Numbers in the Real World Copyright © 2011 Pearson Education, Inc.

Uses and Abuses of Percentages Unit 3A Uses and Abuses of Percentages Copyright © 2011 Pearson Education, Inc.

Brief Review of Percentages p.129 per cent means per 100 P% = P/100 Convert a percentage to a common fraction- replace the % with division by 100 Convert a percentage to a decimal- drop the % a move decimal 2 places to left Convert decimal to percentage- move the decimal 2 places to the right and add % Convert a common fraction to a percentage- convert fraction to decimal, then decimal to percentage Copyright © 2011 Pearson Education, Inc.

Class Notes (13) Express the following numbers in 3 forms: reduced fraction, decimal, and percentage 1. 2/5 2. 30% 3. .20 4. .85 5. 150% 6. 2/3 7. 4/9 8. 1.25 9. 5/8 10. 44% 11. 69% 12. 4.25 Copyright © 2011 Pearson Education, Inc.

Three Ways of Using Percentages As fractions: 15% of the of the 850 students in a school were absent. To describe change: The price of a stock increased 75% from $50 per share. For comparisons: A Mercedes costs 25% more than a Lexus. Copyright © 2011 Pearson Education, Inc.

Class Notes (13) Example 1 Newspaper Survey A newspaper reports that 64% of 1069 people surveyed said that the president is doing a good job. 13. How many said the president is doing a good job? (p.129) Copyright © 2011 Pearson Education, Inc.

Class Notes (14-16) Percentages as Fractions 14. 28 pounds of recyclable trash in a barrel of 52 pounds. 15. 12.0 million metric tons of beef produced annually in the US out of 65.1 million metric tons of beef produced annually worldwide. 16. The median salary for US men in 2007 was $33,196 and the median salary for US women in 2007 was $20,922. Copyright © 2011 Pearson Education, Inc.

Absolute and Relative Change The absolute change describes the actual increase or decrease from a reference value (starting number) to a new value: absolute change = new value – reference value The relative change is a fraction that describes the size of the absolute change in comparison to the reference value: Copyright © 2011 Pearson Education, Inc.

Absolute vs. Relative Change Example: A diversified portfolio grows from $1,500 to $2,250. absolute change = new value – reference value = $2,250 – $1,500 = $750 Present several examples from contemporary issues in the media that could serve as concrete reminders of the difference between the two changes. relative change = = $750 / $1,500 = 0.5 = 50% Copyright © 2011 Pearson Education, Inc.

Class Notes (17) Example 2 Stock Price Rise During a 6 month period, Nokia’s stock doubled in price from $10 to $20. 17. What were the absolute and relative changes in the stock price? (p.130) Copyright © 2011 Pearson Education, Inc.

Class Notes (18) Salary Comparisons Clint’s salary increased from $20,000 to 28,000 over a three-year period. Helen’s salary increased from $25,000 to $35,000 over the same period. 18. Who’s salary increased more in absolute terms? In relative terms? Explain. Copyright © 2011 Pearson Education, Inc.

Class Notes (19) Example 3 World Population Growth World population was 2.6 billion in 1950 and 6.0 billion in 2000. 19. Describe the absolute and relative change in world population from 1950 – 2000. (p.131) Copyright © 2011 Pearson Education, Inc.

Class Notes (20-22) Percentage Change 20. The population of the US increased from 249 million in 1990 to 308 million in 2010. 21. The congressional delegation of California increased from 30 in 1950 to 53 in 2010. 22. The number of daily newspapers in the US was 2226 in 1990 and 1420 in 2010. Copyright © 2011 Pearson Education, Inc.

Class Notes (23) Example 4 Depreciating a Computer You bought a computer three years ago for $1000. Today it is worth only $300. 23. Describe the absolute and relative change in the computer’s value. (p.131) Copyright © 2011 Pearson Education, Inc.

Class Notes (24-25) Percentage Change 24. The number of daily newspapers in the US was 2226 in 1900 and 1420 in 2010. 25. The number of music CDs shipped in the US decreased from 942 million in 2000 to 511 million in 2008. Copyright © 2011 Pearson Education, Inc.

Absolute and Relative Difference The absolute difference is the actual difference between the compared value and the reference value: absolute difference = compared value – reference value The relative difference describes the size of the absolute difference as a fraction of the reference value: Copyright © 2011 Pearson Education, Inc.

Class Notes (26)Example 5 Pay Comparison Average pay for full time wage earners varies from state to state. Recent data (2006 data, compiled in 2009) showed that New York ranked first in average pay at $55,479 per person. S. Dakota had the lowest average pay, at $30,291 per person. 26. Compare average pay in S. Dakota to that in New York in both absolute and relative terms. (p.133) Copyright © 2011 Pearson Education, Inc.

Class Notes (27-28) 27. The gestation period of humans (266 days) is ________% longer than the gestation period of grizzly bears (220 days). 28. The 2009 life expectancy in Canada (81.2 years) is __________% greater than the 2009 life expectancy in Russian (65.9 years). Copyright © 2011 Pearson Education, Inc.

Of versus More Than (or Less Than) If the compared value is P% more than the reference value, it is (100 + P)% of the reference value. If the compared value is P% less than the reference value, it is (100 - P)% of the reference value. Walk through several careful examples from the reading or those you may find in the media to demonstrate the correct use of %s. Copyright © 2011 Pearson Education, Inc.

Class Notes (29-31) Example 6 Salary Difference Carol earns 50% more than William. 29. How many times larger is her income than his? (p.134) 30. Will is 22% taller than Wanda, so Will’s height is _____% of Wanda’s height. 31. The area of Norway is 24% more than the area of Colorado, so Norway’s area is ______% of Colorado’s area. Copyright © 2011 Pearson Education, Inc.

