + 3 0 ( 3)( ) 3 0 or 0 +3 +3 + + 3 Check: The solution set is {}. Page 9 of 30
Math 02, Intermediate Algebra Section 2.5 Solve.. (a) 2 + 2 3 6 Solve.. (b) 2 + 3 5 3 0 2. (a) 2 5 + 3 2. (b) 3 4 + Answers:. (b) { 3 } ; 2. (b) { 4, 3} Page 20 of 30
Math 02, Intermediate Algebra Section 2.5 Solve. 3. (a) a a 2 4 + a 2 a + 4 a + 2 Solve. 3. (b) b +7 b 2 b + b 2 b 4. (a) 3 + 2 + 5 3 3 2 +4 5 4. (b) 3 4 + 2 2 4 2 9 + 2 Answers: 3. (b) { 4, 5}; 4. (b) { 9 } Page 2 of 30
Math 02, Intermediate Algebra Section 2.5 Solve. 5. (a) + 2 Solve. 5. (b) 4 + 3 3 3 3 6. (a) 5 + 2 + 2 3 3 6. (b) 6 y 2 + 7 y 8 y y 8 Answers: 5. (b) {6}; 6. (b) No solution Page 22 of 30
Math 02, Intermediate Algebra Section 2.6 2.6 Applications Eample (a) What number must be subtracted from the numerator and subtracted from the denominator of to get? Number Problems Let the number 4( ) 3(5 ) 4 4 5 3 +4 +4 4 + 0 5 + 4 5 + 5 5 0 + Check: The number is. Solve.. (a) What number must be added to both the numerator and the denominator of 3 23 to get 3? 7 Solve.. (b) What number must be subtracted from both the numerator and the denominator of 3 4 to get 5 7? 2. (a) The denominator of a fraction is 5 more than twice the fraction s numerator, and this fraction is equivalent to 3. Find the original fraction. 2. (b) The numerator of a fraction is 5 less than the fraction s denominator, and this fraction is equivalent to 2. Find the original fraction. Answers:. (b) 6; 2. (b) 5/0 Page 23 of 30
Math 02, Intermediate Algebra Section 2.6 Rate Time Problems Rate of production, rate of travel, and rate of pay are eamples of real world applications of the concept of rate. We are particularly interested in knowing unit rates, that is, a quantity (of something such as production, travel, or pay) per single unit (of something such as time). In this section, we will eamine methods of solving problems involving rates of production and rates of travel. Eample (b) Let s eplore the concept of rate: If a farmer can plow an acre in 5 hours, how many acres can the farmer plow in 0 hours? in 5 hours? in hour? The unit rate is 5 acre per hour. We can use this rate and the formula to answer the preceding questions again. Rate Time Production (Amount of work done) If a farmer can plow an acre in 5 hours, how many acres can the farmer plow in 0 hours? in 5 hours? in hour? Eample (c) Caleb can build a certain brick wall in 7 hours while Callie can build the same brick wall in 6 hours. Working together, how long will it take them to build the brick wall? Let t the time it will take working together Rate Time Production Unit Rate Time (working together) Amount of work done Caleb t t Callie t t t + t 42 ( t + t ) 42 6t + 7t 42 3t 42 t hours Total t + t 3 hours + minutes It will take them about 3 hours & 4 minutes working together. 3 hours & 4 minutes Page 24 of 30
Math 02, Intermediate Algebra Section 2.6 3. (a) A large splitter can split a cord of wood in 9 hours and a small splitter can split a cord of wood in 2 hours. How long will it take both splitters working together to split a cord of wood? Let t large small Rate Time (WT) Total Amount of work done 3. (b) A crane with a large bucket can fill a truck with gravel in 0 hours and a crane with a small bucket can fill the same truck in 6 hours. How long will it take to fill the truck if both cranes are working together? Let t large small Rate Time (WT) Total Amount of work done 4. (a) It takes Juan and Jose 0 hours to paint a house. If Juan can paint the house by himself in 30 hours. How long would it take Jose to paint the house by himself? Let t Juan Jose Rate Time (WT) Total Amount of work done 4. (b) It takes Miss Honey and Miss Peach hour working together to grade a class set of eams. If Miss Honey can grade the eams in 4 hours by herself, how long would it take Miss Peach to grade the eams by herself? Let t Miss Honey Miss Peach Rate Time (WT) Total Amount of work done Answers: 3. (b) 6 hrs & 9 mins; 4. (b) hr & 20 mins Page 25 of 30
