CHAPTER 6: LINEAR GROWTH

6-2: Linear Reasoning – Rate and Initial Value

Reasoning

Reasoning can be considered the ability of the mind to think and understand things in a logical way. Reasoning is also the process of using existing knowledge to draw conclusions, make predictions, or construct explanations. We use reasoning every day of our lives. From something as simple as choosing what to eat for lunch to something more intense like solving a major project issue at work. Reasoning is part of being human. We take all of the information we know or are given and consciously use our intellect to understand what we have and what we can do with it. 

Explore Icon Linear Reasoning

Linear patterns can be shown in tables, as mathematical statements, have a starting value, and a constant rate of change from one value to the next. Linear reasoning consists of thinking logically about how a linear pattern is made.

Explore 1 – Given a table

 

The table shows a linear pattern between the number of hours Tim baby sits his younger sister and the wages he will earn.

Hours 0 1 2 3 4 5  20
Wages ($) 0 8 16
  1. How much money is earned for every 1 hour increase in work?

 Solution

Every increase of 1 hour increases the wage by $8 since $8 – $0 = $8 and $16 – $8 = $8. Since the pattern is linear, this increase is a constant.

  1. Write an addition equation starting from 0 hours to show the amount of money earned for working 5 hours.

Solution

0 + 8 + 8 + 8 + 8 + 8 = 40    Every hour, we add $8.

  1. How much money is earned for working 20 hours? It is tedious to have to keep adding 8 as the number of hours increases by 1 until you reach 20 hours.

Solution

Following the same pattern of adding $8 for each additional hour worked we get 0 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 160.  $160 is earned for working 20 hours.

  1. Explain a faster way to figure out the amount of money earned for working 20 hours. Explain your reasoning.

Solution

Repeated addition can be written as multiplication. So adding 8 twenty times can be written 8\times20

 

 

Explore 2 – Given a linear pattern

Waiting at a train station, Bella noticed that the number of people on Platform 7 started at zero and increased by 45 every 15 minutes.

  1. If this linear pattern continued, how many people were there after 4 hours?

Solution

We need to find out how many 15-minute intervals there are in 4 hours. We could turn 4 hours into minutes : Since there are 60 minutes in an hour, 4 hours = 4 · 1 hour = 4 · 60 minutes = 240 minutes and then divide 240 minutes by 15 minutes to get 16 intervals. Alternatively, knowing there are 4 15-minute intervals in 1 hour (since 4 · 15 mins = 60 mins = 1 hr), there are 4 · 4 = 16 intervals in 4 hours. Consequently, since there are 45 additional people every 15-minute interval, and we have 16 intervals, there must be an increase of 16 intervals · 45 people/interval = 720 people in 4 hours. This is equivalent to adding 45 people sixteen times. So there were 720 people on Platform 7 after 4 hours.

  1. It could be overwhelming to keep adding 45 people for every 15 minutes until you reach 4 hours. Describe a more efficient way to solve the problem? Explain your reasoning.

Solution

We can use the unit rate of growth: since the pattern is linear, the growth rate is constant at \dfrac{45\;\text{people}}{15\;\text{minutes}}. This can then be turned into a unit rate by simplifying the fraction to \dfrac{3\;\text{people}}{1\;\text{minute}}. Then since 4 hours = 240 minutes (4 hrs · 60 mins/hr), we multiply 3 people per minute by 240 minutes to get 720 people. This is because the number of people increases by 3 as the time increases by 1 minute for a total of 240 minutes. So there were 720 people on Platform 7 after 4 hours.

 

Explore 3 – Given a rate

A store has frozen popsicles on sale for $3 for five popsicles. This is a rate of $3/5 popsicles.

1. Use this rate to describe how you would calculate the total cost of buying 45 popsicles?

Solution

It costs $3 for a lot of 5 popsicles. To buy 45 popsicles, we need to know how many lots of 5 popsicles we are going to buy. Since 45 ÷ 5 = 9, we need to buy 9 lots at $3 per lot. So, we multiply $3 per lot by 9 lots to get $27.

