On this page, we propose problems/exercises based on summaries of news articles (in roughly reverse chronological order). While assessment in QL/QR courses is a huge subject of debate, these problems may be a starting point. Please do not hesitate to share similar items with us by email. wbriggs@math.cudenver.edu
According to Census Bureau figures, solo drivers (no passengers) accounted for 73% of all driving commuters in 1990 and 76% of all driving commuters in 2000. The average commute time also increased from 22.4 minutes to 25.5 minuted betwen 1990 and 2000.
1. Estimate the increase in the number of solo drivers between 1990 and 2000.
2. What is the percent increase in commute time between 1990 and 2000?
3. Over the course of a year, how much more time does one person spend commuting in 2000 than in 1990?
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According to Census Bureau figures, in 1990 13.1% of the U.S. population lived in population, while only 12.4% of the population lived below poverty in 2000. Assume populations of 249 million and 281 million in 1990 and 2000, respectively.
1. Express the change in poverty rate between 1990 and 2000.
2. Did the actual number of people living in poverty increase or decease between 1990 and 2000?
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The June 5, 2002 Boulder Daily Camera reports that as a summer drought approaches, a critical city drinking water reservoir is leaking 80 gallons per minute. The leak cannot be fixed immediately because it would require draining the reservoir, which is not practical in view of the impending drought. The report claims that the leak amounts to 120,000 gallons per day, which represents a 10% water loss (fairly typical for a medium city). The reservoir capacity is 9.5 million gallons.
1. Are the two leaking rates cited in the story consistent?
2. Compare the water lost in a day to the water in an olympic-sized swimming pool (50 meters by 20 meters by 2 meters). Assume that 1 cubic foot of water is 7.5 gallons.
3. If the 10% water loss is interpreted as amount of water lost as a percent of water flowing out of the reservoir, how much water flows out of the reservoir each day?
4. What percent of the total reservoir capacity flows out in a day?
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As study done by the Center for Complementary and Alternative Medicine published in the Journal of the American Medical Association concluded that the herbal supplement St. John's Wort is no more effective than a placebo (or an alternative treatment, sertraline) in treating depression. The study claims that as much as 5 percent of the adult population has taken St. John's Wort, spending over $100 million on the supplement in 2000. Major depression affects 9 million adult Americans in any given year.
1. What percent of the American population is affected by depression in any given year?
2. Assuming that 26% of the American population is under 18 years of age, what percent of the adult American population is affected by depression in any given year?
3. Compare the number of people who have taken St. John's Wort with the number of adults who have suffered from depression.
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The University of California, Berkeley Wellness Letter for February 2002 reports that about 10% of people in industrialized countries will develop a kidney stone in their lifetimes, with men four times more likely that women. About half of those who develop a kidney stone will have a recurrence within five years. According to a recent report in the medical journal Lancet, it's a myth that avoiding calcium prevents most kidney stones.
1. If you are a man what is the probability that you will develop a kidney stone in your lifetime?
2. If you are a woman what is the probability that you will develop a kidney stone in your lifetime?
3. If you have already had a kidney stone, what is the probability that you will develop another kidney stone in the next five years?
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In the run-up to the 2002 Superbowl (New England vs, St. Louis), it was reported that ticket prices (probably the cheapest seats) were $400, compared to $12 for the first Superbowl in 1967. Adjusting for inflation, which tickets are more expensive?
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A new gene test developed at Johns Hopkins University shows promising results in providing early detection for colon cancer. Researchers reported that the test gives a positive result in 60% of patients with colon cancer with no false positives. Another similar gene test for colon cancer detects cancer in 70% of patients who actually have cancer with a 5% false positive rate. The base incidence rate for colon cancer is approximately 5 in 10,000.
1. What is the probability that you would receive a false test (either positive or negative) for colon cancer if you were given the first test?
2. What is the probability that you would receive a false test (either positive or negative) for colon cancer if you were given the second test?
