At the supermarket, the price of a steak is proportional to its weight;
at constant speed, the fuel consumption of a car is proportional to the distance traveled.
Each of these describes a proportional relationship.
What does that mean? Under what conditions can we say that a situation displays a proportional relationship?
I. Recognizing a proportional table
Example:
2 × 4 = 8
7 × 4 = 28
0.8 × 4 = 3.2
Definition: A proportional table is a table (formed from two lines of numbers) for which there is a number k so that the numbers on the second line are each equal to k times the numbers on the first line.
The number k is known as the coefficient of proportionality.
II. Applications
A. Example 1: the price of steak
The weights (in pounds) and prices (in dollars) of four packages of steak in a supermarket were recorded, and the results are shown in the table below.
$6.25 ÷ 1.250 lb = $5/lb, so 1.250 lb × $5/lb = $6.25
1.700 lb × $5/lb = $8.50
2.300 lb × $5/lb = $11.50
2.520 lb × $5/lb = $12.60
These calculations demonstrate that the table above is a proportional table. This is how we should understand the phrase, “the price of steak is proportional to its weight.”
Note: The coefficient of proportionality is equal to $5/lb; $5 will be the price of a one-pound steak. This coefficient appears on the package labels in the form “price per pound: $5.”
B. Example 2: the fuel consumption of a car
The graph in this figure shows the fuel consumption of a car as a function of the distance traveled at a constant speed. For example, we can read from it that the car consumes 5 gallons of gas for 100 miles traveled.
All of the points on the graph are aligned with the point of origin. This alignment is characteristic of a proportional situation.
We can highlight this proportional relationship by setting up the table shown below using the values on the graph:
Note: The coefficient of proportionality is equal to 0.05 gallons/mile; that is the gas consumption in gallons for one mile traveled.
C. Example 3: the annual expenditure on a video-rental club
In a video club, you pay $18 for a one-year subscription, and then $2 each time you rent a film.
Is the annual expenditure proportional to the number of films rented?
Look at the table below.
You spend $20 if you rent one film ($18 subscription + $2 for the film).
You spend $22 if you rent two films ($18 subscription + $4 for the two films).
By examining just the first two columns in the table, we can see that it is not a proportional table, since: 20 = 1 × 20 and 22 2 × 20.
We can conclude from this that annual expenditure is not proportional to the number of films rented. However, if the yearly subscription fee were $0 (that is, if membership were free), the annual expenditure would be proportional to the number of films rented.
Read more:
Applying the Distributive Law
Calculating a Numerical Expression (1)
Calculating a Numerical Expression (2)
Determining the Common Factors of Two Integers
Rounding to the Nearest Unit
Writing a Numerical Expression That Corresponds to a Sequence of Operations