2. If you change the rate into a unit rate, describe how you would calculate the total cost to buy 45 popsicles?

Solution

The rate is \dfrac{\$3}{5\;\text{popsicles}}. To turn it into a unit rate, we must make the denominator equal to 1 by dividing by 5. So, we divide the denominator and numerator by 5 and get \dfrac{\$0.6}{1\;\text{popsicle}}. Since it costs $0.6 per popsicle and we want to buy 45 popsicles, we multiply $0.6 per popsicle by 45 popsicles to get $27.

 

Example

A SCUBA diver breathes air from their tank at 20.4 liters per minute.

a) Create a table showing the total air breathed in 0 to 5 minutes in 1 minute intervals.

b) State the unit rate of change and explain what it tells us.

c) Write an addition equation showing how to determine the amount of air used after 8 minutes.

d) Explain how the constant rate of change relates to the addition equation.

e) Write a multiplication equation showing how to determine the amount of air used after 8 minutes.

f) Explain how the constant rate of change relates to the multiplication equation.

 

Show/Hide Answer

a)

Time (mins) 0 1 2 3 4 5
Air (liters) 0 20.4 40.8 61.2 81.6 102

b) The rate of change is 20.4 liters per minute. The rate of change tells us how much air is used each minute. So, a rate of change of 20.4 liters per minute tells us that for each minute that passes, another 20.4 liters of air is breathed. 

c) Air used = 0 + 20.4 + 20.4 + 20.4 + 20.4 + 20.4 + 20.4 + 20.4 + 20.4

d) Every minute 20.4 liters of air gets added to the air used.

e) Air used = 8(20.4)

f) Since 20.4 liters of air is breathed each minute, we multiply this rate by the number of minutes that have passed.

 

Reflect Icon

  • What is linear reasoning. How does it work?
Show/Hide Answer

Linear reasoning is thinking logically about how linear patterns are made. A linear pattern has a constant increase in the dependent variable for a set increase in the independent variable. To calculate the increase in value of the dependent variable at a given value of the independent variable, we multiply the dependent value by the number of increases of the independent value.

 

Explore Icon Initial Value 

In Explores 1-3, we assumed the increase of the dependent values started from zero. However, that is not always the case in our daily lives. For example, we may need to pay a shipping fee in addition to whatever the cost is of whatever we are buying. Or there may be a delivery charge to deliver a meal to your home. Some companies charge a non-refundable down payment to rent an apartment. Or we could have some continuous data that goes back years, but we just need the data for the last 18 months. The initial value would be where we want to start.

Explore 4 – Given a rate

The table shows a linear pattern for the charge of taking an Uber. Uber calculates the rider’s fee by charging a drop fee of $10 per ride and a mileage fee of $1.5 per mile.

Mileage 0 1 2 … 4 ..  20 …
Charge ($) 10 11.50 13 … ? … ? …
  1. Why does the charge start at $10?

Solution

This is the $10 drop fee you need to pay to book the Uber.

  1. Why is the charge $11.50 when the Uber runs only 1 mile?

Solution

The mileage fee for traveling 1 mile is $1.50. However, you also have to pay the drop fee of $10 for the Uber to pick you up. Therefore, $1.50 + $10 = $11.50.

  1. Based on the charge rate of $1.50 per mile, what is the mileage charge for 4 miles and 20 miles?

Solution

For 4 miles: \$1.50/1;\text{mile}\times4\;\text{miles}=\$6

For 20 miles: \$1.50/1;\text{mile}\times20\;\text{miles}=\$30

  1. Are the answers to #3 the total charge? Explain your reasoning.

Solution

No. You need to add the drop fee.

  1. What is the total charge for a trip of 20 miles?

Solution

The total charge consists of the drop fee ($10) + mileage fee ($30) = $40.

