Chapter 3.A

A store is having a quarter​-off (25​% ​off) sale. The sale price for an item with a regular price of ​$200 is​ _______.

$150

State whether the following statement is true or​ false, and explain why. If the statement is​ false, state the true change. You receive a pay cut of 4​%, followed later by a pay raise of 4​%. After the two changes in pay​, your salary is unchanged.

-0.16%

Find the absolute change and the relative change in the following case. The per person consumption of meat in a certain country decreased from 68.1 pounds in 2000 to 51.4 pounds in 2020. The absolute change is _____pounds per person. The relative change is______​%.

-16.7 -24.5 %

Find the absolute change and the relative change​ (as a​ percentage) in the following case. The number of commercial radio stations playing a particular genre of music decreased from 730 in 1990 to 355 in 2020. The absolute change is _____ The relative change is ___​%.

-395 -54.1

1/9 as a decimal and a percentage

0.11 11.11%

4/15 as a decimal and a percentage

0.2667 26.67%

33/100 as a decimal and a percentage

0.33 33%

43% as a fraction and a decimal

0.43 43/100

Compare A and B in three​ ways, where A=44,904,000 is the number of people in a certain country over age 65 in 2010 and B= 84,775,000 is the projected number of people in that country over age 65 in 2050. a. Find the ratio of A to B. b. Find the ratio of B to A. c. Complete the​ sentence: A is​ ____ percent of B.

0.53 1.89 53%

The wholesale price of a TV is 40​% less than retail price.​ Therefore, the wholesale price is​ _____ times the retail price.

0.6

Compare A and B in three​ ways, where A=1.8 million is the 2020 population of City X and B=2.9 million is the 2020 population of City Y. a. Find the ratio of A to B. b. Find the ratio of B to A. c. Complete the​ sentence: A is​ ____ percent of B.

0.62 1.61 62%

70% as a fraction and a decimal

0.7 7/10

4/5 as a decimal and a percentage

0.8 80%

Compare A and B in three​ ways, where A=494 is the population of endangered mountain gorillas in a large preserve in 2010 and B=613 is the population in the same preserve in 2019. a. Find the ratio of A to B. b. Find the ratio of B to A. c. Complete the​ sentence: A is​ ____ percent of B.

0.81 1.24 81%

Compare A and B in three​ ways, where A=946 is the average combined SAT score in State X in 2020 and B=1,173 is the average combined SAT score in State Y in 2020. a. Find the ratio of A to B. b. Find the ratio of B to A. c. Complete the​ sentence: A is​ ____ percent of B

0.81 1.24 81%

5/6 as a decimal and a percentage

0.83 83.33

Compare A and B in three​ ways, where A=90.4% is the 2021 high school graduation rate in State X and B=84.7% is the 2021 high school graduation rate in State Y. a. Find the ratio of A to B. b. Find the ratio of B to A. c. Complete the​ sentence: A is​ ____ percent of B.

1.07 0.94 107

A store is having a 25% off sale.​ Therefore, the original price of an item is​ _______ times as much as the sale price.

1.33

The retail cost of a TV is 45​% more than its wholesale cost.​ Therefore, the retail cost is​ ____ times the wholesale cost.

1.45

180% as a fraction and a decimal

1.8 9/5

5/40 as a fraction, a decimal and a percentage

1/8 0.125 12.5%

In the following​ statement, express the first number as a percentage of the second number. The 2019 population of a certain state was approximately 35.7 million. The 2000 population was approximately 33.3 million. The 2019 population of the state was ​% of the 2000 population.

107.2%

In​ 2020, the median age at first marriage of women in a certain country ​(27.8 years) was​ _____ percent less than the median age at first marriage of men in that country ​(31.5 years).

11.7

275 as a fraction and a decimal

11/4 2.75

111% as a fraction and decimal

111/100 1.11

Between 2007 and​ 2020, the number of twin births in a certain country decreased by 11​%, to approximately 104,130. About how many twin births were there in​ 2007? There were approximately enter your response here twin births in 2007.

117,000

Will is 25% taller than​ Wanda, so​ Will's height is​ ____% of​ Wanda's height.

125

The area of a certain country is 26​% more than the area of California​, so the​ country's area is what percent of California's ​area?

