3.6 Equations and Fractions

Transcription

1 .6 Equations and Fractions EQUATIONS INVOLVING THE MULTIPLICATION OF FRACTIONS In Section 1.5 you learned how to solve equations involving multiplication. Let s take a look at an example as a reminder. Example 1: Divide both sides of each equation by the coefficient so that it isolates the variable. a) N x 6 = 18. b) 8 x N = 40. Answer: a) N x 6 = 18 To isolate the variable we need to divide both sides by the coefficient, 6. N x 6 6 = 18 6 Apply division to both sides. N x 1 = Since 1 is the identity for multiplication, N x 1 becomes just N. N = We now know the value of the variable; it is no longer an unknown value. b) 8 x N = 40 This time, we need to rewrite the left side so that N is written first; in other words, we need to commute the left side. N x 8 = 40 N x 8 8 = 40 8 N x 1 = 5 N = 5 Now we re in a position to divide both side by the coefficient, 8. Apply division to both sides. Since 1 is the identity for multiplication, N x 1 becomes just N. So, N = 5 is the solution. Equations and Fractions page.6 1

2 YTI #1 Divide both sides by the coefficient of the equation so that it isolates the variable. (You might not need all of the lines below.) a) N x 5 = 0 b) x N = 21 Notice that, in isolating the variable, we were able to toward the very end of the process multiply N by just 1; after that step, N was isolated. Of course, the way we were able to get a factor of 1 was to divide by the coefficient of N. If we are, instead, presented with an equation in which the coefficient is a fraction, the process for solving could be exactly the same: divide both sides by the coefficient, as in Example 2. Example 2: Isolate the variable by dividing both sides by the coefficient. N x 2 = 10 Answer: N x 2 = 10 To isolate the variable we need to divide both N x 2 2 = 10 2 N x 2 x 2 = 10 x 2 N x 2 x 2 = 10 1 x 2 sides by the coefficient, 2. Apply division to both sides by using invert and multiply. Multiply the fractions together. N x 6 6 = 0 2 N x 1 = 15 N = 15 Simplify each fraction. Again we get 1, the identity for multiplication; N x 1 becomes just N. So, N = 15 is the solution. Equations and Fractions page.6 2

3 A quicker way to isolate the variable instead of dividing by the fraction would be to multiply by the fraction s reciprocal. Recall from Sec..5 that the product of a fraction and its reciprocal is always 1. Example : Isolate the variable by multiplying both sides by the reciprocal of the coefficient. a) N x 2 = 10. b) 8 5 x N = 16 Procedure: Multiply both sides by the reciprocal of the coefficient of N. Answer: a) N x 2 = 10 2 is the reciprocal of 2, so multiply both sides by 2. N x 2 x 2 = 10 x 2 Make 10 into a fraction: 10 1 N 2 x x 2 = 10 1 x 2 Multiply the fractions together. N x 6 6 = 0 2 Simplify each fraction. N x 1 = 15 Again we get 1 is the identity for multiplication; N x 1 becomes just N. N = 15 So, N = 15 is the solution. b) 8 5 x N = 16 This time, we can commute the left side so that N is written first. both N x 8 5 = 16 The reciprocal of 8 5 is 5 8 sides by 5 8., so multiply Equations and Fractions page.6

4 N x 8 5 x 5 8 = 16 x 5 8 N 40 x 40 = 16 5 x 8 At this point we can cross-cancel the extremes by as factor of 8. 2 N x 1 = 5 x 1 N = 10 So, N = 10 is the solution. YTI #2 Isolate the variable by multiplying both sides by the reciprocal of the coefficient. Use Example as a guide. a) N x 5 2 = 20 b) 4 x N = 9 c) N x 8 = 12 5 d) 1 2 x N = 5 9 e) N x 4 7 = 8 21 f) 4 x N = 6 Equations and Fractions page.6 4