Math 02, Intermediate Algebra Section 2.6 Eample (d) Using a certain hose, a large tank can be filled with water in 6 hours. How many large tanks can be filled Rate Time Production (Number of tanks filled) in hour? in 3 hours? in 6 hours? When the drain is open, the large tank can be completely drained in 0 hours. (Let s view this as the negative of filling the tank.) What has happened to the filling of the tank Rate Time Production (Number of tanks filled) in hour? 0 in 5 hours? in 0 hours? Eample (e) A method of cleaning a certain tank involves filling it while the drain is left open. If the tank can be filled with the drain closed in 6 hours and emptied (when it is not being filled) in 0 hours, how long will it take to fill the tank if the drain is left open during the filling? Let t the time needed to fill the tank when the drain is open. Rate Time Production Unit Rate Time (filling and draining together) Number of tanks filled Fill t t Drain t t Total t + t t + t 30 ( t + t ) 30 5t + ( 3)t 30 2t 30 t 5 hours It will take 5 hours to fill the tank. Page 26 of 30
Math 02, Intermediate Algebra Section 2.6 5. (a) A water tower can be filled in 24 hours. The same water tower can be drained in 40 hours. The tower is designed to send its water to an irrigation system while it is being filled. How long will it take to fill the water tower if it is being drained at the same time it is being filled? Let t Fill Drain Rate Time (WT) Total Number of tanks filled 5. (b) A water tank similar to the one in 5. (a) can be filled in 20 hours and drained in 30 hours. How long will it take to fill the tank if it is being drained at the same time it is being filled? Let t Fill Drain Rate Time (WT) Total Number of tanks filled Answer: 5. (b) 60 hours Page 27 of 30
Math 02, Intermediate Algebra Section 2.6 Eample (f) A car traveling 60 miles per hour will travel how far in Rate Time Distance in hour? in 2 hours? in 0 hours? Eample (g) A boat can travel 8 miles downstream in the same time it takes to travel 2 miles upstream in the same river. If the boat s speed is 45 mph in still water, what is the speed of the current? Let the speed of the current. Rate Time Distance Rate Time Distance upstream 45 2 downstream 45 + 8 8(45 ) 2(45 + ) 80 8 540 + 2 80 540 + 30 270 30 9 The rate of the current is 9 mph. Page 28 of 30
Math 02, Intermediate Algebra Section 2.6 6. (a) A cyclist travels 20 miles westbound with a tail wind in the same amount of time it takes the cyclist to travel 0 miles eastbound on the same road with a head wind. If the cyclist can travel at 8 mph with no wind, what is the speed of the wind? Let w 6. (b) A plane flies 300 miles westbound with a tail wind in the same amount of time it takes the plane to fly 200 miles eastbound on the same route with a head wind. If the plane can travel at 80 mph with no wind, what is the speed of the wind? Let w Rate Time Distance eastbound westbound eastbound westbound Rate Time Distance Answer: 6. (b) 36 mph Page 29 of 30
Math 02, Intermediate Algebra Section 2.6 7. (a) A boat travels upstream 60 miles and then back again in a total of 24 hours. If the speed of the current is 5 mph, find the speed of the boat in still water. Let upstream downstream Rate Time Distance Total 7. (b) A cyclist travels into a head wind 60 miles and then back again to his starting point, enjoying a tail wind on the return. The round trip took him 8 hours. If the speed of the wind was 0 mph, find the speed the cyclist travels when there is no wind. Let head wind tail wind Rate Time Distance Total Answer: 7. (b) 20 mph Page 30 of 30