Eighth Grade Mathematics: Unit Rates and the Constant of Proportionality
Proportional relationships describe physical and economic scenarios where two quantities scale together in a consistent ratio, such as driving at a steady speed, calculating wages earned per hour, or mixing paint according to a fixed recipe. In eighth grade, your understanding of proportionality deepens from simple equivalent ratios into the algebraic concept of the constant of proportionality, symbolized by k. In this chapter, you will master calculating unit rates, identifying proportional relationships from tables and equations, and analyzing how the constant of proportionality governs quantitative change.
The Definition of a Proportional Relationship
Two quantities x and y are said to be in a proportional relationship if the ratio of the dependent variable y to the independent variable x is constant for every non-zero pair of corresponding values.
The Constant of Proportionality (k):
y
k = - or y = k * x (for x != 0)
x
The value k is called the constant of proportionality, or the unit rate. It tells you the exact amount that y changes whenever x increases by exactly 1 unit.
Table Analysis: Determining Proportionality
Time in Hours (x) Earnings in Dollars (y) Ratio y / x
1 18 18 / 1 = 18
2 36 36 / 2 = 18
3 54 54 / 3 = 18
5 90 90 / 5 = 18
Because the ratio y/x produces the identical value of 18 across every data row, the relationship is proportional, and the constant of proportionality is k = 18 dollars per hour. The algebraic equation representing this relationship is y = 18x.
Identifying Non-Proportional Relationships
If even a single data pair in a table produces a different ratio, the relationship cannot be proportional. Furthermore, a relationship is not proportional if there is a non-zero starting fee or initial offset.
Non-Proportional Example:
Miles (x) Taxi Fare (y) Ratio y / x
1 7 7 / 1 = 7.00
2 10 10 / 2 = 5.00 (Ratios are not equal!)
3 13 13 / 3 = 4.33
In the taxi fare example, there is a base dispatch charge of 4 dollars plus 3 dollars per mile (y = 3x + 4). Even though the rate of change is constant (3 dollars per mile), the relationship is not proportional because the ratio y/x varies continuously depending on the value of x.
Unit Rates Involving Fractions and Decimals
In real-world applications, unit rates frequently involve fractional quantities of time, area, or weight. To determine the unit rate when given fractional inputs, divide the fractional change in y by the fractional change in x using standard fraction division.
Example: A painter covers 3/4 of a wall in 2/5 of an hour.
What is the unit rate in walls per hour?
Step 1: Set up the ratio y / x:
(3/4 wall) / (2/5 hour)
Step 2: Multiply by the reciprocal of the denominator:
(3/4) * (5/2) = 15/8 = 1 and 7/8 walls per hour
The unit rate of 15/8 walls per hour represents how much wall the painter would complete in one full hour at that steady pace.
Comparing Proportional Relationships
When comparing two different proportional relationships presented through different representations—such as one in a table, one in an equation, and one in a verbal description—you can immediately compare their steepness and speed by comparing their values of k.
Comparing Runners:
Runner A: Given by equation y = 8.5x (k = 8.5 meters per second)
Runner B: Runs 54 meters in 6 seconds (k = 54 / 6 = 9.0 meters per second)
Comparison: Runner B runs faster because 9.0 m/s > 8.5 m/s.
The greater the value of k, the faster the rate of change, and the steeper the corresponding linear graph will appear on the coordinate plane.
Chapter Exercises
Exercise 1: Determine whether the relationship in the following table is proportional: for x = 4, y = 14; for x = 6, y = 21; for x = 10, y = 35; and for x = 14, y = 49. If it is proportional, state the constant of proportionality k and write the equation relating y to x.
Exercise 2: A motorized drone uses 5/8 of its battery pack in 15 minutes (which is 1/4 of an hour). Find the unit rate of battery consumption in fractions of a battery pack per hour. Write the equation relating battery consumption y to time in hours x.
Exercise 3: Explain in paragraph form why the equation y = 5x represents a proportional relationship, while the equation y = 5x + 2 does not, even though both equations increase by 5 units of y for every 1 unit increase in x.
Solutions and Explanations
Solution 1: To determine whether the table represents a proportional relationship, compute the ratio y/x for each coordinate pair. For the first pair, 14 / 4 simplifies to 7/2 = 3.5. For the second pair, 21 / 6 simplifies by dividing by 3 to 7/2 = 3.5. For the third pair, 35 / 10 simplifies by dividing by 5 to 7/2 = 3.5. For the fourth pair, 49 / 14 simplifies by dividing by 7 to 7/2 = 3.5. Because the ratio y/x is identical for all pairs, the relationship is proportional. The constant of proportionality is k = 7/2 (or 3.5), and the linear equation relating the variables is y = (7/2)x, or y = 3.5x.
Solution 2: To calculate the unit rate of battery consumption per hour, divide the amount of battery used (5/8 of a pack) by the elapsed time in hours (1/4 hour): (5/8) / (1/4). To divide by a fraction, multiply by its reciprocal: (5/8) * (4/1) = 20/8. Simplifying by dividing numerator and denominator by 4 gives 5/2, or 2.5 battery packs per hour. Therefore, the unit rate is 2.5 battery packs per hour, and the algebraic equation is y = 2.5x, where x is measured in hours and y represents battery packs consumed.
Solution 3: A proportional relationship requires that the ratio y/x remains completely constant for all non-zero values of x. For the equation y = 5x, dividing both sides by x gives y/x = 5. No matter what non-zero value is substituted for x, the ratio y/x is always exactly 5 (for instance, when x = 1, y = 5 and 5/1 = 5; when x = 3, y = 15 and 15/3 = 5). Furthermore, when x = 0, y = 0, meaning the relationship passes through the origin. In contrast, for the equation y = 5x + 2, dividing by x gives y/x = 5 + 2/x. Because the term 2/x changes with every different value of x, the ratio y/x is not constant (when x = 1, y = 7 and 7/1 = 7; when x = 2, y = 12 and 12/2 = 6). Additionally, when x = 0, y = 2, meaning it does not pass through the origin. Therefore, while both have a constant rate of change, only y = 5x is proportional.