2.3: Deceptive and Misleading Numbers

Although mathematics covers a whole lot more than numbers, it is still true that arithmetic, including the basic operations on numbers such as integers, fractions, decimals, and percentages, remains one of the most practical and fundamental applications of mathematics that you encounter every day. Many decisions are made, both in large and small scales, based on numbers, figures, statistics, and all sorts of data. Based on various numbers, companies may decide where to build a distribution center or a shopping mall, and government agencies may decide to implement certain policies. Different political parties may present different numbers to push their own agenda or to silence their opponents. Advertisers know that figures can be very effective in persuading consumers. In fact, many of us may not even be aware of how we make our shopping decisions based on numbers presented to us in a variety of forms. This is especially true in the Internet era in which we live since advertisements can be targeted based on our online activities and digital footprints.

It is precisely because of this fact that numbers and figures are frequently manipulated. In this brief section we will look at how certain numbers can be presented to us in misleading or deceptive ways to give certain impressions or to change our mind.

Relative Changes and Differences

As we have seen earlier, percentages can be calculated in different ways to produce different numbers. For instance, suppose the annual tuition is $50,000 at College A and $20,000 at College B. In terms of absolute difference, you can state, truthfully, that the tuition is $30,000 higher at College A than at College B. In terms of relative difference, however, each of the following four statements is true:

• The tuition at College A is 250% of the tuition at College B since \[50,000 \div 20,000 = 2.50 \nonumber \]

• The tuition at College A is 150% more than at College B since \[(50,000 - 20,000) \div 20,000 = 1.50 \nonumber \]

• The tuition at College B is 40% of the tuition at College A since \[20,000 \div 50,000 = 0.40 \nonumber \]

• The tuition at College B is 60% cheaper than at College A since \[(50,000 - 20,000) \div 50,000 = 0.60 \nonumber \]

Here, the first two statements use the tuition at College B as the reference number while the last two use the tuition at College A as the reference number. Note the words like “of,” “more than,” and “cheaper than.” The amount that follows “of” and “than” is the reference number.

Now, 250, 150, 40, and 60 are very different numbers, and yet, each of these numbers can be used to represent the tuition difference between these two colleges. If you are trying to persuade someone to go to College B based on the tuition, you could say, “Hey, College A charges you 250% as much as College B.” And that sounds dramatic.

Similar calculations can be carried out in comparing many things including prices, index numbers, crime rates, populations, and averages.

Example \(\PageIndex{1}\)

According to Wikipedia.org, the population estimates (in 2018) for some states are as follows (rounded to the nearest thousand):

California: 39,557,000

Michigan: 9,996,000

Virginia: 8,518,000

Mississippi: 2,987,000

Wyoming: 578,000

1. The population of Michigan is what percent less than the population of California?
2. The population of California is what percent more than the population of Michigan?
3. The population of Michigan is what percent of the population of California?
4. The population of California is what percent of the population of Michigan?

Solution

1. \((39,557,000-9,996,000) \div 39,557,000=0.7473\). Therefore, the population of Michigan is about 74.7% less than the population of California.
2. \((39,557,000-9,996,000) \div 9,996,000=2.9573\). Therefore, California has about 295.7% more people than Michigan does.
3. \(9,996,000 \div 39,557,000=0.2527\). Therefore, the population of Michigan is about 25.3% of that of California.
4. \(39,557,000 \div 9,996,000=3.957\). Therefore, the population of California is about 396% of that of Michigan.

Of course, they all imply the same fact: California has about 4 times as many people as Michigan.

Try it Now 1

1. Make the same comparison as above (finding four percentage figures) between

   1. Virginia and Mississippi

   2. Michigan and Wyoming

2. As of 2018, California is the most populous state in the union while Wyoming is the least populous. The population of Wyoming is about what percent of that of California? How many times is the population of California as compared to that of Wyoming?

Ever-Changing Reference Numbers

We have seen this earlier also, but here is a very common mistake we need to pay attention to. One thing you may want to remember is this: you cannot simply add percentages when a percentage change is followed by another. This is not a difficult concept, but many people are unaware of this. Here are a few examples.

