Lesson 7

Comparing Relationships with Tables

Let’s explore how proportional relationships are different from other relationships.

7.1: 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:

  1. One that would make more lemonade but taste the same as the original recipe.
  2. One that would make less lemonade but taste the same as the original recipe.
  3. One that would have a stronger lemon taste than the original recipe.
  4. One that would have a weaker lemon taste than the original recipe.

7.2: Visiting the State Park

Entrance to a state park costs \$6 per vehicle, plus \$2 per person in the vehicle.

  1. 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
  2. For each row in the table, if each person in the vehicle splits the entrance cost equally, how much will each person pay?
  3. How might you determine the entrance cost for a bus with 50 people?
  4. Is the relationship between the number of people and the total entrance cost a proportional relationship? Explain how you know.


What equation could you use to find the total entrance cost for a vehicle with any number of people?

7.3: 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
  1. Is Han running at a constant pace? Is Clare? How do you know?
  2. 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)
price
($)
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 \(\displaystyle p=0.75 s\) 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 \(y = k x\), then we are sure it is proportional.