Question

which of the following situations are determined by a proportional relationship? select all that apply. (3,12) and (6,8) x y 2 7 3 10.5 5 17.5 a sports journalist is paid a different amount for each article she writes. x 2.5 6.25 8.75 y 2 5 7 a babysitter charges $16 per hour.

Explanation:

Step1: Recall proportional - relationship formula

A proportional relationship is of the form $y = kx$, where $k$ (the constant of proportionality) is $\frac{y}{x}$ and is constant for all pairs of $x$ and $y$ values.

Step2: Check the pair $(3,12)$ and $(6,8)$

For $(3,12)$, $\frac{y}{x}=\frac{12}{3} = 4$. For $(6,8)$, $\frac{y}{x}=\frac{8}{6}=\frac{4}{3}$. Since $\frac{12}{3}
eq\frac{8}{6}$, this is not a proportional relationship.

Step3: Check the table

$$\begin{array}{|c|c|}\hline x&y\\\hline2&7\\\hline3&10.5\\\hline5&17.5\\\hline\end{array}$$

For $x = 2,y = 7$, $\frac{y}{x}=\frac{7}{2}=3.5$. For $x = 3,y = 10.5$, $\frac{y}{x}=\frac{10.5}{3}=3.5$. For $x = 5,y = 17.5$, $\frac{y}{x}=\frac{17.5}{5}=3.5$. Since $\frac{y}{x}$ is constant ($k = 3.5$), this is a proportional relationship.

Step4: Analyze the sports - journalist situation

Since the sports journalist is paid a different amount for each article she writes, there is no constant ratio between the number of articles and the pay. So, this is not a proportional relationship.

Step5: Check the table

$$\begin{array}{|c|c|}\hline x&y\\\hline2.5&2\\\hline6.25&5\\\hline8.75&7\\\hline\end{array}$$

For $x = 2.5,y = 2$, $\frac{y}{x}=\frac{2}{2.5}=0.8$. For $x = 6.25,y = 5$, $\frac{y}{x}=\frac{5}{6.25}=0.8$. For $x = 8.75,y = 7$, $\frac{y}{x}=\frac{7}{8.75}=0.8$. Since $\frac{y}{x}$ is constant ($k = 0.8$), this is a proportional relationship.

Step6: Analyze the babysitter situation

The babysitter charges $16$ per hour. The relationship between the number of hours ($x$) and the total charge ($y$) is $y = 16x$. Here, the constant of proportionality $k = 16$. So, this is a proportional relationship.

Answer:

The second table, the fourth table, and the babysitter situation.