2 Linear Growth Models
Learn by watching
The links below will launch the video lessons in YouTube
- Constant Numbered Rate of Change (8 minutes 13 seconds)
- Planning and Predicting with Linear Growth Models (5 minutes 18 seconds)
- Linear Growth Model when Rate of Change is Known (bus example) (6 minutes 46 seconds)
- Linear Growth Model when Rate of Change is Known (negative rate example) (7 minutes 25 seconds)
Try Exercises #1-4.
The links below will launch the video lessons in YouTube
- Linear Growth Model when Rate of Change is NOT Known (population example) (9 minutes 59 seconds)
- Linear Growth Model when Rate of Change is NOT Known (negative rate example) (7 minutes 44 seconds)
Try Exercises #5-8.
learn by reading
In a linear growth model, the rate of change must be a constant, set number. Here are some examples of constant rates that you may have come across:
- Washington State’s minimum wage is $15.74 per hour.
- The community band has been losing 3 members every year.
- Once your warranty expires, a car dealership charges $95 per hour for labor on repairs.
- An average motorcycle gets 55 miles per gallon of fuel.
Note that our rate of change is always given in terms of 1 unit of increase:
- Washington State’s minimum wage is $15.74 per (1) hour.
- The community band has been losing 3 members every (1) year.
- Once your warranty expires, a car dealership charges $95 per (1) hour for labor on repairs.
- An average motorcycle gets 55 miles per (1) gallon of fuel.
One of the nice things about having a constant, continuous growth pattern is that it is very straightforward to write an equation or formula to represent the situation. And the nice thing about having a mathematical formula is that you can then make plans or predictions.
Using RATES of change to write linear equation models
Example 1: You want to figure out how much money you are going to be making on a given paycheck. Your pay is calculated based on how many hours you work. You are earning WA state minimum wage ($15.74 per hour). Write a formula for the amount of your paycheck.
The first thing you should think about when writing a formula is what your variables are. Clearly defining these is a step many people like to skip, however, it is really important for communicating your math thinking to others. It also really helps when making predictions so that you substitute values in the correct place.
In this case, let’s let:
n = number of hours worked
P = paycheck amount (in dollars)
For every 1 hour worked, our paycheck is increasing by $15.74. So to find the dollar amount of the paycheck, we simply need to multiply our wage (dollars per hour) by the number of hours worked.
P = 15.74 * n
Once you have a formula, we can then use it to plan and predict.
Example 1a: Suppose you know that you worked 20 hours last week. What would you expect your paycheck to be?
Well, we can substitute 20 in for n in our formula to find the value expected for your paycheck.
P = 15.74 * n
P = 15.74 * 20 = 314.80
Looking back at how we defined our variables, P is the value of the paycheck in dollars.
You can plan for your 20-hour paycheck to be $314.80.
Example 1b: Suppose you are hoping to make $18,000 this year. How many hours would you have to work?
This time, I know the value of the paycheck ($18,000) — so that number needs to be substituted in for the variable, P. I now have enough information to find the number of hours, n.
P = 15.74 * n
18000 = 15.74 * n
This time, I’m going to have to do just a little bit of algebra. The n that I want to find is not by itself in the equation! It is being multiplied by 15.74. So to get the n alone, I am going to divide each side of the equation by 15.74
18000 = 15.74 * n
(18000) / 15.74 = (15.74 * n) / 15.74
1143.6 = n
You will have to work 1143.6 hours this year to end up with $18,000 (this ends up being about 22 hours/week).
(Yes, this problem is a little bit over-simplified. Your take-home paycheck will always be less due to income tax and social security deductions).
Example 2: You want to know how many members to expect in your community band in five years. The band has been losing 3 players each year. Write a formula for the number of players in the band.
Identifying our variables:
n = number of years (?)
P = # of band players
Now wait a minute… I don’t have enough information here! The “losing 3 players a year” is important information — but it only tells us how the band population is changing. If we want to know the number of players in the band (and we do!), we need to have some sort of a starting point and a starting band size to do our calculations from.
Example 2 (now a solvable version): In 2022, your community band had 124 players. The band has been losing 3 players each year. Write a formula for the number of players in the band in a given year.
In this case, we have a starting value. In 2022, our band had 124 players.
Identifying our variables:
n = number of years since 2022
P = # of band players in a given year
Every year since 2022, the band has been losing 3 players. Because the number of players is decreasing, we can write our rate of change as “-3” players per year. If we multiply the -3 by the number of years that have gone by, we will end up with how many players have left the band.
P = 124 + (- 3)n or, more simply, P = 124 – 3n
Example 2a: How many band players would you expect to have in the year 2030?
