CHAPTER 3: RELATIVE MEASUREMENT

3-5: Absolute and Relative Change

“The Only Constant in Life is Change.”- Heraclitus

From climate change to weight gain to yards lost in a football down, humans are constantly measuring change. In many circumstances, we are interested in the change of size, amount, capacity, or magnitude of things in daily life. Change can be expressed in two ways.

  1. Absolute Change. Absolute change is synonymous with difference. Absolute change = new value − reference value. The absolute change describes the actual increase or decrease from a reference value to a new value. For example, if you were to get a raise from $15 per hour to $16.50 per hour, the absolute change would be $1.50 per hour.
  2. Relative Change. Relative change is the ratio of absolute change to the reference value. Relative change = absolute change/reference value. Relative change is a fraction that describes the size of the absolute change in comparison to the reference value. This comparison is a part-to-whole ratio. For example, people may be interested in how the change in some amount compares to the original amount (i.e., the reference value). Relative change can always be written as a percent change by multiplying by 100%.

Explore Icon Percent Change

Since a relative change is a part-to-whole ratio or fraction, we can convert it into a percent by multiplying the ratio by 100%. Consequently, relative change is often referred to as percent change.

 

Explore 1 – Absolute change

Student housing tends to be rented per room. In 2022, the average paid was $550 per room and in 2023 the same room was rented for $695. What is the absolute change in value between the two rental prices?

Solution

The absolute change is simply the difference between the two values. So we subtract: $695–$550 = $145. The absolute change tells us that the cost for a student to rent a room has increased by $145.

 

Explore 2 – Relative Change

You bought a computer three years ago for $1000. Today, it is worth only $300. Describe the absolute and relative change in the computer’s value.

Solution

The absolute change is the difference in values: $300 – $1000 = –$700. This tells us that the computer has lost $700 worth of value since you bought it.

The relative change is the absolute change divided by the original value: –$700 ÷ $1000 = –0.7 = –70% This tells us that the computer has lost 70% of its original value in the time you have owned it.

 

Change

Absolute change = New value – Original Value

Relative change = Absolute change ÷ Original value

 

Explore 3 – Comparing absolute and relative changes

Taylor’s annual salary increased from $42,000 to $49,000 over the last five years. Jo’s annual salary increased from $51,000 to $59,000 over the same period.

  1. Who has the higher absolute salary increase?
  2. Who has the higher relative salary increase?

Solution

  1. To determine the absolute salary increase we subtract the original salary from the new salary:  Taylor’s salary increased by $49,000 – $42,000 = $7000.  Jo’s salary increased by $59,000 – $51,000 = $8000. Consequently, Jo had the higher absolute salary increase.
  2. To determine the relative salary increase, we divide the absolute increase by the original salary: Taylor’s salary increased by $7000 and started at $42,000 so the relative increase is \$7000\div \$42,000=0.1667=16.67\%. Jo’s salary increased by $8000 and started at $51,000 so the relative increase is \$8000\div\$51,000=0.1569=15.69\%. SInce Taylor’s percentage increase was 16.67% compared to Jo’s percentage increase of 15.69%, Taylor has the higher relative salary increase.

 

Explore 4 – Comparing absolute and relative changes

Aasir and Aaron are in competition to win a new Laptop. Their father will buy a new laptop for the one who makes the most progress in their English tests. Aasir improves his test scores from 32 to 66. Aaron took the same two tests and improves from 25 to 55.

Calculate the absolute and relative change for Aasir and Aaron.

Who makes the most progress? Explain your reasoning.

Solution

The absolute increase of Aasir’s score is 66 – 32 = 34. The absolute increase of Aaron’s score is 55 – 25 = 30. SInce 34 > 30, Aasir has the greater absolute change. The relative change of Aasir’s score is 34 ÷ 32 = 1.0625 = 106%. The relative change of Aaron’s score is 30 ÷ 25 = 1.2 = 120%. SInce 120% > 106%, Aaron has the greater percentage (relative) change.

Although Aasir has the higher absolute change, Aaron has the higher percentage change. This means that Aasir made greater progress with points gained, but Aaron made greater progress by percent increase.

When relative change is written as a percent, it is telling us the percent of change from the original value. That change could be an increase or a decrease, so we are really finding the percent increase or the percent decrease from a starting value.

 

Explore 5 – Percent increase

The population of Orem city in Utah was 88,328 in 2010. It was 96,725 in 2019. What is the percent increase in population from 2019 to 2020? Round the percentage to two decimal places.