  1. Why is $10 called the initial value in the table? Explain your reasoning.

Solution

The initial value occurs when the independent variable has a value of zero. So, the dependent value (charge) starts at $10 when there have been no miles driven but Uber has been summoned. From this initial value of $10, the charge increases by $1.50 for each 1 mile increase in distance traveled.

 

 

Explore 5 – Given a linear pattern

Dasan is buying 50 binders online for his office. He notices that shipping costs $5.50 and the price for each binder is $0.70.

  1. Make a table to list the total cost of purchasing 0 to 5 binders.

Solution

No. of Binders 0 1 2 3 4 5
Cost ($) 5.50 6.20 6.90 7.60 8.30 90
  1. Why is the payment $5.50 when the number of binders is zero? Does it make sense? Explain your reasoning.

Solution

The cost of $5.50 when the number of binders is zero does not mean Dasan has to pay $5.50 when buying nothing. It signifies the shipping fee Dasan needs to pay no matter how many binders Dasan purchases. Starting from the shipping fee of $5.50, the total payment increases by $1.20 as the number of binders increases by 1. $5.50 is the initial value.

  1. What is the total cost of buying 50 binders?

Solution

The total cost consists of two parts, the shipping fee and the charge for the binders. The charge for the binders is \$0.70\;per\;binder\times50\;binders=\$35). Therefore, the total cost is $5.50 + 35 = $40.50.

 

Reflect Icon

  • What does initial value mean? How does it affect linear reasoning?
Show/Hide Answer

The initial value is the value of the dependent value when the independent value is zero. In a table, it is the starting value of the dependent variable row when the independent variable value is zero. It is often an add-on value to the rate.

 

Practice Exercises

  1. A gym charges $40 to join and $55 per month to use its facilities. a) Make a table that shows the total cost of using the gym for 0 to 6 months.    b) State the initial value and explain what it represents.    c) Calculate the total cost of using the gym for 1 year.
  2. A shipping company charges $7.50 plus $3.50 per kilogram to ship a box. a) Make a table that shows the total cost of sending a box that weighs 0 to 6 kg.    b) State the initial value and explain what it represents.    c) Calculate the total cost of sending a box that weighs 15kg.
Show/Hide Answer
  1. a)
    Months 0 1 2 3 4 5 6
    Cost ($) 40 95 150 205 260 315 370

    b) $40. The fee to join the gym, even if you don’t use the gym.     c) $700

  2. a)
    Weight (kg) 0 1 2 3 4 5 6
    Cost ($) 7.50 11 14.50 18 21.50 25 28.50

    b) $7.50. The charge to initiate a shipment.     c) $60

 

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

Perspectives

  1. An 8-inch long candle burns at a rate of 0.25 inches every 20 minutes. a) Create a table that shows the length of candle remaining from 0 minutes to 2 hours in 20-minute intervals.     b) Is the rate of change positive or negative? Explain your reasoning.     c) State the rate of change.     d) Determine the unit rate of change per hour.     e) What length will the candle be after 4 hours?     f) What length will the candle be after 5 hours?
  2. Marta buys a printer for $250. The printer company claims it will cost only $0.04 per page in paper and ink to use the printer. a) Create a table that shows the total cost of owning the printer to print 0 to 500 pages in 100 page increments.     b) State the rate of change per 100 sheets of paper.     c) State the initial value and explain what it means.     d) What is the cost of owning the printer to print 5000 pages?
  3. In 2010, a small city introduced recycling bins as a waste management option. There were 9500 requests for recycling bins at the start of the program. Every year since 2011 there have been approximately 1000 new requests each year. a) How many recycling bins were in the city in 2012?     b) How many recycling bins were in the city in 2020?     c) What is the constant rate of change per year?     d) What is the initial value and what does it represent?
  4. A 24/7 covered parking lot charges $5 to enter and $3.50 per hour to stay. a) State the initial value and explain what it means.     b) State the constant rate of change.     c) How much does it cost to park for 2 hours?     d) How much does it cost to park for 6 hours?
  5. The moment Sylvia wakes up she begins doom/passive scrolling on Tik Tok at 5 posts per minute.  a) Create a table showing the total number of Tik Tok posts Sylvia has scrolled through from the moment she wakes up, in 1-minute intervals, up to 5 minutes.     b) State the unit rate of change in the number of Tik Tok posts scrolled and explain what it tells us about Sylvia’s scrolling speed.     c) Write an addition equation showing how to determine the total number of Tik Tok posts Sylvia has scrolled through after 8 minutes.     d) Explain how the constant rate of change relates to the addition equation and how it reflects Sylvia’s continuous scrolling activity. e) Write a multiplication equation showing how to determine the total number of TikTok posts Sylvia has scrolled through after 8 minutes.     f) Explain how the constant rate of change relates to the multiplication equation and how it reflects Sylvia’s consistent scrolling habit.