Example \(\PageIndex{2}\)

The number of students at a city college increased 20% one year, and the number again increased by 30% the following year. Does this mean the two-year increase was 50%?

Solution

These problems can be simplified if you start with a fixed number. For example, let’s assume that the college had 1000 students two years ago. Then, in one year, with a 20% increase, the college had 1200 students. But then, in the second year, the college added 30% of 1200, which is 360. With the addition of 360 students over 1200, the college has 1560 students after two years. That represents an increase of 560 over 1000, which amounts to a 56% (not 50%) increase. (By the way, it turns out that the original number is irrelevant. Regardless of the number you start with, a 20% increase followed by a 30% increase would always result in a total increase by 56% since 1+0.21+0.3=1.56.

Example \(\PageIndex{3}\)

The value of a certain stock jumped up by 50% one day, only to lose 50% the following day. Does it mean that the stock is back to the original value after the two-day period?

Solution

Again, let us consider a specific number to start with. Suppose the stock was valued at $100 two days ago. After jumping up by 50%, the stock was valued at $150 yesterday. Then, after a 50% loss (and 50% of $150 is $75), the stock is now only worth $75, meaning that it lost 25% of the value in the two-day period.

Why do these things happen? It’s because the reference number for the percentage has changed when the value changed. Again, remember that percentages cannot be simply added in these cases.

A change in the reference number can happen in the simplest of cases, as illustrated by the following example.

Example \(\PageIndex{4}\)

1. A local theme park just increased the parking fee by 25%, and now it is $20. How much was the parking fee before?

2. A dining room set just went on sale and is now available at $1400, which is 30% off the regular price. What is the regular price?

Solution

Many people make the error in assuming that $200 is the reference number for 25% in (a) and, similarly, $1400 is the reference number for 30% in (b). However, careful reading suggests that this is not correct.

In (a), the new parking fee is $20 while we do not yet know what the fee was before. In other words, the problem does NOT state the reference number explicitly. However, we DO know that whatever it was, adding 25% of that old parking fee results in the new fee of $20. Now, the old parking fee is of course 100% of itself, so $20 represents 125% of the old parking fee. Hence, if PF represents the old parking fee, we have

\[PF \times 1.25=20 \nonumber \]

\[PF = 20 \div 1.25=16 \nonumber \]

Therefore, the parking fee used to be $16. If you make the common mistake of finding 25% of $20, that would be $5, and deducting $5 from $20 would give you $15. But this is clearly incorrect because adding 25% to $15 would result in $18.75 (as shown below), not $20.

\[15 \times(1+.25)=18.75 \nonumber \]

For (2), the calculation is similar except we subtract (not add) 30% (0.3) from 1. Whatever the regular price of the dining room set was, a 30% discount means that you are still paying 70% of the original price (1-0.3=0.7). Denoting the regular price by RP, we have

\[RP \times(1-.3)=RP \times 0.7=1400 \nonumber \]

\[RP=1400 \div 0.7=2000 \nonumber \]

So the original price of the dining room set was $2000. This makes sense because 30% of $2000 is $600, which is the discount you are getting when you pay $1400 rather than $2000 for it.

Percentages of Percentages

This was mentioned in one example in Section 2.1, but we discuss it again here with an emphasis on avoiding confusion.

Suppose the mayor of your city has worked very hard to fight poverty in your city and, as a result, the poverty rate in your city has gone down from 16% to 12% over the last four years. How much of a drop does that represent?

Of course, you can say that the poverty rate has gone down by 4% (16% – 12%), right? Well, that would be a little ambiguous, and here is why. Generally, when we use percentages to express a relative change, we use the earlier number (in this case, 16%) as the reference number. Therefore, the relative change in this case is

\[(12-16) \div 16=0.25, \nonumber \]

representing a 25% drop in the poverty rate.

So what are we saying? Is 4% equal to 25%? Not really. A more accurate way of describing this is as follows: “The 4% drop represents 25% of the reference number, which is 16%” (because 4 is 25% of 16). Yes, we are dealing with a percentage of a percentage. These numbers are particularly misleading and quite confusing. For that reason, when we refer to the absolute difference in two percentages (such as the 4% drop from 16% to 12%), we refer to that as “points.” In other words, here we can say that the poverty rate has dropped by 4 points (as opposed to the more confusing “4 percent”). 25% then refers to the relative difference from 16% to 12%.