Class Notes (32-34) Example 7 Prices and Sales Sale! A store is having a 25% off sale. 32. How does an item’s sale price compare to its original price? (p.134) 33. The wholesale price of a TV is 40% less than the retail price. Therefore, the wholesale price is _______ times the retail price. 34. A store is having a 50% off sale. Therefore, the original price of an item is _____ times as much as the sale price. Copyright © 2011 Pearson Education, Inc.

Percentages of Percentages When a change or difference is expressed in percentage points, assume it is an absolute change or difference. with the % sign or the word percent, it is a relative change or difference. Example: If a bank increases its interest rate from 4% to 5%, the interest rate increased by 1 percentage point. Copyright © 2011 Pearson Education, Inc.

Class Notes (35-37) Example 8 Newspaper Readership Declines The percentage of adults who report reading a daily newspaper fell from about 78% in 1980 to 49% in 2009. 35. Describe this change in newspaper readership. (p.135) 36. The annual interest rate for Jack’s savings account increased from 2.3% to 2.8% 37. The percentage of Republicans in the House of Representatives decreased from 53.3% in 2007 to 46.4% in 2009. Copyright © 2011 Pearson Education, Inc.

Class Notes (38-39) Care in Wording Assume that 40% of the registered voters in Carson City are Republicans. Read the following questions carefully, and give the most appropriate answers. The percentage of voters registered as Republicans is 25% higher in Freetown than in Carson City. 38. What percentage of the registered voters in Freetown are Republicans? The percentage of voters registered as Republicans is 25 percentage points higher in Freetown than in Carson City. 39. What percentage of the registered voters in Freetown are Republicans? Copyright © 2011 Pearson Education, Inc.

Solving Percentage Problems If the compared value is P% more than the reference value, then and If the compared value is less than the reference value, use (100 – P) instead of (100 + P) in the above calculations. Copyright © 2011 Pearson Education, Inc.

Solving Percentage Problems You purchase a shirt with a labeled (pre-tax) price of $21. The local sales tax rate is 6%. What is your final cost (including tax)? final cost = labeled price + (6% of labeled price) = (100 + 6)%  labeled price = 106%  $21 = 1.06  $21 = $22.26 Copyright © 2011 Pearson Education, Inc.

Class Notes (40-41) Example 10 Tax Calculations You purchase a shirt with a labeled (pre-tax) price of $17. The local sales tax rate is 5%. 40. What is your final cost (including tax)? Your receipt shows that you paid $19.26 for a Blu-ray disc, tax included. 41. The local sales tax rate is 7%. What was the labeled (pre-tax) price of the disc? Copyright © 2011 Pearson Education, Inc.

Class Notes (42-44) You purchase a bicycle with a retail (pre-tax) price of $760. The local sales tax rate is 7.6% 42. What is the final cost? The final cost of your new shoes is $107.69. The local sales tax rate is 6.2%. 43. What was the retail (pre-tax) price? The 2410 wome4n undergraduates at the college comprise 54% of all undergraduates. 44. How many undergraduates attend the college? Copyright © 2011 Pearson Education, Inc.

Example 11 Up 44%, to 10.4% Consider the following statement from the introduction to this unit: The rate of smoking for eighth graders is up 44 percent, to 10.4 percent. What was the previous smoling rate for eight graders. Copyright © 2011 Pearson Education, Inc.

Abuses of Percentages Beware of Shifting Reference Values A 10% pay cut is followed by a 10% pay raise. Copyright © 2011 Pearson Education, Inc.

Example 12 Shifting Investment Value A stockbroker offers the following defense to angry investors: “I admit that the value of your investments fell 60% of its value during my first year on the job. This year, however, their value has increased by 75%, so you are now 15% ahead. Evaluate the stockbroker’s defense. Copyright © 2011 Pearson Education, Inc.

Example 13 Tax Cuts A politician promises, “if elected, I will cut your taxes by 20%, for each of the first three years of my term, for a total cut of 60%. Evaluate the promise Copyright © 2011 Pearson Education, Inc.

Decrease caloric intake by 150% to lose weight. Less than Nothing Decrease caloric intake by 150% to lose weight. We often see numbers that represent large “more than” percentages. However in most cases it is not possible to have a “less than” percentage that is greater than 100%. Copyright © 2011 Pearson Education, Inc.

Example 14 Impossible Sale A store advertises that it will take “150% off” the price of all merchandise. aWhat should happen when you go to the counter to buy a $500 item? Copyright © 2011 Pearson Education, Inc.

Don’t Average Percentages If 70% of the boys and 60% of the girls in a class voted to go to a water park, then 65% of the students in the class voted to go to the water park. It is tempting to say yes, but it would be wrong. Unless both happened to have the same number of kids. Copyright © 2011 Pearson Education, Inc.

Example 15 Batting Average In baseball, a player’s batting average represents the percentage of at-bats in which he got a hit. For example, a batting average of .350 means the player got a hit 35% of the times he batted. Suppose a player had a batting average of .200 and .400. Can we conclude that his batting average for the entire season was .300 (the average of .200 and .400)? Why or why not? Give an example that illustrates your reasoning. Copyright © 2011 Pearson Education, Inc.

3A Homework Class Notes Quick Quiz p. 140:1-10 Exercises p.141 1-16 1 world 108. Percentages 109. Percentage Change 110. Abuse of Percentages Copyright © 2011 Pearson Education, Inc.