The Correct Answer and Explanation is:
The correct answers are (2, 1) and (1, 3).
To determine the reasonable solutions, we first need to create a mathematical inequality based on the problem’s description. Let x represent the number of hot dogs sold, and let y represent the number of bottled waters sold. Since hot dogs cost $4 each and bottled water costs $2 each, the total cost can be represented by the expression 4x + 2y. The problem states that the total sales were “less than $12.” This translates to the inequality:
4x + 2y < 12
Next, we must consider the “reasonable” aspect of the solutions. Since x and y represent physical items being sold, they cannot be negative numbers or fractions. Therefore, both x and y must be non-negative whole numbers (integers ≥ 0).
Now we can test each ordered pair (x, y):
- (-1, 5): This is not a reasonable solution because the number of hot dogs (x) cannot be negative.
- (0, 6): This is a whole number pair. Let’s test it in the inequality: 4(0) + 2(6) = 0 + 12 = 12. The result is 12, which is not less than 12. Therefore, this solution is incorrect.
- (2, 1): This is a whole number pair. Let’s test it: 4(2) + 2(1) = 8 + 2 = 10. The result is 10, which is less than 12. This is a correct and reasonable solution.
- (1, 1.5): This is not a reasonable solution because the number of bottled waters (y) must be a whole number; you cannot sell half a bottle of water.
- (1, 3): This is a whole number pair. Let’s test it: 4(1) + 2(3) = 4 + 6 = 10. The result is 10, which is less than 12. This is also a correct and reasonable solution.
- (2, 2): This is a whole number pair. Let’s test it: 4(2) + 2(2) = 8 + 4 = 12. The result is 12, which is not less than 12. Therefore, this solution is incorrect.