126

Find the absolute change and the relative change in the following case. The estimated number of refugees in a part of the world increased from 6.4 million in 2005 to 20.6 million in 2020. The absolute change is 14.214.2 million. The relative change is ​%.

14.2 million 221.9%

1.46 as a percentage and a fraction

146% 73/50

State whether the following statement is true or​ false, and explain why. If the statement is​ false, state the true change. If the national economy shrank at an annual rate of ​4% per year for three consecutive​ years, then the economy shrank by 16​% over the three​-year period.

15.1

The total​ (after-tax) cost of a laptop computer is ​$1596.54. The local sales tax rate is 6.1​%. What is the retail​ (pre-tax) price? The retail​ (pre-tax) price of the computer is ​$ .

1504.75 100+6.1= 106.1 106.1= 1.061 ( CONVERT TO A DECIMAL) 1596.54/1.061= 1504.7502

1.5555 as a percentage and a fraction

155.56% 14/9 1.5555 X 10= 15.5555 10X= 15.5555 X= 1.5555 ------------- 9X = 14 x= 14/9

In the following​ statement, express the first number as a percentage of the second number. In 2019​, there were 59 million citizens of a particular country under the age of 15​, and a total of 345 million citizens. In 20192019​, % of the citizens of the country were under the age of 15.

17.1

Compare A and B in three​ ways, where A=52,114 is the number of deaths due to a deadly disease in the United States in 1995 and B18,426 is the number of deaths due to the same disease in the United States in 2018. a. Find the ratio of A to B. b. Find the ratio of B to A. c. Complete the​ sentence: A is​ ____ percent of B.

2.83 0.35 283%

100/15 a a fraction, decimal and a percentage

20/3 6.6667 666.67%

The percentage of people accessing the Internet in a given area increased from 61​% in 2005 to 82​% in 2020. Complete parts​ (a) and​ (b) below. The percentage of people accessing the Internet increased by percentage points.

21 34.4

During the past​ decade, the percentage of smokers in a certain country has decreased 27.4​% to 15.6​%. What percentage of the population of that country smoked at the beginning of the​ decade? The percentage of the population that smoked at the beginning of the decade was approximately %.

21.5

In the following​ statement, express the first number as a percentage of the second number. The land area of a country is 4,208,785 km2. The land area of the continent containing that country is 17,819,000 km2. The country has ​% of the land of the continent containing it.

23.6%

2.333 as a percentage and fraction

233.33 7/3

2.333... as a percent and a fraction

233.33% 7/3 10X2.333=23.33 23.33-2.33=21 21/9= 7/3

58% as a fraction to a decimal

29/50 0.58

Over the past​ decade, the percentage of renters in a given area who were​ "rent burdened"​ (meaning they spent more than​ 30% of their pretax income on​ rent) increased from 33% to 36%. Describe this change as an absolute change in terms of percentage points and as a relative change in terms of a percentage. The percentage of renters who were​ "rent burdened" increased by _____ percentage points. The percentage of renters who were​ "rent burdened" increased by ___​%.

3 9.1

18/5 as a decimal and a percentage

3.6 360%

0.6% as a fraction and decimal

3/500 0.006

Answer the question about the following quote from a news source. ​"The unemployment rate has risen more than a percentage point to 8.7​% in February from 6.6 % last​ November." What is the relative change in the unemployment rate expressed as a​ percentage?

31.8

In the following​ statement, express the first number as a percentage of the second number. A particular country has an estimated 4380 nuclear weapons out of an estimated total of 13,125 nuclear weapons worldwide​ (as of 2021​). The country has ​% of the total number of nuclear weapons worldwide

33.4%

In the following​ statement, express the first number as a percentage of the second number. On one particular day there were 4964 deaths attributed to a virus in a particular country and 117,467 new cases. The number of deaths was 4.2% of the total number of new cases.

4.2%

Describe the following change in two​ ways: as an absolute change in terms of percentage points and as a relative change in terms of a percentage. The percentage of a certain political party in a government body decreased from 54.7​% in 2016 to 49.9​% in 2020. The percentage of the political party in the government body decreased by ______percentage points.