3. In the second test, what percentage of the false tests are false positives?
4. Why is a test with a zero false positive rate desirable?
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Imagine that you buy a book in your campus bookstore and pay $58.66. The same day you see the same book in Books in Print for $55. In an attempt to explain the difference, you find out that the price given in Books in Print is a 25% markup from the wholesale price; that is, the retail price is 25% more than the wholesale price. You then visit the campus bookstore and are told that the bookstore determines the retail price by "taking the usual 25%."
1. Based on the 25% markup, what is the wholesale price of your book?
2. How can you explain why the bookstore price is higher than the Books in Print price?
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The following article from the Associated Press (January 4, 2002) is
a fairly typical summary of the Department of Labor's monthly unemployment
reports. Read it carefully and answer the questions that follow.
WASHINGTON (AP) -- The nation's unemployment rate climbed to 5.8 percent in December, highest in more than six years, as businesses cut 124,000 jobs and the year ended with the job market in the depths of recession.
The unemployment rate was up 0.2 percentage points from a revised 5.6 percent in November, the Labor Department reported Friday. Since the recession began in March, businesses have slashed 1.4 million jobs. December's loss of 124,000 jobs reflected continued declines in manufacturing, retail, air transportation and temporary employment services. But the hemorrhaging slowed somewhat last month. Job losses had averaged about 400,000 a month in October and November.
The last time the nation's unemployment rate stood at 5.8 percent was March 1995. It hit 5.9 percent in September 1994. But 2001 was a particularly bad year for manufacturing, which shed 1.3 million jobs -- or about 7 percent of its work force. Many manufacturing companies lost more than one in 10 of their workers last year, mainly in furniture, metals, industrial machinery, textiles and apparel.
A sliver of hope for manufacturing appeared in December's employment report, as the factory workweek increased by 0.4 hour to 40.7 hours and factory overtime rose 0.2 hour to 3.9 hours.
December declines also were heavy in retail with 77,000 job cuts, particularly at general merchandise stores and retailers such as toy stores and jewelry stores, both of which fell short of their typical holiday hiring.
Those declines were tempered slightly by job gains in auto dealerships as free financing beckoned customers. During 2001, retail had added 200,000 jobs by July, but losses since then have left employment down by 73,000 during the year.1. What is the unemployment rate?
2. What was the percentage increase in the unemployment rate between November and December?
3. Are the figures "slashed 1.4 million jobs" and "shed 1.3 million jobs" consistent? Explain.
4. Roughly how many jobs were lost in March through October in 2001?
5. Roughly how many people are in the manufacturing work force?
6. How many retail jobs were lost between July and the end of the year?
7. Given the information in the article, is it possible to estimate the total number of Americans in the work force (those who are either employed or seeking employment)?
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On January 1, 2002, twelve European countries (280 million people) begin to switch their individual currencies to the common euro currency. Citizens have several months (depending on the country) to exchange their old currency to the euro before it becomes worthless. At this moment, here are the exchange rates for the currencies of the twelve countries, plus the United States.
1. Which is worth more, 1 euro or 1 Irish pound?
2. If a Greek girl exchanged 1200 drachmas, how many euros would she receive?
3. Suppose you want to exchange $1000 for euros. How many euros will you receive?
4. Suppose you want to exchange $1000 for euros, but do it by first exchanging dollars for francs and then francs for euros. How many euros will you receive given the exchange rate $1 = 7.2516 francs? Compare the results of questions 3 and 4.
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An article in the New York Times (December 23, 2001) about the unseasonably warm autumn over the eastern part of the country included the following passage.
Another is that the warming trend is destined to continue. This year is expected to be the second warmest on earth since 1860, when temperatures were first measured systematically. Nine of the 10 warmest years have occurred since 1990. And one of the world's leading forecasting agencies, the Hadley Center for Climate Prediction and Research in Britain, says there is a 75 percent chance that next year will be even warmer than this year.
Just how unusual is it that the 1990s were so warm? Assume that the weather for one year is independent of the weather in other years.
1. What is the probability that the ten warmest years on record all occurred in the 1990s?
2. What is the probability that at least one of the ten warmest years on record occurred in the 1990s.
3. What is the probability that nine of the ten warmest years on record occurred in the 1990s (as claimed in the article)? Interpret this result.