Solution

First we determine the absolute increase by subtracting the values: 96,725 – 88,328 = 8397. Now we use that absolute increase to determine the percent increase by dividing it by the original value: 8397 ÷ 88,328 = 0.095 = 9.5% The percentage increase in population is 9.51%.

 

Explore 6 – Percent decrease

Utah received 21.65 inches of rainfall in 2019 and 10.96 inches in 2020. What is the percent decrease in rainfall from 2019 to 2020? Round the percentage to two decimal places.

Solution

First we find the absolute change: 10.96″ – 21.65″ = –10.69″. Notice that this change is negative! This shows that there was a decrease in values.

Now we use the absolute change to determine the percentage change by dividing by the original value: –10.69″ ÷ 21.65″ = –0.494 = –49.4%

The rainfall decreased by 49.4% from 2019 to 2020.

If the change from the reference value to the new value is from a larger number to a smaller number, the absolute change will be a negative number. Consequently, the percentage change, (the absolute change) ÷ (the reference value), will be a negative percentage. To express the percentage change, we either say that the percentage change is a negative percentage, or we say that it has a percentage decrease using a positive percentage. This is because the word decrease substitutes for the negative sign of the negative percentage change. For example, we can either say, “the percentage change is –20%”, or, “it has a percentage decrease of 20%”.

 

Reflect Icon

  • Are the following two statements the same? Statement 1: “The rainfall today is 20% of yesterday’s rainfall.” Statement 2: The rainfall decreased 20% from yesterday to today.
Show/Hide Answer

No. They are not the same. Statement 1 tells that today’s rainfall is yesterday’s rainfall times 20%. For example, if yesterday’s rainfall was 1 inch, today’s rainfall is 1\times20\%=1\times0.2=0.2 inch. Statement 2 tells that today’s rainfall is yesterday’s rainfall minus yesterday’s rainfall times 20%. To illustrate, if yesterday’s rainfall was 1 inch, today’s rainfall is 1-1\times20\%=1-1\times0.2=1-0.2=0.8 inch.

 

Practice Exercises

  1. Aranya wants to go to an NBA Jazz game in Salt Lake City, but she does not have enough money. Aranya’s father told her he would contribute $100 if she makes a 20% increase in her next mathematics test. Aranya just finished the test and scored 84. Aranya’s last math test score was 72. Will Aranya get the $100 from his father? Show your work and explain your reasoning. Round the percentage to two decimal places.
  2. The 7-day average number of COVID-19 cases on January 18, 2020 was 2488 (before the major vaccination move). The 7-day average number of COVID-19 cases on May 18, 2021 was 315 (after the major vaccination move). Find the percent decrease in the number of COVID-19 cases from January 18 to May 18.
  3. Josie bought a house in Orem in 2017. The price was $300,000. Josie checks the value of the house in 2021, and finds it is $450,000. What is the percent increase in house price?
Show/Hide Answer
  1. No. Aranya’s score increased by 16.67% which is less than 20%
  2. The number of COVID-19 cases decreased 87.3% from January 18, 2020 to May 18, 2021.
  3. 50% increase in value.

 

 

In this section, we will take what we have learned and apply the concepts to new situations.

Perspectives

  1. The world population was 3.031 billion in 1960 and 6.922 billion in 2010. a) What is the absolute change in the world population between 1960 and 2010?  b) What is the relative change in the world population between 1960 and 2010?
  2. The population of the Salt Lake City metro area was 1,087,873 in 2010. It was 1,169,000 in 2020. a) Find the absolute and relative change in population growth from 2010 to 2020. b) In what circumstances would you use absolute versus relative change to describe the population growth?
  3. The price of a Big Mac was $2.45 in 1990. It is $3.99 in 2021. What is the percent increase in price?
  4. In 2017, the Utah excise tax on alcoholic beverages was $13.11 per gallon. In 2020, the Utah excise tax on alcoholic beverages was $15.92 per gallon. Calculate the percent increase in excise tax per gallon.
  5. Gasoline demand was measured as 8.9 million barrels per day in the Energy Administration’s (EIA) latest weekly report, up nearly 4% from the week prior.[1] According to this information, what was the prior week’s gasoline demand?
  6. According to the International Union for Conservation of Nature (IUCN), any species whose population decreases 50-70% over 10 years is identified as an endangered species. The population of Prairie Dogs in Utah was 9200 in 1989. It had decreased to 8000 by 2000.[2] Is the Prairie Dog qualified to be an endangered species? Explain your reasoning.
  7. In 2020 the average US corporate tax rate was 25.77%. This compares to a 37.5% corporate tax rate in Puerto Rico.[3] Calculate the percent difference in corporate tax rates between Puerto Rico and the US.
  8. According to NOAA, “Arctic ice normally reaches its largest extent in March or, occasionally, April, with a 1981–2010 average of 6.0 million square miles. It reaches its smallest extent in September, with a 1981–2010 average of 2.4 million square miles. “The 2020 Arctic sea ice minimum extent of 1.88 million square miles was reached on September 15, 2020. This was the second-lowest extent in the 40+ year satellite record. The 2021 extent was 5.70 million square miles, reached on March 21, 2021. [4] a) Determine the percent decrease for the average smallest extent from the average of 1981-2010 to the minimum extent in 2020.  b) Determine the percentage decrease for the average largest extent from the average of 1981-2010 to the maximum extent in 2021.
  9. Use the infographic The Cost of Mobile Internet Around the World to answer the questions:Infographic showing cost of mobile Internet across the world