4.8 8.8

Since 2009​, a company has reduced its annual water usage by 43​%. By what fraction was the 2009 water usage​ reduced?

43/100

4.45 as a percentage and reduced fraction

445% 89/20

4.60 as a percentage and a fraction

460% 23/5

State whether the following statement is true or​ false, and explain why. If the statement is​ false, state the true change. If the sales in your pizza restaurant increase by 8​% one year and decrease by 2​% the following​ year, your sales are up by 6​% over two years. False; if the sales in your pizza restaurant increase by 88​% one year and decrease by 22​% the following​ year, your sales are up by _______% over two years.

5.84

0.54 as a percent and a fraction

54 24/50

In the following​ statement, express the first number as a percentage of the second number. The​ full-time year-round median salary for workers in a certain country was ​$27,992 in​ 2000, compared to ​$50,065 in 2020. The​ full-time year-round median salary in 2000 was % of the​ full-time year-round median salary in 2020.

55.9%

Complete the following sentence. As of​ 2020, the life expectancy​ (at birth) of women in a particular country ​(82.4 ​years) is​ _____ percent greater than the life expectancy of men in that same country ​(77.4 ​years).

6.5

With a circulation of 28.5 million per​ year, a​ top-ranked magazine has 330​% more magazines in circulation than a magazine ranked seventh. What is the circulation of the​ seventh-ranked magazine?

6.6 MILLION

The population of city A is 31​% less than the population of city​ B, so city​ A's population is​ _____% of city​ B's

69

State whether the following statement is true or​ false, and explain why. If the statement is​ false, state the true change. A high school reports that its​ students' SAT scores were down by 6​% in one year. The next​ year, however, the test scores rose by 14​%. The high school principal​ announces, "Overall, test scores have improved by 8​% over the past two​ years."

7.16

22/3 as a decimal and a percentage

7.3333 733.33%

In the following​ statement, express the first number as a percentage of the second number. A certain candidate won 399 electoral votes in an​ election, out of a total of 554.

72

In the following​ statement, express the first number as a percentage of the second number. The 2020 revenue for Company A was ​$379million, compared to ​$525 million for Company B. The 2020 revenue for Company A was ​% of the revenue for Company B.

72.2

0.888 as a percentage and a fraction

88.89% X=8/9

0.90 as a percent and a fraction

90% 9/10

Distinguish between absolute and relative change. Give an example that illustrates how a relative change is calculated. Give an example that illustrates how a relative change is calculated. Choose the correct answer below.

Absolute change describes the actual increase or decrease from a reference value to a new value. Relative change is the size of the absolute change in comparison to the reference value and can be expressed as a percentage. If the new value is 50 and the reference value is​ 20, then relative change 50-20/ 20 X 100= 150%

Distinguish between absolute and relative difference. Give an example that illustrates how to calculate a relative difference. Give an example that illustrates how to calculate a relative difference. Choose the correct answer below.

Absolute difference is the actual difference between the compared value and the reference value. Relative difference describes the size of the absolute difference in comparison to the reference value and can be expressed as a percentage If the compared value is 50 and the reference value is​ 45, then relative difference = 50-45/ 45 X100 = 11.1%

The price of a​ four-star meal, only ​$160 per couple decades​ ago, has since increased 200​%. The current price of a​ four-star meal is which of the​ following?

A​ four-star meal is ​$480 because 2×$160=$320​, which is added to ​$160.

You currently earn ​$600 per​ month, but you are expecting your earnings to rise 10​% per year. In five​ years, what should you expect to be​ earning?

In five​ years, you should expect to be earning somewhat more than ​$660 per month because your earnings rise 10​% per​ year, which are added to the earnings of the previous month.

Suppose you have an​ 80% average in a class going into the final exam. You receive a​ 90% grade on the final exam. In​ general, does it follow that your average for the entire course is​ 85%? Explain.

No, because it is a general rule to not average percentages.

Explain the difference between the terms​ "percent" (%) and​ "percentage points." Give an example of how they can differ for the same situation. Give an example of how​ "percent" (%) and​ "percentage points" can differ for the same situation. Choose the correct answer below

Percent is used to describe a relative change or difference. Percentage points are used to describe an absolute change or difference. If a savings account that previously offered​ 2% interest now offers​ 6% interest, then the percent increase is​ 200% and the percentage point increase is 4. 6-2/2 X100=200%

Describe the three basic uses of percentages. Give a sample statement that uses percentages in each of the three ways. What is one way to use​ percentages? What is a second way to use​ percentages? What is the third way to use​ percentages?