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The annual Monitoring the Future survey conducted by the of University of Michigan's Institute for Social Research revealed that teen smoking rates declined during the past year, as had rates for ecstasy, heroin, and inhalants. Specifically, 12.2% of 8th graders reported smoking in the 30 days before they were surveyed, down from 14.6% the year before. The peak in the 1990s was 21.0 percent in 1996. The survey found 21.3% of 10th graders had smoked in the last 30 days, down from 23.9% the year before and 30.4% in 1996. And 29.5% of 12th graders had smoked in the month before being asked, down from 31.4% in 2000. Their peak in the 1990s came in 1997 at 36.5%. (See original report.)
1. Discuss how percentages are used these statements.
2. What was the percentage change in the smoking rates of 8th, 10th, and 12th graders during the past year. Interpret the result.
3. The report was received as good news by public health officials. Is it possible that the actual number of smoking teens increased during the past year?
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The results of the voting for the top five candidates for college football's Heisman Trophy were as follows:
| First place votes | |||||
| Second place votes | |||||
| Third place votes |
Who won the Heisman Trophy?
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A study that appeared in the Journal of the American Medical Association (December 12, 2001) reports on obesity American Children. According to the study, the percentage of overweight black and Hispanic children "more than doubled" between 1986 and 1998; the percentage of overweight white children increased by 50%. Specifically, in 1986 the percentage of overweight 4-12-year olds was 8%, 10%, and 8% for black, Hispanic, and white children, respectively. In 1998, the percentages had risen to 22%, 22%, and 12%, respectively. Among the reasons given for the increases were increased television watching, increased us of computers and video games, and more reliance on fast food. The study was based on data from the National Longitudinal Survey of Youth, which followed a sample of 8270 children from 1986 to 1998.
1. Discuss the ways in which percentages are used this summary of the study.
2. Given the figures in this summary, is it true that the percentage of overweight black and Hispanic children "more than doubled" and the percentage of overweight white children increased by 50% between 1986 and 1998?
3. Does it follow that the number of overweight black and Hispanic children doubled between 1986 and 1998?
4. Make the reasonable assumption that the number of all children increased by 30% between 1986 and 1998. What was the percent increase in the number of overweight black, Hispanic, and white children over the same time period?
5. Can the rates of increase cited in the study continue indefinitely? Discuss how long could these increases continue.
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One of many conclusions of Census 2000 (reported by Doug Smith in the Los Angeles Times) is 81% of the American over-25 population has graduated from high school, a figure that is expected to increase in the next decade. At the same time, only 75% of those currently in the 18-24 age category have completed high school. Discuss how can these figures be consistent.
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In an interview of Harold Hodgkinson by Karen Arenson (New York Times, August 5, 2001), Hodgkinson says (among many other fascinating things):
"We tend to associate poverty mostly with minorities. But the largest number of poor children are white. Of the poor children [in the United States] in 1999, 8.9 million were white, 4 million were black, and 3.9 million were Hispanic. But the children in poverty represented 16% of all white children, 37% of all black children, and 36% of Hispanic children."
Here are just a few of many possible questions suggested by these facts:
1. Are there more white poor children or black poor children? Explain.
2. Assuming that these figures account for all poor children in the country, what percentage of all poor children were black or Hispanic?
3. How many white children were there in 1999?
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The Bureau of Justice Statistics' National Crime Vicitmization Survey released in June 2001 had good news about crime in the United States. Rates for both property crimes and violent crimes were down in 2000 compared to 1999. In particular, violent crime (such as rape and robbery) was down 15%, which amounts to 1 million fewer crimes in 2000 than in 1999. The overall violent crime rate decreased from roughly 50 crimes per 1000 people (aged 12 and over) in 1993 to 28 crimes per 1000 people in 2000. While the news is good, the fact remains that about 25 million people were victims of violent crimes in 2000. Ironically, the report conflicts with crime statistics released by the FBI, which found that violent crime increased over the same period by 0.1%.
1. If a 15% decrease in violent crimes amounted to 1 million fewer crimes in 2000 than in 1999, how many violnet crimes were there in 1999?