5 You have seen two different types of coefficients in these equations. When the coefficient is a whole number, it is easier to divide; when the coefficient is a fraction, it is easier to multiply by its reciprocal. APPLICATIONS In Section 2.7 you were introduced to some situations that require using multiplication. At that time, we explored the formula (Number of Same Parts) x Part = Whole. or Repetitions x Part = Whole. Some of the situations you ll see here will be dividing something the whole into equal parts. Also remember that the whole and the parts are always of equal units of measure. For example, a dozen cookies can be divided easily between 4 children. The question might be, How many cookies will each child get? It s like asking for the number of cookies per child. You probably know that each child will receive cookies, but we ll use this simple situation to illustrate the proper way to approach the problem. Notice that the word dividing is used. We will still set up the equation using multiplication, but in order to isolate the variable we will need to divide to find the answer. Also, remember that the part can be identified by the words each and per. WHEN THE PART IS UNKNOWN Example 4: Procedure: If a dozen cookies is to be divided evenly among four children, how many will each child receive? In this case, the whole is the total number of cookies. The part, the unknown, is how many cookies each child will receive. (Notice that both the whole and the part are measured in cookies.) Since the part is shared (divided evenly) among 4 children, the number of times the part is repeated is 4. Equations and Fractions page.6 5

6 Answer Heading: Let N = the number of cookies each child receives. 4 x N = 12 Commute and divide. N x 4 4 = 12 4 N x 1 = N = So, Each child will receive cookies. It may seem that the last example is a bit too easy, especially since it doesn t involve dividing fractions. Consider this next example, though; similar, but different. Example 5: Andy has a strip of wood that is 9 4 inches wide and needs to be divided into smaller strips of equal width. How wide will each smaller strip be? Procedure: Picture yourself in the situation. You ve got a strip of wood that is 9 4 inches wide. You are going to cut it (evenly) into smaller pieces. Here, we re given a width 4 inches and we re asked to find a width. This must mean that the widths (in inches) are the part and the whole. The larger (original) piece must be the whole (total width of 9 4 inches); Each smaller strip is a part, the width is unknown, and the part is repeated times (there are to be smaller strips) Answer Heading: Let N = the width of each smaller strip. x N = 9 4 Commute and divide by the coefficient. Equations and Fractions page.6 6

7 N x = 9 4 Change this to invert and multiply N x 1 x 1 = 9 4 x 1 by the reciprocal of the coefficient. N x 1 = N = 9 12 Reduce by a factor of. 4 So, Each strip will be 4 inches wide. Example 6: A mother has a sick child and needs to divide up the liquid cough medicine into 6 equal doses. If she has 4 be in each dose? of a cup of cough syrup, how much cough syrup will Procedure: Here we have 4 cup of cough syrup, and we re asked to find the amount of cough syrup (number of cups) in each dose. Cups is the common measure between the part and the whole. So, the whole is the 4 cup of cough medicine. The part (the amount of cough syrup in each dose) is unknown, and it is repeated 6 times (there are to be 6 doses) Answer: Heading: Let N = the amount (cups) of cough syrup in each dose. 6 x N = N x 6 6 = Commute and divide by the coefficient. 6 Change this to invert and multiply by the reciprocal of the coefficient. Equations and Fractions page.6 7

8 N x 6 1 x 1 6 = 4 x 1 6 N x 1 = 24 Reduce by a factor of. N = 1 8 So, Each dose will be 1 8 cup. Solve each of the following situation problems. Follow the outline one step at a time. YTI # Gail has a stack of flyers that need to be folded, and she asks Barbara and Jan to help her. They decide to measure the height (thickness) of the stack; it turns out a) Identify the whole: to be 9 4 inches high. How high (how thick) should each person s stack be if they divide it evenly ways? b) Identify the part: c) Identify the number of repetitions: d) Write the heading e) Write and solve the equation: Equations and Fractions page.6 8

