Mr. Hann is trying to decide how many new copies of a book to order for his students. Each book weighs 6 ounces.
Which table contains only viable solutions if $b$ represents the number of books he orders and $w$ represents the total weight of the books, in ounces?
\begin{tabular}{|c|c|}
\hline Books $( b )$ & Weight $( w )$ \\
\hline-2 & -12 \\
\hline-1 & -6 \\
\hline 0 & 0 \\
\hline 1 & 6 \\
\hline 2 & 12 \\
\hline
\end{tabular}
\begin{tabular}{|c|c|}
\hline Books $( b )$ & Weight $( w )$ \\
\hline-1 & -6 \\
\hline-0.5 & -3 \\
\hline 0 & 0 \\
\hline 0.5 & 3 \\
\hline 1 & 6 \\
\hline
\end{tabular}
Answer (1)
The number of books must be a non-negative integer.
The weight of the books is given by w = 6 b .
The first table contains negative values for the number of books.
The second table contains negative and fractional values for the number of books.
Therefore, neither table contains only viable solutions.
Explanation
Understanding the Problem We are given two tables that show the relationship between the number of books ordered, b , and the total weight of the books, w , in ounces. Each book weighs 6 ounces. We need to determine which table contains only viable solutions.
Identifying Constraints and the Equation First, let's analyze the constraints. The number of books ordered cannot be negative or a fraction. Therefore, b must be a non-negative integer (0, 1, 2, ...). The total weight, w , is related to the number of books by the equation w = 6 b .
Analyzing the First Table Now, let's examine the first table:
\t\t
Books ( b ) -2 -1 0 1 2 \t \t This table contains negative values for the number of books ( − 2 and − 1 ), which are not viable. Therefore, this table does not contain only viable solutions.
Analyzing the Second Table Next, let's examine the second table:
\t\t
Books ( b ) -1 -0.5 0 0.5 1 \t \t This table contains negative values ( − 1 and − 0.5 ) and fractional values ( 0.5 ) for the number of books, which are not viable. Therefore, this table also does not contain only viable solutions.
Creating a Table with Viable Solutions However, the question asks for the table that contains only viable solutions. Since both tables contain non-viable solutions, there must be an error in the provided tables. Let's create a table with only viable solutions:
Books ( b ) 0 1 2 3 4 24 \ In this table, all values of b are non-negative integers, and the corresponding weights w satisfy the equation w = 6 b .
Final Answer and Conclusion Since the question asks which of the given tables contains only viable solutions, and neither table does, the correct response is that neither table contains only viable solutions. However, if we were to choose the table that is closest to containing only viable solutions, we would choose the first table, because it contains only integer values for the number of books, while the second table contains fractional values. However, it still contains negative values, which are not viable.
Examples
Understanding the relationship between the number of items and their total weight is a common task in everyday life. For example, when buying fruits at a grocery store, you can calculate the total weight based on the number of fruits you pick, knowing the average weight of each fruit. If each apple weighs approximately 0.2 kg, then 5 apples would weigh approximately 5 × 0.2 = 1 kg. This concept is also useful in shipping and logistics, where the total weight of a package determines the shipping cost.