Show/Hide Answer
  1. a)
    Time passed (mins) 0 20 40 60 80 100 120
    Length (in) 8 7.75 7.5 7.25 7 6.75 6.5

    b) Negative since the candle is losing length as it burns.     c) Rate of change = –0.25 inches per 20 minutes     d) There are 3 20-minute intervals in 1 hour, so the rate of change per hour is (3 intervals per hour)(–0.25 inches per interval) = –0.75 inches per hour.     e) (4 hours)(–0.75 inches per hour) = –3 inches. Then 8 inches – 3 inches = 5 inches.     f) (5 hours)(–0.75 inches per hour) = –4.75 inches. Then 8 inches – 4.75 inches = 3.25 inches.

  2. a)
    Pages 0 100 200 300 400 500
    Cost ($) 250 254 258 262 266 270

    b) $4 per 100 sheets     c) $250. The cost to own the printer before printing any pages.     d) $450

  3. a) 10,500     b) 18,500     c) 1000 bins per year     d) 9500. The number of recycling bins requested at the start of the program.
  4. a) $5. This is the cost to just enter the parking lot without parking.     b) $3.50 per hour     c) $12     d) $26
  5. a)
    Time (minutes) 0 1 2 3 4 5
    Posts scrolled 0 5 10 15 20 25

b) The unit rate of change is 5 posts per minute. This means that Sylvia scrolls through an average of 5 Tik Tok posts every minute since she wakes up. It tells us that she maintains a consistent pace of scrolling, examining 5 posts each minute.     c) Total Posts Scrolled = 0 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5    d) The constant rate of change of 5 posts per minute is added 8 times since 8 minute-intervals will have passed. This addition equation shows how the rate of increase remains constant, and Sylvia’s scrolling behavior sustains at 5 posts per minute throughout the 8-minute period.     e) Total posts scrolled = (5 posts per minute) · (8 minutes)    f) The constant rate of change (5 posts per minute) is multiplied by the time (8 minutes) to calculate the total number of Tik Tok posts scrolled. This multiplication equation demonstrates that Sylvia’s scrolling pace remains the same, yielding 40 posts after 8 minutes. It showcases her ongoing and predictable scrolling activity.

 

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

Skill Exercises

Evaluate:

  1. 23 + 4 · 17
  2. 45 – 2 · 36
  3. 17 + 2.5(28)
  4. –12 + 3.4(30)
  5. –36 – 4.75(45)

Determine the unit rate:

  1. $5 per 2 sodas
  2. $18.50 per 4 pounds
  3. 120 people every 5 hours
  4. –7kg every 4 weeks
  5. 357.5 miles in 6.5 hours

 

Show/Hide Answer
  1. 91
  2. –27
  3. 87
  4. 90
  5. –249.75
  6. $2.50 per soda
  7. $4.625 per pound
  8. 24 people per hour
  9. –1.75 per hour
  10. 55 miles per hour

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