9 f) Write a sentence answering the question. In the next few exercises, use the same outline, as above, as needed. YTI #4 Hank works for a landscaper. Their most recent job requires them to prepare, plant and cultivate 2 5 of an acre of land. The owner of the company is assigning Hank and others (4 workers in all) to the job and wants them to divide the land up evenly amongst themselves. What part of an acre will each employee be responsible for? Write a sentence answering the question. YTI #5 A huge soccer field is 8 of a mile long. Sandra works for the city s Parks and Recreation Department, and it is her job to mark off the fields appropriately. She needs to mark off 6 fields (of equal length) before the start of the soccer season. How long will each field be? Equations and Fractions page.6 9

10 Write a sentence answering the question. WHEN THE NUMBER OF REPETITIONS IS UNKNOWN Every situation presented in this section thus far has involved the formula (Number of Same Parts) x Part = Whole. or Repetitions x Part = Whole. To this point, the part has been the unknown. We have known the whole and the number of repetitions and needed to find the part, so we say, Let N = the part. In other situations, we might know the whole and the size of the part we wish to obtain, but we might not know the number of repetitions of the part that will be needed. Consider the following Equations and Fractions page.6 10

11 Example 7: Andy needs small strips of wood that are only 8 inches wide. How many of these strips of wood can he get from a board that is 15 4 inches wide? Procedure: Put yourself in Andy s position. He has a board that is 15 4 inches wide and needs to cut it up into smaller strips. The board is the whole 15 4 inches and the width of each strip 8 inches is the part. Both of these are in inches and are known values, so the unknown value is the number of repetitions, which is mentioned in the question. Answer: Heading: Let N = the number of strips. N x 8 = 15 4 Multiply both sides by the reciprocal of the coefficient. N x 8 x 8 = 15 4 x 8 Simplify the right side by cross-canceling; reduce the extremes by a factor of and the means by a factor of 4. N x 1 = 5 1 x 2 1 N = 10 1 = 10 So, Andy can get 10 smaller strips out of the board. Equations and Fractions page.6 11

12 YTI #6 At a bicycle making factory, one bicycle can be assembled in 2 of an hour. The factory is in operation 24 hours a day. How many bicycles can the factory make in 24 hours? a) Identify the whole: b) Identify the part: c) Identify the number of repetitions: d) Write the heading e) Write and solve the equation: f) Write a sentence answering the question. In the next two exercises, use the same outline, as above, as needed. YTI #7 Marie works in a candy shop. The shop advertises home made candies, and one of their specialties is a brick of fudge. Marie is given a 6-inch brick of fudge, and she needs to cut it into smaller bricks each having a width of 4 How many smaller bricks of fudge can she cut from the 6-inch brick? inches. Equations and Fractions page.6 12

13 Write a sentence answering the question. YTI #8 At a gift wrapping booth, each package gets a ribbon bow made of 4 foot of ribbon. If a spool has 24 feet of ribbon on it, how many bows can be made form one spool? Write a sentence answering the question. WHEN THE WHOLE IS UNKNOWN There are some situations where the whole is unknown. Of course, when that occurs, the variable, N, represents the whole. Example 8: At a bookstore, Devra needs to place 2 copies of the new best-selling paperback book tightly together on a shelf. If each book is 5 4 of the shelf will the 2 books take up? inches thick, how many inches Procedure: As Devra is placing the books on the shelf, the part, the width (thickness) of each is repeated over and over, 2 times in all. Taken together, they will add up to the whole, which is unknown. (The common unit of measure is inches, so the part and the whole will be in inches.) Equations and Fractions page.6 1

14 Answer: Heading: Let N = the total number of inches. The part is 5 4 inches, and it is repeated 2 times; so the equation is 2 x 5 4 = N x 4 = N Reduce the extremes by a factor of x 5 1 = N 40 = N So, The 2 books will take up 40 inches on the shelf. YTI #9 Katie uses 5 6 yards of fabric for every doll dress she makes. How many yards of fabric is needed to make 12 doll dresses? a) Identify the whole: b) Identify the part: c) Identify the number of repetitions: d) Write the heading e) Write and solve the equation: f) Write a sentence answering the question. Equations and Fractions page.6 14

