4.4: Corequisite- Percentages and Ratios, Absolute and Relative Change, Rates
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By the end of this lesson, you should understand that
- population density is a ratio of the number of people per unit area.
- proportionality can be used to compare population densities.
- the concept of proportional representation in voting.
- how representation in the U.S. Congress is allocated.
- a relative change is different from an absolute change.
- a relative measure is always a comparison of two numbers.
By the end of this lesson, you should be able to
- calculate population densities.
- determine if two population density ratios are proportional to each other.
- compare and contrast populations via their densities.
- calculate a unit rate.
- solve a proportion by first finding a unit rate and then multiplying appropriately.
- calculate a relative change.
- explain the difference between relative change and absolute change.
PROBLEM SITUATION 1: HOW THE CENSUS AFFECTS THE HOUSE OF REPRESENTATIVES
Every 10 years, the United States conducts a census. The census tells us how many people live in each state. You can also find out how much populations have changed over time from the census data. The original purpose of the census was to decide on the number of representatives each state would have in the House of Representatives. Census data continue to be used for this purpose, but now there are many other uses for the data. For example, governments may use the data to plan for public services such as fire stations and schools.
In this problem situation, your instructor will ask your team to explore data from one of two census regions: 1) South Atlantic States or 2) Mountain States. There are tables on the next page that show the population of each state in these census regions in 2010 and 2020. You will be asked to calculate the population growth (in people) as a percentage for each state in your assigned region, and for the region as a whole. You will examine how this affects the number of representatives each state has in the House of Representatives.
The absolute change in a state’s population tells us by how many people the population has changed. The relative change is the change as it compares to the earlier population. Often relative change is given as a percentage. See the table below for an example using the population change in California from 2010 and 2020.
| 2020 Population (in millions) | 2010 Population (in millions) | Absolute Change (in millions) |
Relative Change |
|
| California | 39.58 | 37.27 | 39.58-37.27=2.31 | 2.3137.27*100%=6.2% |
Use the following data to answer Questions 1–6 for your assigned census region.
South Atlantic States
| 2020 Population | 2010 Population | |
| Florida | 21,570,527 | 18,801,310 |
| Georgia | 10,725,274 | 9,687,653 |
| North Carolina | 10,453,948 | 9,535,483 |
Mountain States
| 2020 Population | 2010 Population | |
| Arizona | 7,158,923 | 6,392,017 |
| Colorado | 5,782,171 | 5,029,196 |
| Nevada | 3,108,462 | 2,700,551 |
(1) Match the states below with the absolute change in population.
|
|
(2) Match the states below with the relative change (in percentage form) in the population.
|
|
(3) Which of the three states from each region changed the most in absolute population?
Mountain states:
South Atlantic states:
(4) Which of the three states from each region changed the largest relative increase in population?
Mountain Region states:
South Atlantic states:
(5) Explain why the lists in Question 3 and Question 4 are not the same.
(6) Use the previous information to calculate the following absolute and relative changes in total population.
(a) The absolute change in total population for the three Mountain Region states is
___________ people.
(b) The relative change in total population for the three Mountain Region states is _________ %.
(c) The absolute change in total population for the three South Atlantic Region states is
____________ people.
(d) The relative change in total population for the three South Atlantic Region states is
____________ %.
(7) Let’s now consider a state we have not explored yet: Illinois. Interestingly, the population of the state of Illinois actually decreased between 2010 and 2020. Illinois had a population of 12,864,380 in 2010. Its population was 12,822,739 in 2020.
(a) What was the absolute change in Illinois’ population?
(b) What was the relative change in Illinois’ population? Write a statement that describes the relative change in Illinois’ population between 2010 and 2020. Think about this on your own first before sharing with your group. Round your answer to the nearest hundredth of a percent.