2. If a 15% decrease in violent crimes resulted in a crime rate of 28 crimes per 1000 people in 2000, what was the crime rate in 1999?
3. The 2000 violent crime rate for men was 33 crimes per 1000 people, while for women the rate was 23 crimes per 1000. If the overall violent crime rate for men and women in 2000 was 28 crimes per 1000, approximately what percentage of violent crimes were committed by men?
4. Speculate on why the results of the BJS study and the FBI studies differed.
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A study conducted by the University of Rochester and reported in the June issue of Pediatrics, was summarized in an Associated Press new article (June 5, 2001). According to the AP story,
"the study involved nationally representative data on 5,398 children ages 6 to 16 who participated in a health survey from 1988 to 1994. Iron deficiency was found in 3% of the children overall, representing 1.2 million school-age children. It occurred in 8.7% of the girls ages 12 to 16, including 7% without anemia. Average math scores for iron-deficient children with or without anemia were about six points lower than those with normal iron levels. Among adolescent girls, the difference in scores was more than eight points. The average math score for normal youngsters was 93.7, 87.4 for iron-deficient children without anemia and 86.4 for those with anemia. The highest score recorded was 151. "
1. Interpret the statement, "iron deficiency was found in 3% of the children overall, representing 1.2 million school-age children."
2. Interpret the statement, "it occurred in 8.7% of the girls ages 12 to 16, including 7% without anemia."
3. Interpret the test scores for adolescent girls and for all children.
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An article in the Boulder (Colorado) Daily Camera cites a statewide study and reports that
"Nearly 40% of Colorado middle school students have experimented with tobacco -- a rate 6% higher than the national average. Of the sixth through eighth graders surveyed a year ago, 39.4% reported having used tobacco compared to 33.5% nationally. Thirty-two percent of these students said they tried cigarettes and 28% had done so before the age of 11."
What's wrong with the figures reported in the article?
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A survey sponsored by the Consumer Federation of America and Primerica (telephone survey of 1,010 adults 18 and older) revealed that 27% of those polled believe that their best chance of building wealth for retirement is playing lotteries. Among households with annual incomes of $35,000 or less, the percentage is 40%. People were also asked how much they could save by investing $25 per week for 40 years at an annual percentage rate of 7%. Fewer than one-third responded that more than $150,000 could be saved.
1. What are the chances of winning the grand prize in a "typical" lottery that requires matching 6 numbers drawn for a barrel of 40 numbers?
2. How much could you save by investing $25 per week for 40 years at an annual percentage rate of 7%?
3. Discuss your interpretation of these two calculations.
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The article Phantom Numbers Haunt the War on Drugs (New York Times, April 20, 1997) states that
"children who smoke marijuana are 85 times more likely to use cocaine than those who don't. This finding in 1994 by the National Center on Addiction and Substance Abuse at Columbia University is often cited as proof that marijuana leads to harder drugs."
The study reported that 17% of the marijuana users interviewed said they had tried cocaine. Only 0.2% of those who had not used marijuana said they had tried cocaine. Put another way, though, 83% of the marijuana smokers, or nearly five out of six, said they hadn't tried cocaine, which may undercut marijuana's threat as a "gateway drug."
1. How did the researchers determine the factor 85?
2. How did the researchers determine the figure 83%?
3. What percentage of non-marijuana smokers did not use cocaine?
4. With the given figures, is it possible to determine the percentage of cocaine users who once used marijuana?
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During the first two months of 1997, reports from Albania revealed that a widespread pyramid scheme had led to the financial ruin of thousands of Albanians. In this country, it is not uncommon to receive mail invitations to join (illegal) pyramid schemes. What is a pyramid scheme? Here is an example of a local pyramid scheme as described in a mail invitation that I recently received. To join, you must send as many copies of the invitation letter as possible to friends and strangers alike. All of those who join in the first month because of your letter become your descendants. Your descendants will also invite as many people as possible; Those new members become your second descendants. If all goes according to plan, in five months you will have five generations of descendants. Likewise, you have a first ancestor (who invited you to join), a second ancestor (who invited your ancestor to join), and so on.
