Question

A student is making trail mix for each serving size in the table given. Serving Size 1 2 3 4 Cups of Nuts 2 3 4 5 Cups of Fruit 4 6 8 10 Part A. Is the relationship between Cups of Nuts and Cups of Fruit in each serving linear? If so, determine the constant rate of change. Part B. Is the relationship between Cups of Nuts and Cups of Fruit in each serving also proportional? If so, determine the constant ratio.

Answer 1

The relationship between Cups of Nuts and Cups of Fruit is linear with a constant rate of change of 2, and it is also proportional with a constant ratio of 1/2 for each serving size.

Step-by-step explanation:

The student is inquiring whether the relationship between Cups of Nuts and Cups of Fruit is linear, and if it is also proportional, including the constant rate of change and constant ratio.

Part A: Linearity and Rate of Change

To determine if the relationship is linear, we calculate the rate of change between the servings. A linear relationship would mean a constant rate of change. Between each serving size (1 to 2, 2 to 3, and so on), the increase in cups of fruit is consistent at +2 cups for every +1 cup of nuts.

Rate of change = ∆ Cups of Fruit / ∆ Cups of Nuts = 2/1 = 2

Thus, the relationship is linear with a constant rate of change of 2.

Part B: Proportionality and Ratio

To determine if the relationship is proportional, we check if the ratio between Cups of Nuts and Cups of Fruit is consistent across all servings. The ratio remains constant at 1/2 (Cups of Nuts/Cups of Fruit) for each serving size. Therefore, the relationship is proportional with a constant ratio of 1/2.