This is where defining your variables earlier is incredibly helpful. Because our starting value of 124 players occurred in 2022, when we go to substitute our “n” in the equation, we can’t just put 2030 in for the year. What we really are interested in is how many years have gone by between 2022 and 2030 (when we’ve been losing our players!). Subtracting the years, we find that we are looking for the number of band members 2030 – 2022 = 8 years after our known starting value. So we substitute n = 8 into our formula.
P = 124 – 3n
P = 124 – 3(8) = 124 – 24 = 100 players.
In the year 2030, we would expect our band to have 100 players.
Example 2b: The band will lose access to the venue if they drop below 75 players. If the current decrease in players continues at the same rate, when would you no longer be able to use the venue?
We already have our formula. In this case we know the number of players will be 75. So put 75 in for P in the formula.
P = 124 – 3n
75 = 124 – 3n
This is leaves us with just a single variable, and we can solve for n. Remember when solving algebra equations to get a variable alone, we always get rid of any added or subtracted values first. Then we can get rid of multiplied and divided values. (Why? We are “undoing” our order of operations, so we do our PEMDAS[1] work in backwards priority). So in this case, since we want to get the n alone, we will subtract 124 from each side
75 = 124 – 3n
(75) – 124 = (124 – 3n) – 124
-49 = -3n
Then we can divide each side by -3 to finish getting the n alone.
-49 = -3n
(-49) / -3 = (-3n) / (-3)
16.33 = n
Now that we found n what does this mean? Looking back at how we defined our variables, n was the number of years since 2022. So 16.33 years after the year 2022, we will no longer be able to use the venue. What year will that be? Well 2022 + 16.33 = 2038.33.
If we continue to lose players at this same rate, then sometime during the year 2038, our community band will drop to 75 players and we will lose access to the venue.
Linear equations all have this same pattern. You have some sort of starting value. Then we can add the constant numbered rate-of-change multiplied by how often that change happens.
GENERAL LINEAR EQUATION MODEL
You may have noticed that our answer to Example 1 did not have a starting value (P=15.74n). So did we mess up? Well, let’s think about the minimum wage paycheck situation again here in context. If n is the number of hours we worked, and P is the dollar amount of our paycheck, then our starting value (when n = 0 hours worked) will be $0. If we don’t work, we don’t get paid! So technically our equations would have been:
P = 0 + 15.74n
Of course, adding 0 doesn’t change anything! So P = 15.74n was a great way to (correctly) write the linear equation model.
Example 3: Once your warranty expires, a car dealership charges $95 per hour for labor on repairs. The dealership will also charge you the cost of the parts. If a new transmission costs $2000, write an equation model to represent the final amount of your repair bill.
n = number of hours of labor
P = cost of repair bill (in dollars).
a = starting amount (the cost for 0 hours of labor). For this repair, a = $2000 (the cost of the transmission part).
b = the rate of change. In this case b = $95 dollars per hour.
Example 4: An average motorcycle gets 55 miles per gallon of fuel. Write an equation model to represent how far you would expect to travel after filling up at the gas station?
Let:
n = number of gallons of fuel
P = number of miles the motorcycle will travel
a = starting amount (the number of miles for 0 gallons of gas). Since we cannot drive without fuel, a = 0 miles.
b = the rate of change. Our average motorcycle gets b = 55 miles per gallon
P = 0 + 55n or just P = 55n
Exercises (Part 1): Try these!
For each of the following situations:
- label the units for your variables (n = ? and P = ?)
- write a linear equation model of the form P = a + bn to fit the situation
- use your equation model to answer the questions that follow (Be sure to show your work — if evaluating, write what you are typing into your calculator; if solving, show the steps to solve the equation).
1) A donor for a popular charity gifts $1500. As part of a fundraising event, they promise to also donate an additional $10 for every banquet ticket sold.
a) If 230 banquet tickets are sold, how much money will the donor be giving to the charity?
b) If the maximum seating capacity at the banquet is 500 people, what is the maximum possible dollar contribution from this donor?
c) If the charity needs $4750 to meet their expenses, how many tickets to the banquet will they need to sell?
2) You are planning a vacation and would like to save up enough money to go. You figure that hotel costs and food for the trip will be $500. Current fuel prices are $4.25 per gallon.
a) You want to stay close to home, and decide you do not want to stop to fill up for gas until you get back home. If your fuel tank will hold 12.5 gallons, how much should you plan to save for your vacation?
b) You decide you can budget $850 for this trip. How many gallons of gas can you use?
c) Using your answer to part (b), how many miles away can you plan to travel (remember you have to get to your destination AND back home)? Assume your car gets 22 miles per gallon.
