Which statement of variation describes the table of values below?

\begin{tabular}{|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & 3 & 4 & 6 & 8 & 12 \\
\hline
[tex]$y$[/tex] & 16 & 12 & 8 & 6 & 4 \\
\hline
\end{tabular}

A. [tex]$y$[/tex] varies directly as [tex]$x$[/tex]

B. [tex]$y$[/tex] varies jointly as [tex]$x$[/tex]

C. [tex]$y$[/tex] varies inversely as [tex]$x$[/tex]

D. [tex]$y$[/tex] varies directly and inversely



Answer :

To determine the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the given table, we need to analyze how [tex]\( y \)[/tex] changes as [tex]\( x \)[/tex] changes.

[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 3 & 4 & 6 & 8 & 12 \\ \hline y & 16 & 12 & 8 & 6 & 4 \\ \hline \end{array} \][/tex]

One way to understand this is by finding the product of each pair [tex]\( (x, y) \)[/tex]. If [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex], then the product [tex]\( x \cdot y \)[/tex] should be constant.

Let's calculate the product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] for each pair:

1. For [tex]\( x = 3 \)[/tex] and [tex]\( y = 16 \)[/tex]:
[tex]\[ 3 \cdot 16 = 48 \][/tex]

2. For [tex]\( x = 4 \)[/tex] and [tex]\( y = 12 \)[/tex]:
[tex]\[ 4 \cdot 12 = 48 \][/tex]

3. For [tex]\( x = 6 \)[/tex] and [tex]\( y = 8 \)[/tex]:
[tex]\[ 6 \cdot 8 = 48 \][/tex]

4. For [tex]\( x = 8 \)[/tex] and [tex]\( y = 6 \)[/tex]:
[tex]\[ 8 \cdot 6 = 48 \][/tex]

5. For [tex]\( x = 12 \)[/tex] and [tex]\( y = 4 \)[/tex]:
[tex]\[ 12 \cdot 4 = 48 \][/tex]

Since the product [tex]\( x \cdot y \)[/tex] is constant (48) for all pairs, we can conclude that [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex].

Therefore, the correct statement of variation is:

C. [tex]\( y \)[/tex] varies inversely as [tex]\( x \)[/tex]