2.3.1: Comparing Relationships with Tables
Lesson
Let's explore how proportional relationships are different from other relationships.
Exercise \(\PageIndex{1}\): Adjusting a Recipe
A lemonade recipe calls for the juice of 5 lemons, 2 cups of water, and 2 tablespoons of honey.
Invent four new versions of this lemonade recipe:
- One that would make more lemonade but taste the same as the original recipe.
- One that would make less lemonade but taste the same as the original recipe.
- One that would have a stronger lemon taste than the original recipe.
- One that would have a weaker lemon taste than the original recipe.
Exercise \(\PageIndex{2}\): Visiting the State Park
Entrance to a state park costs $6 per vehicle, plus $2 per person in the vehicle.
-
How much would it cost for a car with 2 people to enter the park? 4 people? 10 people? Record your answers in the table.
number of people in vehicle total entrance cost in dollars \(2\) \(4\) \(10\) Table \(\PageIndex{1}\) - For each row in the table, if each person in the vehicle splits the entrance cost equally, how much will each person pay?
- How might you determine the entrance cost for a bus with 50 people?
- Is the relationship between the number of people and the total entrance cost a proportional relationship? Explain how you know.
Are you ready for more?
What equation could you use to find the total entrance cost for a vehicle with any number of people?
Exercise \(\PageIndex{3}\): Running Laps
Han and Clare were running laps around the track. The coach recorded their times at the end of laps 2, 4, 6, and 8.
Han's run:
| distance (laps) | time (minutes) | minutes per lap |
|---|---|---|
| \(2\) | \(4\) | |
| \(4\) | \(9\) | |
| \(6\) | \(15\) | |
| \(8\) | \(23\) |
Clare's run:
| distance (laps) | time (minutes) | minutes per lap |
|---|---|---|
| \(2\) | \(5\) | |
| \(4\) | \(10\) | |
| \(6\) | \(15\) | |
| \(8\) | \(20\) |
- Is Han running at a constant pace? Is Clare? How do you know?
- Write an equation for the relationship between distance and time for anyone who is running at a constant pace.
Summary
Here are the prices for some smoothies at two different smoothie shops:
Smoothie Shop A
| smoothie size (oz) | price ($) | dollars per ounce |
|---|---|---|
| \(8\) | \(6\) | \(0.75\) |
| \(12\) | \(9\) | \(0.75\) |
| \(16\) | \(12\) | \(0.75\) |
| \(s\) | \(0.75s\) | \(0.75\) |
Smoothie Shop B
| smoothie size (oz) | prize ($) | dollars per ounce |
|---|---|---|
| \(8\) | \(6\) | \(0.75\) |
| \(12\) | \(8\) | \(0.67\) |
| \(16\) | \(10\) | \(0.625\) |
| \(s\) | \(???\) | \(???\) |
For Smoothie Shop A, smoothies cost $0.75 per ounce no matter which size we buy. There could be a proportional relationship between smoothie size and the price of the smoothie. An equation representing this relationship is \(p=0.75s\) where \(s\) represents size in ounces and \(p\) represents price in dollars. (The relationship could still not be proportional, if there were a different size on the menu that did not have the same price per ounce.)
For Smoothie Shop B, the cost per ounce is different for each size. Here the relationship between smoothie size and price is definitely not proportional.
In general, two quantities in a proportional relationship will always have the same quotient. When we see some values for two related quantities in a table and we get the same quotient when we divide them, that means they might be in a proportional relationship—but if we can't see all of the possible pairs, we can't be completely sure. However, if we know the relationship can be represented by an equation is of the form \(y=kx\), then we are sure it is proportional.
Practice
Exercise \(\PageIndex{4}\)
Decide whether each table could represent a proportional relationship. If the relationship could be proportional, what would the constant of proportionality be?
-
How loud a sound is depending on how far away you are.
distance to listener (ft) sound level (dB) \(5\) \(85\) \(10\) \(79\) \(20\) \(73\) \(40\) \(67\) Table \(\PageIndex{6}\) -
The cost of fountain drinks at Hot Dog Hut.
volume (fluid ounces) cost ($) \(16\) \($1.49\) \(20\) \($1.59\) \(30\) \($1.89\) Table \(\PageIndex{7}\)
Exercise \(\PageIndex{5}\)
A taxi service charges $1.00 for the first \(\frac{1}{10}\) mile then $0.10 for each additional \(\frac{1}{10}\) mile after that.
Fill in the table with the missing information then determine if this relationship between distance traveled and price of the trip is a proportional relationship.
| distance traveled (mi) | price (dollars) |
|---|---|
| \(\frac{9}{10}\) | |
| \(2\) | |
| \(3\frac{1}{10}\) | |
| \(10\) |
Exercise \(\PageIndex{6}\)
A rabbit and turtle are in a race. Is the relationship between distance traveled and time proportional for either one? If so, write an equation that represents the relationship.
Turtle’s run:
| distance (meters) | time (minutes) |
|---|---|
| \(108\) | \(2\) |
| \(405\) | \(7.5\) |
| \(540\) | \(10\) |
| \(1,768.5\) | \(32.75\) |
Rabbit's run:
| distance (meters) | time (minutes) |
|---|---|
| \(800\) | \(1\) |
| \(900\) | \(5\) |
| \(1,107.5\) | \(20\) |
| \(1,524\) | \(32.5\) |
Exercise \(\PageIndex{7}\)
For each table, answer: What is the constant of proportionality?
| \(a\) | \(b\) |
|---|---|
| \(2\) | \(14\) |
| \(5\) | \(35\) |
| \(9\) | \(63\) |
| \(\frac{1}{3}\) | \(\frac{7}{3}\) |
| \(a\) | \(b\) |
|---|---|
| \(3\) | \(60\) |
| \(5\) | \(600\) |
| \(8\) | \(960\) |
| \(12\) | \(1440\) |
| \(a\) | \(b\) |
|---|---|
| \(75\) | \(3\) |
| \(200\) | \(8\) |
| \(1525\) | \(61\) |
| \(10\) | \(0.4\) |
| \(a\) | \(b\) |
|---|---|
| \(4\) | \(10\) |
| \(6\) | \(15\) |
| \(22\) | \(55\) |
| \(3\) | \(7\frac{1}{2}\) |
(From Unit 2.1.2)
Exercise \(\PageIndex{8}\)
Kiran and Mai are standing at one corner of a rectangular field of grass looking at the diagonally opposite corner. Kiran says that if the the field were twice as long and twice as wide, then it would be twice the distance to the far corner. Mai says that it would be more than twice as far, since the diagonal is even longer than the side lengths. Do you agree with either of them?
(From Unit 1.1.4)