Question

key concept
two quantities x and y have a proportional relationship if all the ratios $\frac{y}{x}$ for related pairs of x and y are equivalent.
a proportion is an equation that states that two ratios are equivalent.
$\frac{3}{120}=\frac{5}{200}$
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)	3	5	9
tickets sold (t)	240	400	720

Explanation:

Step1: Recall the definition of proportional relationship

Two quantities \(x\) and \(y\) have a proportional relationship if \(\frac{y}{x}\) is constant for all related pairs of \(x\) and \(y\).

Step2: Calculate \(\frac{y}{x}\) for each pair in question 4

For \(x = 2,y = 5\), \(\frac{y}{x}=\frac{5}{2}=2.5\).
For \(x = 3,y = 7.5\), \(\frac{y}{x}=\frac{7.5}{3}=2.5\).
For \(x = 5,y = 12.5\), \(\frac{y}{x}=\frac{12.5}{5}=2.5\).
For \(x = 8,y = 18\), \(\frac{y}{x}=\frac{18}{8}=2.25\).
Since \(\frac{y}{x}\) is not the same for all pairs, \(x\) and \(y\) do not have a proportional - relationship.

Step3: Analyze the equilateral - triangle relationship in question 5

The perimeter \(P\) of an equilateral triangle with side length \(s\) is \(P = 3s\). Then \(\frac{P}{s}=3\) for all equilateral triangles. So the relationship between the perimeter and the side - length of equilateral triangles is proportional.

Step4: Calculate \(\frac{y}{x}\) for each pair in question 6

For \(h = 3,t = 240\), \(\frac{t}{h}=\frac{240}{3}=80\).
For \(h = 5,t = 400\), \(\frac{t}{h}=\frac{400}{5}=80\).
For \(h = 9,t = 720\), \(\frac{t}{h}=\frac{720}{9}=80\).
Since \(\frac{t}{h}\) is constant (\(80\)), the relationship between the number of tickets sold and the number of hours is proportional.
To find the number of tickets sold in 8 hours, we use the constant of proportionality. If \(\frac{t}{h}=80\), then when \(h = 8\), \(t=80\times8 = 640\).

Answer:

1. \(x\) and \(y\) do not have a proportional relationship because \(\frac{y}{x}\) is not the same for all pairs (\(\frac{5}{2}=2.5,\frac{7.5}{3}=2.5,\frac{12.5}{5}=2.5,\frac{18}{8}=2.25\)).
2. The relationship between the perimeter and the side - length of equilateral triangles is proportional because for an equilateral triangle with perimeter \(P\) and side - length \(s\), \(P = 3s\) and \(\frac{P}{s}=3\) (constant).
3. The relationship between the number of tickets sold and the number of hours is proportional. 640 tickets were sold in 8 hours.