Example \(\PageIndex{5}\)

1. The approval rate of a senator has gone up from 44% to 49%. How many points has the rate increased by? What percentage has the approval rate increased by?
2. The fatality rate of a certain disease has gone down from 22% to 10%. How many points has the rate gone down by? What percentage drop does this represent?

Solution

1. The rate has gone up by 5 points. However, one can also say that the rate has gone up by about 11.4% since \[(49-44) \div 44=0.1136 \nonumber \]
2. The rate has dropped by 12 points. Also, the rate has gone down by about 54.5% (a little more than half) because \[(10-22) \div 22=-0.5454 \ldots \nonumber \]

This number is negative, representing a decrease, not an increase.

There are a few other things to remember when you are presented with numbers or other quantitative information. In addition to these percentage-related statements, watch out for things like the following:

• What is the unit? 20,000,000,000 bytes may sound really big, but this is the same as 20 GB. Similarly, 15,000 Rupiahs (the currency of Indonesia) may sound like a lot of money, but it is worth only about a dollar in the U.S. (as of June, 2019).

• What is the reference number? Many things could happen with a small sample size. For instance, it may not be all that unusual if a basketball player makes 1 out of 2 free throws (50%) in a given game, but it is a completely different story if the same player makes only 100 out of 200 free throws (still 50%) over a long period of time. In other words, the same percentage could mean something quite different if the denominator (the reference number) is large (or small).

• What is the absolute difference? How big or small is the original number? For example, according to CDC (Center for Disease Control) data, the number of measles cases reported in the U.S. more than tripled (grew by more than 200%) in 2013 (the number went up from 55 in 2012 to 187 in 2013). That sounded quite alarming and dramatic (and, in a sense, it was); however, those were very small numbers compared to the dire situation the country is facing in 2019, where the number exceeded 1,000 just in the first six months of the year. Sometimes the percentages do not tell the whole story, and you may want to ask more questions.

There are many other ways in which numbers can deceive. Some of those will be discussed in later chapters on finance, statistics, growth, and politics. What has been presented in this section is just a small portion of number manipulation. The message is clear: we must all be numerically literate and quantitatively intelligent to make wise decisions in this data-driven world.

Exercises

1. You invested some money in a start-up company, and now your investment has earned 60% and is worth $6,720. How much was your original investment?

2. A college football ticket can be purchased at $28 after a 20% student discount is applied. How much is the ticket without the student discount?

3. The unemployment rate in a certain small town has changed from 4% to 7% during the last ten years. The rate went up by how many percentage points? By what percent?

4. Suppose that the survival rate of a certain form of cancer increased from 33% to 59%. Express this change in terms of points and percentage as above.

5. Out of 230 racers who started the marathon, 212 completed the race, 14 gave up, and 4 were disqualified. What percentage did not complete the marathon?

6. Patrick left an $8 tip on a $50 restaurant bill. What percent tip is that?

7. Ireland has a 23% VAT (value-added tax, similar to a sales tax). How much will the VAT be on a purchase of a €250 item?

8. Employees in 2012 paid 4.2% of their gross wages towards social security (FICA tax), while employers paid another 6.2%. How much will someone earning $45,000 a year pay towards social security out of their gross wages?

9. A project on Kickstarter.com was aiming to raise $15,000 for a precision coffee press. They ended up with 714 supporters, raising 557% of their goal. How much did they raise?

10. Another project on Kickstarter for an iPad stylus raised 1,253% of their goal, raising a total of $313,490 from 7,511 supporters. What was their original goal?

11. The population of a town increased from 3,250 in 2008 to 4,300 in 2010. Find the absolute and relative (percent) increase.

12. The number of CDs sold in 2010 was 114 million, down from 147 million the previous year.¹ Find the absolute and relative (percent) decrease.

13. A company wants to decrease their energy use by 15%.

    1. If their electric bill is currently $2,200 a month, what will their bill be if they’re successful?

    2. If their next bill is $1,700 a month, were they successful? Why or why not?

14. A store is hoping an advertising campaign will increase their number of customers by 30%. They currently have about 80 customers a day.

