1. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Section 4.4 Using Percentages

2. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Objectives o Identify and calculate percentage increase and percentage decrease o Identify common mistakes with percentages

3. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Service Tipping In most countries, tipping your server is expected. The U.S. has the highest expectations for tipping of any other country at 15-20%. Suppose you want to always tip 20%. The easiest way to quickly calculate the tip amount is to start with 10% of the bill and then double it. Remember from earlier math courses that to find 10% of a number, you move the decimal point one place to the left.

4. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Service Tipping (cont.) For instance, 10% of $25.00 = $2.50 10% of $349.00 = $34.90 10% of 1275.00 = $127.50 Once you know 10% of the total amount, you can find twenty percent by doubling the amount. So we have 20% of $25.00 = 2($2.50) = $5.00 20% of $349.00 = 2($34.90) = $69.80 20% of 1275.00 = 2($127.50) = $255.00

5. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Service Tipping (cont.) The last bill is quite large, so 20% seems rather much for a tip. Instead, you decide to tip only 15%. No problem. You still know your base amount of 10%, so to find 15% we need to add half that amount again. (10%) (5%)

6. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #1 Suppose your lunch bill is $9.84. Approximately how much would a 15% tip be? 20%? Answer: $1.48, $1.97

7. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Sales Tax Sales tax on goods in the United States is not typically included in the advertised sticker price. Instead, it is usually added on at the checkout register. Sales tax typically ranges from about 3% to 7% depending on the state. One way to find the price including tax is to calculate the tax amount and then add it back on to the original price. However, there is an easier method. Suppose the tax that is to be added is 4.2%. You can simply calculate 100% + 4.2% = 104.2% of the original price.

8. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Sales Tax (cont.) Finding the total price of a $25.00 item with 4.2% sales tax using both methods gives the following equivalent results. Calculating sales tax and then adding it on: ($25.00)(0.042) = $1.05 $25.00 + $1.05 = $26.05 Calculating 104.2% of the item: ($25.00)(1.042) = $26.05

9. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #2 Find the total price of a new 32" TV that sells for $219.99 if purchased in Georgia, where the sales tax is 4%, and if purchased in Kentucky, where the sales tax is 6%. Answer: Georgia: $228.79, Kentucky: $233.19

10. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Sale Prices When finding sale prices you can either think of it in terms of the “tipping” method or the “sales tax” method. Since sale discounts usually occur in increments of 5%, we like to think of it in the same manner as the “tipping” method – just subtracting instead of adding at the end!

11. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Sale Prices (cont.) For example, suppose you found the perfect sofa for your new apartment. The store is having its annual End of Season sale and offering 40% off on its entire stock. Estimate the discounted price of your dream sofa if the original price was $496.00. We start just as we did with tipping and find our base amount of 10% by moving the decimal point one place to the left. 10% of $496.00 = $49.60

12. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Sale Prices (cont.) Since this is an in-store estimate for yourself, we can simplify the calculation by rounding this discount to $50. So, if 10% is approximately $50 off, 40% would equate to four $50 discounts, giving you $200 off. Your dream sofa now only costs approximately $296.00. Another way to estimate the discount is to round the cost of the sofa from $496.00 to $500.00. Then, 10% of $500 would be $50.00. A discount of 40% would then equate to four $50.00 discounts, or $200 off.

13. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #3 Find the sale price of a new sweater that is 40% off the original price of $88.00. Answer: $52.80

14. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Absolute Change The absolute change between two amounts is the absolute value of the difference between the two numbers.

15. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Percentage Change

16. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Finding the Percentage Change An on-campus organization is trying to increase its monthly attendance. a.If after a big recruiting push, the meeting attendance increased from 25 to 38 people, what was the percentage change in attendance? b.During the third meeting, only 31 people attended. What was the percentage change from the second meeting to the third?

17. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Finding the Percentage Change (cont.) c.For advertising purposes, the organization decides to promote the overall increase in attendance for the two-month span since it has increased from the original number of attendees. Calculate the percentage change from meeting 1 to meeting 3.

18. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Finding the Percentage Change (cont.) Solution a.To find the percentage change, we need to know the absolute change in attendance between the first meeting and the second meeting. Use subtraction to find this value.

19. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Finding the Percentage Change (cont.) Next, substitute the appropriate values into the equation to find the percentage change. Since the attendance had an increase from the original amount, we say there was a 52% increase in attendance.

20. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Finding the Percentage Change (cont.) b.Now, if at the next meeting only 31 people attend, what is the percentage change from meeting 2 to meeting 3? This time, the absolute change in attendance is as follows.

21. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Finding the Percentage Change (cont.) Substituting this into our equation, we have Since this was a decline in attendance, we say there was an 18.42% decrease from meeting 2 to meeting 3.

22. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Finding the Percentage Change (cont.) c.Recall that, in meeting 1, there were 25 people in attendance and in meeting 3, there were 31 people. So we need to subtract these numbers to find the absolute change in attendance.

23. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Finding the Percentage Change (cont.) So, our percentage change equation becomes Since this was an increase in attendance from meeting 1 to meeting 3, we say there was a 24% increase overall.

24. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Percentage Faux Pas Percentages are a very useful and informative tool. However, percentages are commonly used incorrectly and lead people down the wrong path. Three of the more common misuses involving percentages are: o Comparing percentages of groups of very different sizes o Averaging percentages o Shifting reference points

25. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Comparing Percentages of Groups of Very Different Sizes Suppose you were told that 75% of people living in Alaska favor free skis for school children while only 40% of Californians do. While it is true that a greater proportion of Alaskans favor free skis, it would be wrong to conclude from these percentages that more Alaskans are in favor of the free skis than Californians. Alaska’s population in 2010 was around 710,000 while California’s population was 37,000,000. Given that information, we can find the actual number of people in favor of free skis in both states.

26. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Comparing Percentages of Groups of Very Different Sizes (cont.) Alaska: 75% of 710,000 = (0.75)(710,000) = 532,500 California: 40% of 37,000,000 = (0.40)(37,000,000) = 14,800,000 So, we can see that 75% of Alaskans is a much, much smaller number than 40% of Californians. Comparing 40% to 75% when the populations are not even in the same ballpark is misleading. In fact, 2% of California’s population is already greater than the entire population of Alaska.

27. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #4 Even if the total populations are very different sizes, there are times when percentages/proportions are more appropriate for comparison. Can you think of some examples? Answer:Answers may vary. An example is the percentage of voters in a demographic area.

28. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Averaging Percentages Another “no-no” when manipulating percentages is trying to average them. In general, averaging percentages almost always yields incorrect answers. There are exceptions, but to avoid confusion or misuse, stay away from averaging percentages. Consider our previous example about approval for free skis to school children in Alaska and California. What percentage of people in both states favor the idea? Since there are two states, what is wrong with simply averaging the two percentages?

29. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Averaging Percentages (cont.) If we averaged the two percentages together we would have However, if we add all the people in favor of the idea together, we still have less than half of the population of California. So, the average percentage of 57.5% cannot possibly be correct.

30. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Averaging Percentages (cont.) Here is the correct method. So, the percentage of people in both states in favor of the free skis idea would be

31. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Shifting Reference Points Campus police report that car break-ins on campus decreased by 25% this year compared to only a 20% decline the previous year. They promoted the fact that their law enforcement efforts reduced the number of break-ins more this year than in the previous year. The following is a breakdown of the campus report, including a column to show the percentage change from year to year.

32. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Shifting Reference Points (cont.) Table 1: Campus Car Break-In Report YearCar Break-InsPercentage Change 200910 2010820% decrease 2011625% decrease Consider the underlying numbers rather than the percentages to confirm the assertion made by the campus police.

33. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Shifting Reference Points (cont.) Solution We can see that, just as the police department claims, car break-ins are decreasing by a larger percentage each year. However, notice that the absolute reduction in the number of car break ‑ ins remains exactly 2 fewer each year. Since the number of break-ins reduces from year to year, the absolute difference of 2 becomes a larger percentage of the shrinking overall number of break-ins.

34. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Shifting Reference Points (cont.) In other words, the reference point that we are comparing the difference to changes each year, causing the percentage decrease from year to year to become larger.