3) A city planning department wants to fix up their roads. The cost to bring out a truck for the day is $125. The company charges $40 per pothole.
a) If there are 17 potholes on a main stretch of road, how much will the city have to pay to get all of the potholes repaired?
b) If the city budget only has $500 allocated for road repairs, how many potholes will they be able to fix?
4) Your “Good-to-Go” account has $100 on it. The “Good-to-Go” Pass allows you to cross the WA-520 toll bridge. It costs $6.25 to drive across the bridge during rush hour.
a) How much money will be on your account if you need to use the bridge to drive to work every day for a week? (Assume you work 5 days a week and have to cross the bridge twice each day to get to work and back home).
b) How many times can you cross the bridge before you have to add more money to your account?
writing linear equation models when you don’t know the rate of change
If we want to write a linear equation model (P = a + bn), it is our responsibility to come up with the a and the b that fit our given situation. Sometimes, however, we are not given the rate of change, b. In these cases, as long as we are given some pieces of information about a value other than the starting point, we can figure out the value of b to use. Remember that the rate of change must given in terms of 1 unit of increase
Example 5: The population of the city of Kirkland, WA in 1990 was 45,104 people. By 2021, there were 92,107 residents. Assuming a linear growth pattern, write an equation model representing the population growth of Kirkland.
The general form for a linear equation model is: P = a + bn. To come up with a linear equation model, we need to describe what our variables P and n are. Then we need to identify the values of a and b that are specific to our problem. Remember that a is our starting value, while b describes the rate of change.
In this situation, we are just given 2 different populations of Kirkland measured at different times. We might as well use one of these known values as our starting point. I highly recommend choosing the “oldest” population as the starting point — that way we don’t have to unnecessarily involve negative numbers 🙂 So if we use the fact that there were 45,104 people in Kirkland in 1990 as our starting point, then:
n = number of years since 1990
P = population of Kirkland
and a = 45,104 (our starting population when n = 0 years since 1990….)
Because we know two data points, and we are assuming linear growth, we can figure out the rate of change.
In this case, the population grew from 45,104 to 92,107. This represents a change (increase) of 92,107 – 45,104 = 47,003 people. However, this isn’t our rate, because this change happened over more than one year! Because we are assuming linear growth, we should have the same number of new people each year (a constant-numbered rate of change). This means we can simply divide the increase increase in people by how many years went by — in this case, 2021 – 1990 = 31 years.
So our rate of change = 47003 people / 31 years = 1516.2 people per year, and we will use b = 1516.2 to describe the rate of change for the Kirkland population.
We can use the following equation to model the growth of Kirkland since 1990:
P = 45,104 + 1516.2n
n = number of years since 1990
P = population of Kirkland
a = 45,104 (Kirkland’s population in 1990)
b = 1516.2 new people in Kirkland per (1) year
Once we have our equation model, we are ready to make plans and predictions!
Example 5a: Estimate the population of Kirkland in 2025.
P = 45,104 + 1516.2n
n = number of years since 1990. I want to know the population in 2025.
2025 – 1990 = 35 years since our “starting value” of 1990
P = 45,104 + 1516.2 * 35 = 50,396 people
Based on our equation model, we predict the population in Kirkland in 2025 to be 50,396 people.
Example 5b: Predict when Kirkland will have a population of 150,000 people.
P = 45,104 + 1516.2n
This time we want to find out when (so we are looking for our time, n) when Kirkland’s population will be 150,000 people. This time, we are substituting our value in for P.
150,000 = 45,104 + 1516.2n
I’m going to have to do some algebra to solve for n.
150,000 = 45,104 + 1516.2n
Start by subtracting 45,104 from each side of the equation.
(150,000) – 45,104 = (45,104 + 1516.2n) – 45,104
104,896 = 1516.2n
Then divide each side by 1516.2 from each side of the equation to finish getting the n alone.
(104,896) / 1516.2 = (1516.2n) / 1516.2
69.2 = n
Go back to how your variables were defined before presenting your answer. What does this result mean? Well, n = number of years since 1990. So 69.2 years after 1990 would be 1990 + 69.2 = 2059.2.
Assuming population continues at this same rate, then sometime during the year 2059, Kirkland’s population will reach 150,000 people.
Example 6: As open educational resources have become more popular as a free-to-low-cost option, the market share of the textbook publishing industry has been steadily dropping following a linear pattern. In 2013, textbook publishing companies brought in $4.81 billion dollars. By 2020, those same companies were only bringing in $3.10 billion dollars.[2] If this same rate continues, when can we expect textbook publisher earnings to drop below $2 billion dollars?
Even though this question doesn’t specifically ask us to come up with a linear growth model, we need one so that we can make the requested prediction.