a) What is the difference between the amount we pay in the US compared to France?

b) Calculate the percent difference between the amount we pay in the US compared to France.

c) What is the difference between the amount we pay in the US compared to India?

d) Calculate the percent difference between the amount we pay in the US compared to India.

e) What is the difference between the amount we pay in the US compared to South Korea?

f) Calculate the percent difference between the amount we pay in the US compared to South Korea.

  1. From 2021 to 2022, BYU increased tuition for LDS Church members from $5,970 to $6,120. During the same period, UVU increased tuition for Utah residents from $5,906to $6,010.

    a) Calculate the absolute tuition increases for each university. Which university had the higher absolute tuition increase?

    b) Calculate the relative tuition increases for each university. Which university had the higher relative tuition increase?

 

Show/Hide Answer
  1. a) 3.891 billion. b) 128.4%
  2. a) 81,127; 7.5%  b) Answers vary. Depends on the story you are trying to tell.
  3. 63%
  4. 21.4%
  5. 8.56%
  6. No. A 13% decrease is not enough.
  7. PR tax is 31.28% greater than US tax.
  8. a) 27.7% decrease b) 5% decrease
  9. a) U.S. pays $5.39 more than France     b) 2345.5% higher in U.S. than in France      c) U.S. pays $5.45 more than India      d) 3205.9% higher in U.S. than in India      e) U.S. pays $6.93 less than South Korea      f) 123.3% lower in U.S. than in South Korea
  10. a) BYU tuition increased by $150;  UVU tuition increased by $104;  BYU had the higher tuition increase    b) BYU: 2.5% increase;  UVU: 1.8% increase;  BYU had the higher percent increase in tuition

 

Skills IconIn this section, we will use what we have learned so far to practice skill problems.

 

Skill Exercises

Find the absolute change and relative change (round to the nearest whole percentage) from the first value to the second value.

  1. 10, 14
  2. 15,  10
  3. 12.99, 15.89
  4. 3.29, 2.20
  5. 4.4, 14.8
  6. 20.99, 10.99
  7. 3%, 5.5%
  8. 54%, 30%

Find the percentage change given the original value and an absolute increase/decrease.

  1. The original number is 10 with an absolute increase of 5.
  2. The original number is 10 with an absolute increase of 15 .
  3. The original number is 10 with an absolute decrease of 5.
  4. The original number is 10 with an absolute decrease of 15
  5. The original number is 2.99 with an absolute increase of 3.29
  6. The original number is 5.89 with an absolute decrease of 2.
Show/Hide Answer
  1. 50%
  2. 150%
  3. –50%
  4. –150%
  5. 10%
  6. –34%
  7. The absolute change is 4. The relative change is 40%.
  8. The absolute change is –5. The relative change is –33%.
  9. The absolute change is 2.9. The relative change is 22%.
  10. The absolute change is –1.09. The relative change is –33%.
  11. The absolute change is 10.4. The relative change is 236%.
  12. The absolute change is –10. The relative change is –48%.
  13. The absolute change is 2.5%. The relative change is 83%
  14. The absolute change is –24%. The relative change is –44%

 

 

 

  1. https://gasprices.aaa.com/gasoline-demand-continues-to-grow-as-pump-prices-fluctuate/ ↵
  2. https://www.iucnredlist.org/species/6090/22260975#population ↵
  3. https://taxfoundation.org/corporate-tax-rates-around-the-world-2020/ ↵
  4. https://www.climate.gov/news-features/understanding-climate/climate-change-minimum-arctic-sea-ice-extent ↵
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