Question

1. determine the correct category to indicate whether each table represents a proportional or nonproportional relationship. select the correct answer from each row.
2. maya and her friends are each saving money to buy a new video game. the relationship between the number of weeks and the amount saved is represented for maya and each friend.

which statement is true about the relationships represented?
a. both allie and maya are saving money at the same rate.
b. allie started with an initial amount of money already saved at 0 weeks.
c. both maya and victor are saving money at the same rate.
d. allie and mayas savings are represented by proportional relationships.

Explanation:

Step1: Recall proportional - relationship condition

A relationship is proportional if $\frac{y}{x}$ is constant for all pairs of $(x,y)$.

Step2: Check first table in question 3

For the first table: $\frac{5}{10}=0.5$, $\frac{6}{12}=0.5$, $\frac{11.5}{23}=0.5$. Since $\frac{y}{x}$ is constant, it is proportional.

Step3: Check second table in question 3

For the second table: $\frac{7}{1}=7$, $\frac{9}{4}=2.25$, $\frac{23}{5}=4.6$. Since $\frac{y}{x}$ is not constant, it is non - proportional.

Step4: Analyze Maya's savings in question 4

For Maya, the rate of change between week 1 ($122$) and week 5 ($210$): $\text{Rate}=\frac{210 - 122}{5 - 1}=\frac{88}{4}=22$. Between week 5 and week 6: $\text{Rate}=\frac{232-210}{6 - 5}=22$. But at $x = 0$, $y
eq0$ (it has an initial non - zero amount), so non - proportional.

Step5: Analyze Victor's savings in question 4

Victor's graph is a straight - line through the origin (implied by the linear graph starting from $(0,0)$), so it is proportional with a rate determined by the slope of the line.

Step6: Analyze Allie's savings in question 4

Allie has $y = 22x$, which is in the form $y=kx$ (proportional relationship) with $k = 22$.

Step7: Evaluate answer choices in question 4

• Option A: Maya has an initial amount, Allie starts from $0$. Their rates are the same but the relationships are different in nature, so false.
• Option B: Allie's equation is $y = 22x$, which means she starts with $0$ at $x = 0$, so false.
• Option C: Victor's graph is proportional (through origin) and Maya has an initial non - zero amount. But their rates are calculated as follows: For Maya, rate between points is $22$. For Victor, since it's a straight - line through origin, assume two points $(x_1,y_1)$ and $(x_2,y_2)$ on the line. The slope (rate) is also $22$ (by observing the linear nature and calculating slope between two points on the line). This option is true.
• Option D: Maya has an initial non - zero amount, so her relationship is non - proportional, false.

Answer:

1. First table: Proportional; Second table: Nonproportional
2. C. Both Maya and Victor are saving money at the same rate.