Proportional vs. Non-Proportional - Pre Algebra

1. Apples cost 2 dollars per pound with no other charges. Is the cost proportional or non-proportional to the weight?

   Proportional. Cost $= 2 \times$ pounds, so the ratio of cost to pounds is always 2.

2. Classify the pairs $(2, 3)$, $(4, 6)$, $(10, 15)$ as proportional or non-proportional.

   Proportional. Every ratio $\frac{y}{x}$ equals 1.5.

3. Classify $y = 4x + 3$ as proportional or non-proportional.

   Non-proportional. When $x = 0$, $y = 3$, so the graph does not pass through the origin.

4. A taxi charges a 3 dollar base fare plus 2 dollars per mile. What change would make the total fare proportional to the miles driven?

   Remove the base fare (make it 0 dollars). Then fare $= 2 \times$ miles, which is proportional.

5. Classify the pairs $(1, 5)$, $(2, 8)$, $(3, 11)$ as proportional or non-proportional.

   Non-proportional. The ratios $\frac{y}{x}$ are 5, 4 and about 3.67, which are not equal.

6. Why is a line with a constant rate of change not always proportional?

   A proportional line must also pass through the origin. A constant rate with a nonzero starting value is linear but non-proportional.

7. A gym charges a 20 dollar joining fee plus 10 dollars per month. Is the total cost proportional or non-proportional to the number of months?

   Non-proportional. The joining fee means the cost is 20 dollars at 0 months, not 0.

8. A linear equation has the form $y = mx + b$. For which values of $b$ is it non-proportional?

   Any $b \ne 0$. A nonzero $b$ means the line crosses the $y$-axis away from the origin.

9. What quick test decides whether a set of $(x, y)$ pairs is proportional?

   Compute $\frac{y}{x}$ for each pair with $x \ne 0$. The relationship is proportional only if every ratio is the same.

10. Classify $y = \frac{x}{5}$ as proportional or non-proportional.

    Proportional. It has the form $y = kx$ with $k = \frac{1}{5}$.

11. Classify $y = x^2$ as proportional or non-proportional.

    Non-proportional. The ratio $\frac{y}{x}$ is 1 at $x = 1$ and 2 at $x = 2$, so it is not constant.

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