Exercise numbers appearing in color are answered in the Selected Answers appendix.
SEMESTER GPA CREDITS WEIGHT NUMERICAL EQUIVALENT
1 2 3
Total:
3. In your first semester in college, you took 13 credit hours and earned a GPA of 2.13. In your second semester, your GPA of 2.34 was based on 12 credit hours. You calculated the third semester’s GPA in Problem 8.
a. Explain why the calculation of your overall GPA for the three semesters requires a weighted average.
b. Calculate your overall GPA for the three semesters.
4. You are concerned about passing your economics class with a C– (70) average. Your grade is determined by averaging your exam scores. So far, you have scores of 78, 66, 87, and 59 on four exams. Each exam is based on 100 points. Your economics teacher uses the simple average method to determine your average.
a. What is your current average for the four exams?
b. What is the lowest score you could achieve on the fifth exam to have at least a 70 average?
5. Suppose you took 15 credit hours last semester. You earned an A– in English (3 hours), a B in mathematics (4 hours), a C+ in chemistry (3 hours), a B+ in health (2 hours), and a B– in history (3 hours). Calculate your GPA for the semester.
COURSE
LETTER GRADE
NUMERICAL EQUIVALENT
CREDIT
HOURS WEIGHT
WEIGHT ⴛ NUM. EQUIV.
6. Your history professor discovers an error in his calculation of your grade from last
semester. Your newly computed history grade is a B+. Use this new grade and the information in Exercise 5 to recalculate your GPA.
7. In baseball, weighted averages may lead to surprising results. For example, this happened in 1995 and 1996 in the comparison of batting averages for Derek Jeter of the New York Yankees and David Justice of the Atlanta Braves. In 1995, Derek Jeter made 12 hits and each hit that he made was weighted because he went up to bat 48 times in 1995. Therefore, his batting average
was to the nearest thousandth.
a. In 1995, David Justice of the Atlanta Braves went up to bat 411 times, so each of his 1995 hits was weighted He made 104 hits. Calculate his batting average and record your answer to the nearest thousandth in the table in part c.
b. In the 1996 baseball season, Jeter made 183 hits in 582 times at bat. Justice made 45 hits in 140 at bats. Calculate each of their batting averages for 1996 and record in the table.
c. Other statistics that could be of interest to ballplayers, managers, team owners, and fans would be batting averages over two years, over three years, over entire careers. For example, over the 2-year period, 1995 and 1996, Justice had hits in at bats. Therefore, his batting average combining the two years is Similarly, calculate Jeter’s combined batting average combining the 2 years. Record your result in the table.
1
551 # 149 = .270.
411 + 140 = 551 104 + 45 = 149
1 411. 1
48 # 12 = 12 , 48 = .250 1
48
d. According to the statistics in the preceding table, who was the better hitter in 1995? In 1996?
Give a reason for each answer.
e. Is David Justice a better hitter according to the combined 1995–1996 baseball seasons?
BATTING AVERAGE
1995 1996 1995 & 1996 COMBINED DEREK JETER .250
DAVID JUSTICE .270
f. As it turned out, these contradictory results continued for Jeter and Justice into the 1997 baseball season. In 1997, Jeter had 190 hits in 654 at bats; Justice hit 163 times in 495 at bats. Calculate their batting averages to the nearest thousandth and record in the following table.
h. What conclusions would you draw from the results you recorded in part g?
You may want to research these curious results further to find out what else baseball followers and statisticians have to say about this result (which is known as the Simpson-Yule Paradox in statistics).
BATTING AVERAGE FOR COMBINED DATA FOR 1995 THROUGH 1997 TOTAL HITS TOTAL AT BATS BATTING AVERAGE DEREK JETER 12 + 183 + 190 = 385 48 + 582 + 654 = 1284 385>1284 = .300
DAVID JUSTICE 104 + 45 + 163 = 312 411 + 140 + 495 = 1046 312>1046 = .298
BATTING AVERAGES 1995 1996 1997 DEREK JETER .250 .314
DAVID JUSTICE .253 .321
g. Use the appropriate data from parts a, b, and c to determine the total number of hits and at bats for each player. Then use those totals to determine the batting averages over the three-year period. Record your results here.