    1. How many customers will they have if their campaign is successful?

    2. If they increase to 120 customers a day, were they successful? Why or why not?

15. An article reports “attendance dropped 6% this year, to 300.” What was the attendance before the drop?

16. An article reports “sales have grown by 30% this year, to $200 million.” What were sales before the growth?

17. Are these two claims equivalent, in conflict, or not comparable because they’re talking about different things?

    1. “16.3% of Americans are without health insurance” ²

    2. “only 55.9% of adults receive employer provided health insurance” ³

18. Are these two claims equivalent, in conflict, or not comparable because they’re talking about different things?

    1. “We mark up the wholesale price by 33% to come up with the retail price”

    2. “The store has a 25% profit margin”

19. Are these two claims equivalent, in conflict, or not comparable because they’re talking about different things?

    1. “Every year since 1950, the number of American children gunned down has doubled.”

    2. “The number of child gunshot deaths has doubled from 1950 to 1994.”

20. Are these two claims equivalent, in conflict, or not comparable because they’re talking about different things? ⁴

    1. “75 percent of the federal health care law’s taxes would be paid by those earning less than $120,000 a year”

    2. “76 percent of those who would pay the penalty [health care law’s taxes] for not having insurance in 2016 would earn under $120,000”

21. Are these two claims equivalent, in conflict, or not comparable because they’re talking about different things?

    1. “The school levy is only a 0.1% increase of the property tax rate.”

    2. “This new levy is a 12% tax hike, raising our total rate to $9.33 per $1000 of value.”

22. Are the values compared in this statement comparable or not comparable? “Guns have murdered more Americans here at home in recent years than have died on the battlefields of Iraq and Afghanistan. In support of the two wars, more than 6,500 American soldiers have lost their lives. During the same period, however, guns have been used to murder about 100,000 people on American soil” ⁵

23. A high school currently has a 30% dropout rate. They’ve been tasked to decrease that rate by 20%. Find the equivalent percentage point drop.

24. A politician’s support grew from 42% by 3 percentage points to 45%. What percent (relative) change is this?

25. Marcy has a 70% average in her class going into the final exam. She says "I need to get a 100% on this final so I can raise my score to 85%." Is she correct?

26. Suppose you have one quart of water/juice mix that is 50% juice, and you add 2 quarts of juice. What percent juice is the final mix?

27. Find a unit rate: You bought 10 pounds of potatoes for $4.

28. Find a unit rate: Joel ran 1500 meters in 4 minutes, 45 seconds.

29. Solve: \(\dfrac{2}{5}=\dfrac{6}{x}\)

30. Solve: \(\dfrac{n}{5}=\dfrac{16}{20}\)

31. A crepe recipe calls for 2 eggs, 1 cup of flour, and 1 cup of milk. How much flour would you need if you use 5 eggs?

32. An 8 ft length of 4 inch wide crown molding costs $14. How much will it cost to buy 40 ft of crown molding?

33. Four 3-megawatt wind turbines can supply enough electricity to power 3000 homes. How many turbines would be required to power 55,000 homes?

34. A highway had a landslide, where 3,000 cubic yards of material fell on the road, requiring 200 dump truck loads to clear. On another highway, a slide left 40,000 cubic yards on the road. How many dump truck loads would be needed to clear this slide?

35. Convert 8 feet to inches.

36. Convert 6 kilograms to grams.

37. A wire costs $2 per meter. How much will 3 kilometers of wire cost?

38. Sugar contains 15 calories per teaspoon. How many calories are in 1 cup of sugar?

39. A car is driving at 100 kilometers per hour. How far does it travel in 2 seconds?

40. A chain weighs 10 pounds per foot. How many ounces will 4 inches weigh?

41. The table below gives data on three movies. Gross earnings is the amount of money the movie brings in. Compare the net earnings (money made after expenses) for the three movies. ⁶

Table \(\PageIndex{1}\)

