Each of the following represents an inverse variation relationship. List the relationships in order from least to greatest based on constant of proportionality. mathbf(x) & mathbf(y) 1 & 6 3 & 2 6 & 1 12 & 0.5 [ equiv y=(2.5)/(x) ]

# Explanation<br />Inverse variation describes a relationship where the product of two variables is constant. The general form of an inverse variation is \(y = \frac{k}{x}\), where \(k\) is the constant of proportionality. To determine the constant of proportionality for each relationship given in the table, we can multiply each pair of \(x\) and \(y\) values.<br /><br />Let's calculate the constant of proportionality (\(k\)) for each pair:<br /><br />1. For \(x = 1\) and \(y = 6\), \(k = x \cdot y = 1 \cdot 6 = 6\).<br />2. For \(x = 3\) and \(y = 2\), \(k = x \cdot y = 3 \cdot 2 = 6\).<br />3. For \(x = 6\) and \(y = 1\), \(k = x \cdot y = 6 \cdot 1 = 6\).<br />4. For \(x = 12\) and \(y = 0.5\), \(k = x \cdot y = 12 \cdot 0.5 = 6\).<br /><br />The given equation \(y = \frac{2.5}{x}\) suggests a constant of proportionality (\(k\)) of \(2.5\), which is not directly represented in the table but is provided for comparison.<br /><br /># Answer<br />Upon calculating the constant of proportionality for each relationship in the table, we find that each pair yields the same constant, \(k = 6\). Therefore, when listing the relationships based on the constant of proportionality, we observe that all the given relationships in the table have the same constant of proportionality, \(k = 6\), which is greater than the constant \(2.5\) provided in the equation. <br /><br />However, since the task is to list the relationships in order from least to greatest based on the constant of proportionality, and all provided relationships have the same constant (\(k = 6\)), which is greater than the constant provided in the equation (\(k = 2.5\)), the order based on the constant of proportionality from least to greatest is:<br /><br />1. The relationship represented by the equation \(y = \frac{2.5}{x}\) with \(k = 2.5\).<br />2. All the relationships listed in the table, as they all have a constant of proportionality of \(k = 6\).