1. To add or subtract fractions, you must write them in equivalent form with common denomina-tors. However, to multiply or divide fractions, you do not need a common denominator. Why is this reasonable?
2. The operation of division can be viewed from several different points of view. For example, has at least two meanings:
• Write 24 as the sum of some number of 3s.
• Divide 24 into three equal-sized parts, whose size you must determine.
These interpretations can be applied to fractions as well as to whole numbers.
a. Calculate by answering this question: 2 can be written as the sum of how many ?
b. Calculate by answering this question: If you divide into two equal parts, how large is each part?
c. Do your answers to parts a and b agree with the results you would obtain by using the proce-dures for dividing fractions reviewed in this cluster? Explain.
1 5 1
5 , 2
1 2s 2 , 12
24 , 3
Cluster 2 What Have I Learned?
Exercise numbers appearing in color are answered in the Selected Answers appendix.
Cluster 2 How Can I Practice?
Exercise numbers appearing in color are answered in the Selected Answers appendix.
1. You are in a golf tournament. There is a prize for the person who drives the ball closest to the green on the sixth hole. You drive the ball to within 4 feet inches of the hole and your nearest competitor is 4 feet inches from the hole. By how many inches do you win?
2. One of your jobs as the assistant to a weather forecaster is to determine the average thickness of the ice in a bay on the St. Lawrence River. Ice fishermen use this report to determine if the ice is safe for fishing. You must chop holes in five different areas, measure the thickness of the ice, and take the average. During the first week in January, you record the following measurements:
4, and inches. What do you report as the average thickness of the ice in this area?
3. You and two others in your family will divide 120 shares of a computer stock left by a relative who died. The stock is worth per share. If you decide to sell your portion of the stock, how much money will you receive?
4. You are about to purchase a rug for your college dorm room. The rug’s length is perfect for your room. The width of the rug you want to purchase is feet. If you center the rug in the middle of your room, which is 10 feet wide, how much of the floor will show on each side of the rug?
5. A plumber has feet of plastic pipe. She uses feet for the sink line and feet for the wash-ing machine. She needs approximately feet for a disposal. Does she have enough pipe left for a disposal?
6. Perform the following operations. Write the result in simplest terms or as a mixed number.
a. 5
e. 1
7. At the end of the semester, the bookstore will buy back books that will be used again in courses the next semester. Usually, the store will give you one-sixth of the original cost of the book. If you spend $243 on books this semester and the bookstore will buy back all your books, how much money can you expect to receive?
8. a. The driving distance between Buffalo, New York, and Orlando, Florida, is approximately 1178 miles. If your average speed is 58.5 miles per hour, calculate the total driving time.
b. The driving distance between Erie, Pennsylvania, and Daytona Beach, Florida, is approximately 1048 miles. If your average speed is 68.2 miles per hour, calculate the total driving time. (Round to the nearest tenths place.)
c. If you need to make the trip in part b in 14 hours, calculate the average speed needed.
Use .r = d t
Cluster 3 Comparisons and Proportional Reasoning
During the 2008–2009 National Basketball Association (NBA) season, LeBron James of the Cleveland Cavaliers, Kobe Bryant of the Los Angeles Lakers, Dwayne Wade of the Miami Heat, and Carmelo Anthony of the Denver Nuggets were among the NBA’s highest scorers.
The statistics in the table represent each player’s 3-point field goal totals for the entire season.
Activity 1.6
2. Write ratios in fraction, decimal, and percent formats.
3. Determine equivalence of ratios.
PLAYER
NUMBER OF 3-POINT FIELD GOALS MADE
NUMBER OF 3-POINT FIELD GOALS ATTEMPTED
James 132 384
Bryant 118 336
Wade 88 278
Anthony 63 170
1. Using only the data in column 2, Number of 3-Point Field Goals Made, rank the players from best to worst according to their field goal performance.
2. You can also rank the players by using the data from both column 2 and column 3. For example, LeBron James made 132 3-point field goals out of the 384 he attempted. The 132 successful baskets can be compared to the 384 attempts by dividing 132 by 384.
You can represent that comparison numerically as the fraction or, equivalently, as the decimal 0.344 (rounded to thousandths). Complete the following table, and use your re-sults to determine another ranking of the four players from best to worst performance.
132
James 132 384 132 out of 384 132
384 0.344
Bryant Wade Anthony