Percentages can be used to express a fraction of something. For​ example, out of​ 15,000 employees,​ 3% lost their jobs. Percentages can be used to describe a change in something. For​ example, the cost of milk rose​ 5% within the past month. Percentages can be used to compare two things. For​ example, the car costs​ 25% more but gets​ 10% more miles per gallon.

Give an example to explain​ why, in​ general, it is not legitimate to average percentages.

Suppose that​ 30% of the female students and​ 40% of the male students in a college history class are from a major city. If there are 50 female students and 30 male students in the​ class, then 15 of the female students and 12 of the male student are from a major city. Of the 80 students in the class in​ total, 36.25 % are from a major city. The percentage of students from a major city is not equal to the average of​ 30% and​ 40%, which is​ 35%, because the number of female and male students in the class Differs

Determine whether the following claim could be true. Explain your answer. Restaurant prices have increased 122​% in the last 20 years.

The claim could be true because an increase of 122​% would more than nbspdouble the original​ prices, which is possible.

Determine whether the following claim could be true. Explain your answer. Hao is 185​% taller than his son.

The claim could be true because if the father is 185​% ​taller, then his height is more than double his​ son's height.

Determine whether the following claim could be true. Explain your answer. ​George's computer is 300​% faster than​ Nancy's.

The claim could be true because if​ George's computer is 300​% faster than​ Nancy's, then it is 4 times​ Nancy's computer​ speed, which makes sense.

termine whether the following claim could be true. Explain your answer. By turning off her lights and closing her windows at​ night, Maria saved 116​% on her monthly energy bill.

The claim could not be true because her new monthly energy bill would be -16​% of the previous monthly energy bill. This is impossible since the bill cannot be negative.

Determine whether the following claim could be true. Explain your answer. Through hard​ training, Maria improved her 10​-kilometer running time by 110%.

The claim could not be​ true, because her new running time would be negative 10​% of her previous running​ time, and running times must be positive.

Suppose the value of a home changed by −32​% over the past five years. Which of the following is​ true?

The house decreased in value over the past five years because the relative change is negative.

Consider the statement​ "The interest rate on auto loans has increased​ 50% over the past decade and now stands at 15​%." What can you​ conclude?

The interest rate a decade ago was 10​% because​ 50% of 10 is 5​, and 10+5=15.

Explain the difference between the key words​ 'of' and​ 'more than' when dealing with percentages. How are their meanings​ related? How are their meanings​ related?

The key word​ 'more than' is used to express the relative change between the referenced value and the compared value. The key word​ 'of' is used to express the ratio of the compared value to the referenced value. If the compared value is​ P% more than the reference​ value, it is ​(100+​P)% of the reference value.

The price of a movie ticket increased from ​$25 to ​$30. What does this mean about the new​ price?

The new price is​ 120% of the old price because the price increases by​ 20%, which is added to​ 100%. 30/25= 1.2 1.2X100= 120

Decide whether the statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). If the tax rate increases by​ 10% every​ year, it will rise by​ 100% in 10 years.

The statement does not make sense because each year there is a different reference value.

Decide whether the following statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Explain your reasoning. If John earns​ 20% more than Mary​ does, then Mary must earn​ 20% less than John does.

The statement does not make sense because if John earns​ 20% more than​ Mary, then Mary must earn approximately​ 16.7% less than John does.

Decide whether the following statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Explain your reasoning. ​I've decreased my caloric intake by 129​%, which has helped me lose weight.

The statement does not make sense. My new caloric intake would be −29​% of my previous​ intake, and a positive caloric intake is necessary for survival.

Assuming the given information is​ accurate, determine whether the following statement is true. Provide an explanation. 50​% of the hotels have a​ restaurant, and 10​% have a swimming​ pool, so 60​% of the hotels in town have a restaurant or a pool.

The statement is not true because some of the hotels with a restaurant may also have a pool.

