Identifying Proportional Relationships in Tables - Seventh Grade Mathematics

A table of values reveals a proportional relationship when every pair of corresponding non-zero values maintains a constant ratio between the dependent and independent variables.

The Mathematical Definition of Proportionality

Two quantities are proportional if their values maintain a constant ratio across all instances. In any proportional relationship between an independent variable x and a dependent variable y, dividing each y-value by its corresponding x-value always produces the exact same numerical quotient, designated by the constant k.

               THE PROPORTIONAL TEST FOR DATA TABLES
                       y_1 / x_1 = y_2 / x_2 = ... = y_n / x_n = k
      If any pair produces a different ratio, the table is NOT proportional.

If this quotient varies for even a single data pair, the relationship is non-proportional, regardless of whether the numbers follow another predictable linear or curved pattern.

Testing Tables for Proportionality

To test a data table for proportionality, calculate the ratio y/x for every single row. Let us analyze two distinct tables to compare their underlying mathematical structures.

TABLE A: Distance Covered Over Time
+-------------------+-------------------+-----------------------+
| Time, x (hours)   | Distance, y (mi)  | Ratio: y / x          |
+-------------------+-------------------+-----------------------+
| 2                 | 50                | 50 / 2 = 25           |
| 3                 | 75                | 75 / 3 = 25           |
| 5                 | 125               | 125 / 5 = 25          |
| 8                 | 200               | 200 / 8 = 25          |
+-------------------+-------------------+-----------------------+
Conclusion: Proportional. Constant ratio k = 25 miles per hour.

TABLE B: Taxi Cab Fare Over Distance
+-------------------+-------------------+-----------------------+
| Miles, x          | Cost, y (dollars) | Ratio: y / x          |
+-------------------+-------------------+-----------------------+
| 1                 | 7                 | 7 / 1 = 7.00          |
| 2                 | 11                | 11 / 2 = 5.50         |
| 4                 | 19                | 19 / 4 = 4.75         |
| 6                 | 27                | 27 / 6 = 4.50         |
+-------------------+-------------------+-----------------------+
Conclusion: NOT proportional. Ratios change across rows.

In Table A, every ratio y/x simplifies to twenty-five. The relationship is proportional. In Table B, although the cost increases by four dollars for every additional mile (a constant rate of change), the ratios y/x are not equal because of a three-dollar base pickup fee. The presence of a non-zero starting value destroys strict proportionality.

Multiplicative vs Additive Relationships

Proportional relationships are fundamentally multiplicative. The output is generated by multiplying the input by a constant factor: y equals k times x.

               MULTIPLICATIVE vs ADDITIVE PATTERNS
   Proportional (Multiplicative):  y = 3 * x
   x = 1 -> y = 3      (3 / 1 = 3)
   x = 2 -> y = 6      (6 / 2 = 3)
   x = 3 -> y = 9      (9 / 3 = 3)

   Non-Proportional (Additive):    y = x + 3
   x = 1 -> y = 4      (4 / 1 = 4)
   x = 2 -> y = 5      (5 / 2 = 2.5)
   x = 3 -> y = 6      (6 / 3 = 2)

In the additive relationship y equals x plus three, although both variables increase at identical rates, the ratio y/x continuously diminishes. Proportionality requires that doubling the input doubles the output, tripling the input triples the output, and zero input produces exactly zero output.

Finding Missing Values in Proportional Tables

Once you confirm that a table is proportional by calculating the constant of proportionality k, you can use k to find any missing entries.

+-------------------+-------------------+
| Cups of Flour (x) | Cookies Baked (y) |
+-------------------+-------------------+
| 2                 | 36                |  --> k = 36 / 2 = 18 cookies/cup
| 3                 | 54                |  --> Check: 54 / 3 = 18
| 5                 | ?  [A]            |  --> y = 18 * 5 = 90
| ?  [B]            | 144               |  --> x = 144 / 18 = 8
+-------------------+-------------------+

To find an unknown y-value, multiply the given x-value by k. To find an unknown x-value, divide the given y-value by k.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. A table records the volume of gasoline purchased and total cost: three gallons cost nine dollars and seventy-five cents, five gallons cost sixteen dollars and twenty-five cents, and eight gallons cost twenty-six dollars. Determine whether the relationship between cost and gallons is proportional by testing all ratios.

Exercise 2. A student examines a table where x equals two gives y equals seven, and x equals four gives y equals nine. Is this relationship proportional? Explain why or why not using exact mathematical ratios.

Exercise 3. The relationship between the weight of bulk granola in pounds (x) and the price in dollars (y) is proportional. Two pounds of granola cost five dollars and fifty cents. Use this relationship to determine the cost of seven pounds of granola and the weight of granola that can be purchased for twenty-two dollars.

Exercise 4. Explain why a table representing the perimeter of an equilateral triangle relative to its side length is proportional, while a table representing the area of a square relative to its side length is not proportional.

Solutions

Solution to Exercise 1. Calculate the ratio of cost to gallons for each row: nine point seventy-five divided by three equals three point twenty-five; sixteen point twenty-five divided by five equals three point twenty-five; twenty-six divided by eight equals three point twenty-five. Because all three pairs yield the identical ratio of three dollars and twenty-five cents per gallon, the relationship is proportional, with a constant of proportionality k equal to 3.25.

Solution to Exercise 2. Calculate the ratio y/x for both pairs: seven divided by two equals three and five-tenths (3.5), whereas nine divided by four equals two and twenty-five hundredths (2.25). Because three and five-tenths does not equal two and twenty-five hundredths, the ratios are not constant. Therefore, the relationship is not proportional.

Solution to Exercise 3. First determine the constant of proportionality k by dividing cost by weight: five point fifty divided by two equals two point seventy-five dollars per pound. To find the cost of seven pounds, multiply seven by two point seventy-five: seven times two point seventy-five equals nineteen dollars and twenty-five cents. To find how many pounds can be purchased for twenty-two dollars, divide twenty-two by two point seventy-five: twenty-two divided by two point seventy-five equals eight pounds of granola.

Solution to Exercise 4. The perimeter of an equilateral triangle is given by P equals three times s, where s is the side length. For any side length s, the ratio P divided by s is three s divided by s, which is always the constant three. Doubling the side length doubles the perimeter, making it proportional. In contrast, the area of a square is given by A equals s squared. The ratio A divided by s is s squared divided by s, which equals s. This ratio is not constant; it changes with the size of the square. For side lengths one, two, and three, the areas are one, four, and nine, producing ratios of one, two, and three. Hence, area is not proportional to side length.