Now you are in the game. Each month, you send $20 to the :"organizers" (whose identity is never revealed). The organizers keep $5; the remaining $15 is split into five $3 parcels and sent to your first through fifth ancestors. Here is where you stand to make the money. Each month each person in your five generations of descendants also sends $20 to the organizers. You receive your $3 share of each $20 parcel. We will estimate how much money you make each month.
1. Assume that in the first month, you recruited five first descendants. How much money would you receive at the end of the first month from those first descendants?
2. Now assume that in the second month, each of those five first descendants recruited five first descendants (which would be your second descendants). How much money would you receive at the end of the second month from those two generations of descendants?
3. Now assume that in the third month, each of your second descendants recruited five first descendants (which would be your third descendants). How much money would you receive at the end of the third month from those three generations of descendants?
4. Now assume that in the fourth month, each of your third descendants recruited five first descendants (which would be your fourth descendants). How much money would you receive at the end of the fourth month from those four generations of descendants?
5. Now assume that in the fifth month, each of your fourth descendants recruited five first descendants (which would be your fifth descendants). How much money would you receive at the end of the fifth month and thereafter from those five generations of descendants?
6. Are there any drawbacks to this scheme other than the fact that it is illegal? Would you participate in the scheme, even if it were legal?
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According to the National Cancer Institute (see also a report in the Journal of the American Medical Association, ?? 1997), "melanoma is the most serious skin cancer, with 40,300 new cases and 7300 deaths expected this year (1997). The incidence has been rising about 4% each year since 1973."
1. How many new cases of melanoma were there in 1996?
2. How many new cases of melanoma were there in 1995?
3. Find a general relation that gives the number of new melanoma cases for any year between 1973 and 1997.
4. If the same 4% growth rate continues into the future, how many new cases of melanoma will there be in the year 2010?
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The July 5, 1996 edition of the New York Times ran the following correction:
A headline in some editions July 4 incorrectly characterized Boris Yeltsin's margin over Gennady Zyuganov in the Russian presidential runoff. Yeltsin won by 13 percentage points, but had 33 percent more votes than Zyuganov.
Although we are left to speculate as to the nature of the error, it clearly involved a misuse of percentages.
1. What was the percentage of the votes actually received by Yeltsin and Zyuganov?
2. Since this was a runoff election (only two candidates), are these figures consistent?
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During the last week of March, 1996, the US government announced that a spike flood would be released through the Grand Canyon from Glen Canyon Dam. The flood had been suggested by ecological and biological groups for many years as a way of restoring the Colorado River and the habitats along its banks. The release of water averaged 25,800 cubic feet per second and lasted a week. The total volume of the reservoir behind the 710-foot high Glen Canyon Dam is estimated to be 27 million acre-feet or 33 billion cubic meters. What fraction of the total water supply was released during this spike flood? Comment on whether this release would compromise the other uses of Glen Canyon water in the near future (agriculture and hydroelectric power).
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The December 18, 1995 issues of TIME Magazine has an article (p. 61) called Death and the Maven about a 101-year old spinster, Anne Scheiber, who left $22 million to Yeshiva University upon her death. This fortune was accumulated through shrewd and patient investment of a $5000 nest egg over the course of 50 years. Using a calculator, or any other means, estimate the average interest rate at which Miss Scheiber's funds must have grown.
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An issue of the NY Times (probably 1996) reported the following facts under the headline In China, Rapid Social Changes Bring a Surge in the Divorce Rate:
"The divorce rate in Beijing leapt to 24.4 percent in 1994, more than double the 12 percent rate just four years ago, Beijing Youth Daily reported this month. Although statistics can be misleading -- the divorce rate is measured by comparing the number of marriages and divorces in a given year -- officials say it is rising all over China, and faster in cities than in the countryside. The national divorce rate is now 10.4 percent, still far behind the United States, where the divorce rate rose sharply in the 1970's to around 50 percent, where it has remained."
1. Are the 24.4% divorce rate and the 10.4% divorce rate quoted for China inconsistent?
2. By what percent did the divorce rate itself increase?
3. Is the method used to measure divorce rate reliable?
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