Question

do you understand?

1. essential question how are proportional quantities described by equivalent ratios?
2. look for relationships how do you know if a relationship between two quantities is not proportional?
3. reasoning if the ratio $\frac{y}{x}$ is the same for all related pairs of $x$ and $y$, what does that mean about the relationship between $x$ and $y$?

do you know how?

1. use the table below. do $x$ and $y$ have a proportional relationship? explain.

$x$	2	3	5	8
$y$	5	7.5	12.5	18

1. each triangle is equilateral. is the relationship between the perimeter and the side length of the equilateral triangles proportional? explain.

1 in. 2 in. 3 in.

1. is the relationship between the number of tickets sold and the number of hours proportional? if so, how many tickets were sold in 8 hours?

hours ($h$)	tickets sold ($t$)
3	240
5	400
9	720

Explanation:

4.

Step1: Calculate the ratio for each pair

For the first pair $x = 2,y = 5$, the ratio $\frac{y}{x}=\frac{5}{2}=2.5$.
For the second pair $x = 3,y = 7.5$, the ratio $\frac{y}{x}=\frac{7.5}{3}=2.5$.
For the third pair $x = 5,y = 12.5$, the ratio $\frac{y}{x}=\frac{12.5}{5}=2.5$.
For the fourth pair $x = 8,y = 18$, the ratio $\frac{y}{x}=\frac{18}{8}=2.25$.

Step2: Check for proportionality

Since the ratio $\frac{y}{x}$ is not the same for all pairs of $x$ and $y$ values, $x$ and $y$ do not have a proportional relationship.

Step1: Recall the formula for the perimeter of an equilateral triangle

The perimeter $P$ of an equilateral triangle with side - length $s$ is given by $P = 3s$.

Step2: Calculate the ratios

For a side - length $s_1=1$ in, the perimeter $P_1 = 3\times1=3$ in, and the ratio $\frac{P_1}{s_1}=\frac{3}{1}=3$.
For a side - length $s_2 = 2$ in, the perimeter $P_2=3\times2 = 6$ in, and the ratio $\frac{P_2}{s_2}=\frac{6}{2}=3$.
For a side - length $s_3=3$ in, the perimeter $P_3=3\times3 = 9$ in, and the ratio $\frac{P_3}{s_3}=\frac{9}{3}=3$.
Since the ratio of the perimeter to the side - length is always 3 for all equilateral triangles, the relationship is proportional.

Step1: Calculate the ratio for each pair

For the first pair $h = 3,t = 240$, the ratio $\frac{t}{h}=\frac{240}{3}=80$.
For the second pair $h = 5,t = 400$, the ratio $\frac{t}{h}=\frac{400}{5}=80$.
For the third pair $h = 9,t = 720$, the ratio $\frac{t}{h}=\frac{720}{9}=80$.
Since the ratio $\frac{t}{h}$ is the same for all pairs, the relationship is proportional.

Step2: Find the number of tickets sold in 8 hours

We know that the ratio $\frac{t}{h}=80$, so when $h = 8$, then $t=80\times8 = 640$.

Answer:

No, because the ratios $\frac{y}{x}$ are not equal for all pairs of $x$ and $y$ values in the table.

5.