15 In the next two exercises, use the same outline, as above, as needed. YTI #10 Nate s dog, Banjo, eats 2 pound of dog food each day. How many pounds of dog food does Banjo eat each month (0 days)? Write a sentence answering the question. YTI #11 Tim is a janitor for an elementary school. It takes him 1 4 hour to clean each classroom at the end of the school day. How long (how many hours) should it take Tim to clean all 28 classrooms? Write a sentence answering the question. Equations and Fractions page.6 15

16 Answers to each You try it Exercise Section.6 YTI #1: a) N x 5 = 0 b) x N = 21 N = 6 N = 7 So, N = 6 is the solution So, N = 7 is the solution YTI #2: a) N x 5 2 = 20 b) 4 x N = 9 N = 8 N = 12 So, N = 8 is the solution So, N = 12 is the solution c) N x 8 = 12 5 d) 1 2 x N = 5 9 N = 2 5 N = 10 9 So, N = 2 5 is the solution So, N = 10 9 is the solution e) N x 4 7 = 8 21 f) 4 x N = 6 N = 2 N = 8 So, N = 2 is the solution So, N = 8 is the solution YTI #: a) Whole: 9 4 YTI #4: Whole: 2 5 b) Part: N Part: N c) Repetitions Repetitions 4 Equations and Fractions page.6 16

17 d) Heading: Let N = the thickness Heading: Let N = the part of e) x N = of each person s stack N = an acre each is responsible for x N = = 4 N = 2 20 = 1 10 So, Each person s stack should be 4 inches high. So, Each will be responsible for 1 10 of an acre. YTI #5: Whole: 8 YTI #6: Whole: 24 Part: N Part: 2 Repetitions 6 Repetitions N Heading: Let N = the part of an acre each is responsible for 6 x N = N = Heading: Let N = the number of bicycles 8 N 2 x = N = 6 So, Each field will be 1 16 of a mile long. So, They can make 6 bicycles in 24 hours. YTI #7: Whole: 6 YTI #8: Whole: 24 Part: 4 Part: 4 Equations and Fractions page.6 17

18 Repetitions N Repetitions N Heading: Let N = the number of smaller bricks of fudge Heading: Let N = the number of bows N x 4 = 6 N x 4 = 24 N = 8 N = 2 So, She can cut 8 smaller bricks of fudge. So, 2 bows can be made from one spool. YTI #9: Whole: N YTI #10: Whole: N Part: 5 6 Part: 2 Repetitions 12 Repetitions 0 Heading: Let N = the number of yards of fabric Heading: Let N = the number of yards of fabric 12 5 x 6 = N 0 2 x = N N = 10 N = 20 So, Katie needs 10 yards of fabric. So, Banjo eats 20 pounds of dog food each month. YTI #11: Whole: N Part: 1 4 Repetitions 28 Heading: Let N = the number of yards of fabric 28 x 1 4 = N N = 7 So, It should take Tim 7 hours to clean the rooms. Equations and Fractions page.6 18

19 Section.6 Focus Exercises 1. Isolate the variable by multiplying both sides by the reciprocal of the coefficient. (You might not need all of the lines below.) Use Example as a guide. a) N x 11 5 = 22 b) N x 4 15 = 16 9 c) 8 9 x N = 24 d) 10 7 x N = For each application, use the outline described in this section. Be sure to write out the heading, set up and solve the equation, and write a sentence answering the question. 2. Ramon has written a book that contains colorful graphs and charts. His computer printer can print out the entire book in 2 hour. He needs to print out 6 copies for his team of editors. How long will it take to print out 6 copies of his book? Equations and Fractions page.6 19

20 Sentence:. The Music Room gives guitar lessons 12 hours each day. Each lesson lasts 4 lessons are there each day? hour. How many Sentence: 4. At an amusement park, employees rotate from location to location so that they have a variety of duties to do each day. One location, a hot dog cart, is open for eight hours. If 8 different employees share the 6 hours that the cart is open, how much time does each employee spend there? Equations and Fractions page.6 20

21 Sentence: Equations and Fractions page.6 21