Movie	Release Date	Budget	Gross earnings
Saw	10/29/2004	$1,200,000	$103,096,345
Titanic	12/19/1997	$200,000,000	$1,842,879,955
Jurassic Park	6/11/1993	$63,000,000	$923,863,984

42. For the movies in the previous problem, which provided the best return on investment?

43. Your chocolate milk mix says to use 4 scoops of mix for 2 cups of milk. After pouring in the milk, you start adding the mix, but get distracted and accidentally put in 5 scoops of mix. How can you adjust the mix if:

    1. There is still room in the cup?

    2. The cup is already full?

44. A recipe for sabayon calls for 2 egg yolks, 3 tablespoons of sugar, and ¼ cup of white wine. After cracking the eggs, you start measuring the sugar, but accidentally put in 4 tablespoons of sugar. How can you compensate?

45. The Deepwater Horizon oil spill resulted in 4.9 million barrels of oil spilling into the Gulf of Mexico. Each barrel of oil can be processed into about 19 gallons of gasoline. How many cars could this have fueled for a year? Assume an average car gets 20 miles to the gallon, and drives about 12,000 miles in a year.

46. The store is selling lemons at 2 for $1. Each yields about 2 tablespoons of juice. How much will it cost to buy enough lemons to make a 9-inch lemon pie requiring ½ cup of lemon juice?

47. A piece of paper can be made into a cylinder in two ways: by joining the short sides together, or by joining the long sides together.⁷ Which cylinder would hold more? How much more?

Figure \(\PageIndex{1}\)

48. Which of these glasses contains more liquid? How much more?

49. The flash of lightning travels at the speed of light, which is about 186,000 miles per second. The sound of lightning (thunder) travels at the speed of sound, which is about 750 miles per hour.

    1. If you see a flash of lightning, then hear the thunder 4 seconds later, how far away is the lightning?

    2. Now let’s generalize that result. Suppose it takes \(n\) seconds to hear the thunder after a flash of lightning. How far away is the lightning, in terms of \(n\)?

50. Sound travels about 750 miles per hour. If you stand in a parking lot near a building and sound a horn, you will hear an echo.

    1. Suppose it takes about ½ a second to hear the echo. How far away is the building? ⁸

    2. Now let’s generalize that result. Suppose it takes \(n\) seconds to hear the echo. How far away is the building, in terms of \(n\)?

51. It takes an air pump 5 minutes to fill a twin sized air mattress (39 by 8.75 by 75 inches). How long will it take to fill a queen sized mattress (60 by 8.75 by 80 inches)?

52. It takes your garden hose 20 seconds to fill your 2-gallon watering can. How long will it take to fill

    1. An inflatable pool measuring 3 feet wide, 8 feet long, and 1 foot deep.⁹

    2. A circular inflatable pool 13 feet in diameter and 3 feet deep.¹⁰

53. You want to put a 2" thick layer of topsoil for a new 20'x30' garden. The dirt store sells by the cubic yards. How many cubic yards will you need to order?

54. A box of Jell-O costs $0.50, and makes 2 cups. How much would it cost to fill a swimming pool 4 feet deep, 8 feet wide, and 12 feet long with Jell-O? (1 cubic foot is about 7.5 gallons)

55. You read online that a 15 ft by 20 ft brick patio would cost about $2,275 to have professionally installed. Estimate the cost of having a 18 by 22 ft brick patio installed.

56. I was at the store, and saw two sizes of avocados being sold. The regular size sold for $0.88 each, while the jumbo ones sold for $1.68 each. Which is the better deal?

Figure \(\PageIndex{2}\)

58. The grocery store has bulk pecans on sale, which is great since you’re planning on making 10 pecan pies for a wedding. Your recipe calls for 1¾ cups pecans per pie. However, in the bulk section there’s only a scale available, not a measuring cup. You run over to the baking aisle and find a bag of pecans, and look at the nutrition label to gather some info. How many pounds of pecans should you buy?

Figure \(\PageIndex{3}\)

59. Soda is often sold in 20 ounce bottles. The nutrition label for one of these bottles is shown to the right. A packet of sugar (the kind they have at restaurants for your coffee or tea) typically contain 4 grams of sugar in the U.S. Drinking a 20 oz soda is equivalent to eating how many packets of sugar?¹¹

Some More Things to Ponder…

60. Suppose you hear someone say, “The number of homicides in my school district has quadrupled over the last two years.” What does this mean? What possible scenarios can you think of? Is there anything else you would want to know regarding this claim?