Here, we are given two different values and times. Using the earliest value as my “starting amount”, I’m going to let:
n = number of years since 2013
P = publisher market share (in billions of dollars)
[NOTE – By defining my P value to be using units of “billions of dollars”, I can make my life easier by not having to write in all those zeros for my calculations. If you defined P to be in “dollars”, then you’d need to include the additional zeros. I like the shorter method better. I just need to remember that any results I get will have units of “billions of dollars”. This is why it is such a good idea to clearly define your variables before you start!]
Because of this, I can let:
a = 4.81 (market share in billions of dollars in 2013)
In this problem, again, I wasn’t given the rate of change. So I’m going to have to calculate it. Fortunately they’ve given me a second know data point, so I have enough information to do that. I know that the values are decreasing over time, so I should be using a negative rate of change.
The difference between 4.81 and 3.10 is $1.71 billion dollars. This decrease takes place between 2013 and 2020, which is 2020-2013 = 7 years. We can figure out the rate of change by dividing our $1.71 billion dollars by 7 years to get the linear rate of change.
b = $1.71 billion dollars / 7 years = $0.244 billion dollar decrease per year.
So b = -0.244
Putting all that together, I get my general linear equation:
P = a+bn
P = 4.81 + (-0.244)n, or, more simply:
P = 4.81 – 0.244n
Now that I have a functioning linear equation model, I can use it to predict any year or market share value I am interested in. Example 6 is asking us to find when the publisher earnings are going to drop below $2 billion dollars. Remember that I defined my variables to be in terms of billions of dollars, so I have to let P = 2. Then I can solve for n to find my “when”.
P = 4.81 – 0.244n
2 = 4.81 – 0.244n
To get the n alone, first get rid of any added or subtracted values. Here, to get rid of the 4.81, I’m going to have to subtract 4.81 from each side of the equation.
(2) – 4.81 = (4.81 – 0.244n) – 4.81
-2.81 = -0.244n
Lastly, divide each side of the equation by -0.244 to finish getting the n alone.
(-2.81) / -0.244 = (-0.244n) / -0.244
11.5 = n
Going back to how we defined our variables…. n is the number of years since 2013. So 11.5 years after 2013 would be 11.5 + 2013 = the year 2024.5.
Assuming rates continue, about half way through the year 2024, the market share of textbook publishers will drop below $2 billion dollars.
Exercises (Part 2): Try These!
For each of the following situations:
- label the units for your variables (n = ? and P = ?)
- write a linear equation model of the form P = a + bn to fit the situation (Be sure to include any work you used to calculate b).
- use your equation model to answer the questions that follow (Be sure to show your work — if evaluating, write what you are typing into your calculator; if solving, show the steps to solve the equation).
5) A warehouse outside of a factory currently has an inventory of 1245 boxes. After an 8-hour work day, the warehouse has 2000 boxes. Assume the warehouse was being filled at a constant (linear) rate.
a) How many boxes per hour is the factory able to provide to the warehouse?
b) What would be the inventory at the end of a 40-hour work week?
c) How long will it take to fill the warehouse to its 50,000 box capacity?
6) In the year 2007, a FOREVER stamp cost cost $0.41. In 2023, the cost of a FOREVER stamp was $0.63. Assume that the cost of stamps increased at a constant (linear) rate.
a) If price increases continue at the current rate, how much will a FOREVER stamp cost in 2035?
b) In what year would you expect a FOREVER stamp to cost one dollar?
7) In January of 2021, there were 980,000 games available on the Apple App Store. By July of 2021, there were 984,200 games available.[3] If we assume that the number of available games is steadily increasing at a constant (linear) rate,
a) How many games does this pattern predict will be available in January 2022?
b) At this rate, when will there be 1,000,000 games available for purchase in the Apple App Store?
8) In 2002, newspaper publishers made $46.2 billion dollars. As digital media has gotten more popular, the newspaper industry has been steadily losing money. By 2020, newspaper publisher revenue had decreased to $22.2 billion dollars.[4] Assuming the revenue is declining at a constant (linear) rate,
a) How much money would you expect newspaper publisher to make in the year 2025?
b) Based on this model, when would you expect newspaper revenue to drop below $1 billion dollars?
- PEMDAS is an acronym to remember the order of mathematical operations. P = parentheses; E = exponents; MD = multiplication and division (whatever comes first, left to right), and AS = addition and subtraction (whatever comes first, left to right). ↵
- Hanson, Melanie. “Average Cost of College Textbooks” EducationData.org, July 15, 2022, https://educationdata.org/average-cost-of-college-textbooks ↵
- https://www.statista.com/statistics/268251/number-of-apps-in-the-itunes-app-store-since-2008/ ↵
- https://www.census.gov/library/stories/2022/06/internet-crushes-traditional-media.html ↵
