Mr. Ishimoto ordered x new math books and y new workbooks for his class. The total weight of the box of books cannot be more than 50 pounds. If each math book weighs 3.2 pounds and each workbook weighs 0.8 pounds, which inequality represents the maximum number of each type of book that can be shipped in a single box?
A. 3.2x + 0.8y < 50
B. 3.2x + 0.8y ≤ 50
C. 0.8x + 3.2y < 50
D. 0.8x + 3.2y ≤ 50
The Correct Answer and Explanation is:
The correct answer is B. 3.2x + 0.8y ≤ 50.
Explanation:
To solve this problem, we need to set up an inequality that describes the weight of the books in the box.
- Define variables:
- Let x represent the number of math books.
- Let y represent the number of workbooks.
- Weight of the books:
- Each math book weighs 3.2 pounds. So, the total weight of x math books is 3.2x pounds.
- Each workbook weighs 0.8 pounds. So, the total weight of y workbooks is 0.8y pounds.
- Total weight:
The total weight of all the books in the box is the sum of the weight of the math books and the workbooks, which can be represented as:
[
\text{Total weight} = 3.2x + 0.8y
] - Maximum weight limit:
The total weight cannot exceed 50 pounds. This gives us the inequality:
[
3.2x + 0.8y \leq 50
]
The symbol “≤” (less than or equal to) is used because the weight can be equal to 50 pounds, but it cannot exceed this limit.
Why the other options are incorrect:
- Option A: 3.2x + 0.8y < 50
This inequality suggests that the weight must be strictly less than 50 pounds, which does not allow for a total weight of exactly 50 pounds. Since the problem doesn’t specify that the weight must be strictly less, this option is not valid. - Option C: 0.8x + 3.2y < 50
This inequality swaps the coefficients of x and y. It incorrectly suggests that the workbooks (0.8 pounds each) weigh more than the math books (3.2 pounds each), which contradicts the given information. - Option D: 0.8x + 3.2y ≤ 50
Like option C, this inequality reverses the coefficients of x and y, resulting in an incorrect weight distribution. The correct inequality should have the weight of math books (3.2 pounds each) multiplied by x, and the weight of workbooks (0.8 pounds each) multiplied by y.
Thus, B is the only correct choice, as it accurately represents the situation with the proper weight assignments and allows the total weight to be up to 50 pounds.