61. Is it possible for the price of a concert ticket to increase by 150%? To decrease by 150%? Why or why not?

62. The sun is about 93 million miles from the earth while the moon is about 240,000 miles away (on the average). What percentages can you use to compare these two distances?

63. Your friend, who lives in a typical suburban neighborhood in your town, just says, “The average household income for last year in my neighborhood was $12 million.” Can this be true? What else would you want to know? What could have happened (assuming this statement is true)?

64. The population of the U.S. is about 309,975,000, covering a land area of 3,717,000 square miles. The population of India is about 1,184,639,000, covering a land area of 1,269,000 square miles. Compare the population densities of the two countries.

65. The GDP (Gross Domestic Product) of China was $5,739 billion in 2010, and the GDP of Sweden was $435 billion. The population of China is about 1,347 million, while the population of Sweden is about 9.5 million. Compare the GDP per capita of the two countries.

66. In June 2012, Twitter was reporting 400 million tweets per day. Each tweet can consist of up to 140 characters (letter, numbers, etc.). Create a comparison to help understand the amount of tweets in a year by imagining each character was a drop of water and comparing to filling something up.

67. The photo sharing site Flickr had 2.7 billion photos in June 2012. Create a comparison to understand this number by assuming each picture is about 2 megabytes in size, and comparing to the data stored on other media like DVDs, iPods, or flash drives.

68. If your favorite shoes are on sale with a 30% discount, and if you get an additional employee discount of 10%, can you get the shoes at 40% off? Why or why not?

69. How would you react if your boss tells you the following? “I will have to give you a 50% pay cut now, but I will give you a permanent 60% pay raise next week.” (Start with some fixed pay now and see what happens. Then explain why this is counter-intuitive.)

70. The Walden University had 47,456 students in 2010, while Kaplan University had 77,966 students. Complete the following statements:

    1. Kaplan’s enrollment was ___% larger than Walden’s.

    2. Walden’s enrollment was ___% smaller than Kaplan’s.

    3. Walden’s enrollment was ___% of Kaplan’s.

71. In the 2012 Olympics, Usain Bolt ran the 100m dash in 9.63 seconds. Jim Hines won the 1968 Olympic gold with a time of 9.95 seconds.

    1. Bolt’s time was ___% faster than Hines’.

    2. Hine’ time was ___% slower than Bolt’s.

    3. Hine’ time was ___% of Bolt’s.

72. A store has clearance items that have been marked down by 60%. They are having a sale, advertising an additional 30% off clearance items. What percent of the original price do you end up paying?

73. Which is better: having a stock that goes up 30% on Monday than drops 30% on Tuesday, or a stock that drops 30% on Monday and goes up 30% on Tuesday? In each case, what is the net percent gain or loss?

74. Define a “millionaire.” Does it depend on what currency is used?

75. You are concerned that your friend is taking too much ibuprofen for her headache, but she says, “I never take any more than 1.3 grams of the stuff. Some people take like 900 mg.” What would you say to her?

76. Which is harder? Walking 1,000,000 inches or walking 20 miles? Does one “sound” harder than the other?

Reference

1. http://www.cnn.com/2010/SHOWBIZ/Musi...les/index.html
2. http://www.cnn.com/2012/06/27/politi...are/index.html
3. http://www.politico.com/news/stories/0712/78134.html
4. http://factcheck.org/2012/07/twistin...th-care-taxes/
5. www.northjersey.com/news/opin...tml?c=y&page=2
6. www.northjersey.com/news/opin...tml?c=y&page=2
7. vimeo.com/42501010
8. vimeo.com/40377128
9. http://www.youtube.com/watch?v=DIkwefReHZc
10. http://www.youtube.com/watch?v=p9SABH7Yg9M
11. http://www.youtube.com/watch?v=62JMfv0tf3Q

Contributors and Attributions

• Saburo Matsumoto
  CC-BY-4.0