Assuming the given information is​ accurate, determine whether the following statement is true. Provide an explanation. The class is 70​% women and 40​% of the women have blond​ hair, so blond women comprise 70​%X40​%=28​% of the class.

The statement is true because blond women comprise 40​% of women in the class and women comprise 70​% of the class.​ Hence, blond women comprise 40 % times 70 % equals 28 % of the class.

Decide whether the following statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Explain your reasoning. My bank increased the interest rate on my savings account​ 100%, from 3.2% to 6.4​%.

The statement makes sense because if the interest rate increased by​ 100%, to 6.4​%, then the previous rate was 3.2​%, which makes sense.

Decide whether the following statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Explain your reasoning. The rate of return on our fund increased by​ 50%, to​ 15%.

The statement makes sense because if the rate of return on our fund increased by​ 50%, to​ 15%, then the previous rate is​ 10%, which makes sense.

Decide whether the following statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Explain your reasoning. We found that these rare cancers were​ 700% more common in children living near the toxic landfill than in the general population.

The statement makes sense because the number of rare cancers found in children living near the toxic landfill is 8 times the number of rare cancers found in the general​ population, which is possible.

Decide whether the following statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Explain your reasoning. The price of tuition has quadrupled since my parents went to school—​that's a 300​% increase in price.

The statement makes sense. A 300% increase in price corresponds to a price that is 400% of the original price.

Decide whether the following statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Explain. In many European​ countries, the percentage change in population has been negative in recent decades.

The statement makes sense. A percentage change in population can be negative if the data reflects that.

At a particular​ college, 95​% of the undergraduates are female. If 3515 women attend the​ college, how many total undergraduates attend the​ college?

The total number of undergraduates is 3700.

Assume that​ 30% of city employees in City A ride the bus to work. Consider the following two​ statements: The percentage of city employees who ride the bus to work is​ 10% higher in City B than in City A. The percentage of city employees who ride the bus to work is 10 percentage points higher in City B than in City A. For each​ case, state the percentage of city employees in City B who ride the bus to work. Briefly explain why the two statements have different meanings.

When the percentage of city employees who ride the bus to work is​ 10% higher in City B than in City​ A, what percentage of city employees in City B ride the bus to​ work? enter your response here​% 33% When the percentage of city employees who ride the bus to work is 10 percentage points higher in City B than in City​ A, what percentage of city employees in City B ride the bus to​ work? 40% Percentage points describe an absolute change or​ difference, and percent​ (or %) describes a relative change or difference.

The United States has seen the federal debt grow considerably and the total debt now is over​ $17 trillion dollars. On a single day in October of​ 2013, the debt grew by a record amount in a single day when the federal government borrowed​ $328 billion dollars to fund measures from the past five months. The previous record was set two years ago when​ $238 billion dollars was added to the national debt.

​$2.55 trillion 17,000,000,000,000 X .15= 2.5E+12

Ever dream of traveling into​ space? For a mere​ $75,000, you may be able to do just that with the pace of a leisurely balloon ride. Think this is​ impossible? One​ company, World View​ Enterprises, has plans to offer such a ride as early as the year 2015. Their current​ competitor, Virgin​ Galactic, is planning similar rides via a rocket for​ $250,000. The ride is estimated to reach supersonic speed and reach a height of over​ 65,000 feet. Describing the supersonic ride Virgin Galactic noted that their planned​ ride's "speed and altitude are unusual per the standards of standard plane travel​ - the average commercial jet flies at about​ 30,000 feet and at about 550 miles per​ hour, or Mach 0.84​ (subsonic)" (Barber,​ 2013). Admittedly, the ride is intended for those of both financial means and those desiring an adrenaline rush.

​30% 250000/75000= 0.3 0.3X100=30

Clint's salary increased from ​$24,000 to ​$36,000 over a​ three-year period.​ Helen's salary increased from ​$26,000 to ​$39,000 over the same period. Whose salary increased more in absolute​ terms? In relative​ terms? Explain.

​Helen's salary increased​ more, because her salary increased by ​$13000. This is more than ​$12000​, the amount that​ Clint's salary increased. Neither​ person's salary increased​ more, because both of their salaries increased by 50%.