PROBLEM SITUATION 2: USING RATIOS TO MEASURE POPULATION DENSITY
Earth’s human population has grown from about 1 billion people to about 7 billion in the last two centuries. However, populations in different regions do not always grow uniformly. For example, populations tend to increase in areas where people already live close enough to one another to find mates. On the other hand, crowded populations decrease when deadly diseases, such as smallpox or Ebola virus, sweep through them. In this lesson, you will compare geographic regions by their population densities. The population density of a geographic region is a ratio of the number of people living in that region to the area of the region. Population density ratios are typically reported in a way that makes it easy to compare the ratio to a standard area measurement, such as one square mile or one square kilometer.
Example: Imagine 200 people standing in an open field that measures one mile by one mile (also known as one square mile). The population density of the open field could be thought of as 200 people per one square mile. This can be expressed as a ratio. The expression below shows the population density of the open field as a ratio in fraction form.
\[\dfrac{200\;people}{1\;square\;mile} \nonumber\]
(8) A proportion is the equality of two ratios. Quantities are proportional if they satisfy a proportion. Picture the open field from the example that has a population of 200 people per one square mile. Then, picture a second open field. The length of each side of the second open field is double the length of each side of the first open field (making it 2 miles by 2 miles). The amount of people standing in the second open field is double the amount of people standing in the first open field. Is the population density of the second field equal to the population density of the first field? Explain.
(9) An island in the archipelago nation known as the Maldives is threatened by global warming. The island, called the Ari Atoll, is approximately one square mile in size, and has a population of 1,260 people. What is the population density per square mile?
(10) Imagine that global warming caused flooding in Ari Atoll, so the island is now half of its former size. However, the same number of people still live in Ari Atoll.
(a) Before doing any calculations, decide if you think the population density of Ari Atoll increased, decreased, or stayed the same after the flooding. Explain your answer.
(b) Calculate the population density of Ari Atoll after the flood.
(11) Suppose that since everyone is now living closer together, the birth rate on Ari Atoll increases rapidly. Soon the population has quadrupled. What is the population density now?
(12) Picture the open field from the example that contains 200 people and measures one square mile. Now picture another open field that measures five miles by five miles and has exactly the same population density as the first open field.
(a) How many people are standing in this field?
(b) Write a mathematical expression that shows that the population density of the first open field and the population density of the second open field are equal to each other.
(13) Consider a tiny island in the Atlantic Ocean. There are 50 survivors shipwrecked there. When they first arrived on the island, they all set up camp near the waterfall in the middle of the island. About six months later, a professor who was among the survivors claims the island can only support a population density of one person per acre (an acre is 43,560 square feet, which is a little less than the size of an American football field).
If the survivors then decide to space their campsites out evenly across the island, how will this affect the overall population density of the island?
PROBLEM SITUATION 3: VOTING FOR THE HOUSE OF REPRESENTATIVES
The U.S. Congress consists of two bodies, the Senate and the House of Representatives. For legislation to be enacted, it must pass in both of these bodies. The House of Representatives consists of a fixed number of 435 voting representatives. These representatives are allocated to each state proportionally, based on population. (Recall that a proportion is the equality of two ratios. Quantities are proportional if they satisfy a proportion.) In other words, this means that the higher a state’s population, the more representatives it will have. The aim of this method of allocation is for U.S. voters to have relatively equal representation in the House. (Note: Each state must have at least one representative.)
(14) Representatives were last allocated in 2020, when the population of the U.S. was approximately 330 million. Find the average number of people represented by a single U.S. representative in 2020. Write your answer as a ratio in fraction form.
(15) Determine how many representatives should be allocated to the following states:
Ohio (2020 population 11,799,000):
Wyoming (577,000):
California (39,538,000):
(16) The U.S. population in 2020 was approximately 330,000,000. Based on current trends, the population of the U.S. is expected to grow to 352,427,000 in 2030 when the next U.S. Census is conducted (and representatives are reallocated). However, the population of Ohio is expected to remain approximately the same. Will Ohio keep the same number of representatives? If not, how many will they now have?
(17) All members of the U.S. House of Representatives will be up for reelection in 2026. The population of the U.S. is expected to be approximately 340 million in 2027 when the congressional session begins.
Find, as a unit rate, the average number of people who will be represented by a single U.S. representative in 2027.