Identify Proportional Relationships - 7th Grade Math

Which situation shows a proportional relationship between the two quantities?
Answer: A store sells notebooks for $5 each, so the total cost is $5$ times the number of notebooks. Proportional relationships have form $y = kx$ where one quantity equals a constant times the other, graphing as lines through origin ( $0,0$ ) with no initial value or starting fee. A store selling notebooks for $5$ each gives total cost = $5 \times$ number of notebooks, or $y=5x$ , which is proportional (when $x=0$ notebooks, $y=$0; ratios are constant at $5$ /notebook). The correct answer identifies the notebook scenario as proportional since total cost equals price per unit times quantity with no initial fee. Choice A has initial $4$ fee giving $y=2x+4$ (not proportional due to +4 term), B has initial 3cm giving $y=3-2x$ (not proportional due to +3 term), and D has variable growth rates $1$ cm then $2$ cm (ratios not constant, not proportional). Real-world proportional relationships: (1) unit rates with no initial fees ( $cost = price \times quantity$ ), (2) constant speeds from rest ( $distance = speed \times time$ ), (3) recipes or mixtures in fixed ratios. Non-proportional: initial fees, starting values, or changing rates prevent the $y=kx$ form required for proportionality.
A car rental company charges a one-time registration fee plus a daily rate. The total cost for 3 days is $85, for 5 days is $125, and for 7 days is $165. Based on this information, which statement about the relationship between days rented and total cost is correct?
Answer: The relationship is not proportional because the graph would not pass through the origin due to the registration fee For a proportional relationship, the ratio between quantities must be constant AND the relationship must pass through the origin (0,0). While the daily rate is constant ($20), the one-time registration fee ($25) means when days = 0, cost ≠ 0. Therefore, the graph doesn't pass through the origin, making it non-proportional. Choice A confuses constant rate with proportionality. Choice B incorrectly equates linear relationships with proportional ones. Choice D is wrong because the daily rate is actually constant.
A coordinate plane shows three lines, M, N, and P. Line M crosses the y-axis at (0, 2). Line N crosses the y-axis at (0, -1). Line P passes through the origin (0, 0), and every point on Line P has the same ratio of y to x. Which statement correctly identifies the proportional relationship?
Answer: Line P is proportional because it passes through the origin and maintains constant ratios A relationship is proportional only if its graph is a straight line through the origin (0,0). Line P passes through the origin, so it represents a proportional relationship, matching choice C. Line M crosses the y-axis at (0,2), not the origin, so it isn't proportional even though it's a straight line, and Line N crosses at (0,-1), so it fails the same test. Choice A confuses a steeper slope with proportionality, and choice D wrongly assumes every straight line with a positive slope is proportional.
Two students each made a graph of a relationship. Student 1's graph is the line $y=4x$ . Student 2's graph is the line $y=4x+5$ . Which statement is true?
Answer: Only Student 1's relationship is proportional because its graph passes through $(0,0)$ . Tests identifying proportional relationships by checking equivalent ratios in tables (y/x constant for all pairs) or verifying graphs pass through origin (straight line through (0,0)). Proportional relationship y=kx has constant ratio k: in table, calculate y/x for each pair (10/2=5, 20/4=5, 30/6=5 all equal → k=5 constant → proportional), on graph plots as straight line through origin (0,0) (proportional must have y-intercept=0, form y=kx not y=kx+b). Student 1 has y=4x (passes through (0,0), proportional) while Student 2 has y=4x+5 (passes through (0,5), not proportional due to +5 constant term). Answer B correctly identifies that only Student 1's relationship is proportional because its graph passes through (0,0). Answer A wrongly claims both are proportional just for being straight lines, Answer C incorrectly says Student 2 is proportional (it's not, due to the +5), and Answer D nonsensically claims proportional graphs can't have slopes (they must have slopes, that's the constant k). Graph method: y=4x passes through origin (when x=0, y=0), making it proportional with k=4. But y=4x+5 passes through (0,5), not origin, so it's linear but not proportional. The +5 is a y-intercept that prevents the relationship from being proportional - proportional requires y=kx form with no added constant.
A water tank is being filled at a constant rate. After 2 minutes, there are 50 gallons. After 4 minutes, there are 70 gallons. After 6 minutes, there are 90 gallons. A student graphs this data and concludes the relationship is not proportional because "the line doesn't pass through the origin." What additional information would help verify this conclusion?
Answer: The amount of water in the tank at the start of filling (when time = 0 minutes) To determine if a relationship is proportional, you need to know if it passes through (0,0). The student correctly identified that the relationship appears non-proportional, but knowing the initial amount at t=0 would confirm this. From the pattern (10 gallons/minute increase), at t=0 there would be 30 gallons, confirming non-proportionality. Choice B about tank capacity is irrelevant to proportionality. Choice C is unnecessary since the rate is clearly constant (10 gal/min). Choice D wouldn't change the conclusion about proportionality.
Examine the graph showing three different relationships. If you were to extend each line, which would represent a proportional relationship?
Answer: Line C, because extending it backward would pass through the origin at (0,0) To identify proportional relationships from partial graphs, extend the line backward to see if it would pass through (0,0). Line C, when extended, would pass through the origin. Lines A and B would have positive y-intercepts when extended backward. Choice A confuses consistent rate with proportionality. Choice B confuses correlation strength with proportionality. Choice D is incorrect; the issue isn't negative y-intercepts but rather any non-zero y-intercept.
Two students are comparing relationships between x and y. Student A has data where the ratio y/x equals 2.5 for every point, but one of the data points is (0, 3). Student B has data with the points (1, 2), (2, 4.2), (3, 5.7), and (4, 8), so the ratio y/x is not the same for every point. Which student has a proportional relationship?
Answer: Neither student, since each one fails one of the two requirements for a proportional relationship A proportional relationship must have a constant ratio y/x for every point, and it must pass through the origin (0, 0). Student A's ratio is constant at 2.5, but the point (0, 3) shows that when x = 0, y is 3, not 0, so Student A's relationship doesn't pass through the origin. Student B's points give ratios of 2, 2.1, 1.9, and 2, which are not all the same, so Student B fails the constant ratio requirement. Since each student fails one of the two required conditions, neither has a proportional relationship. Choice A is wrong because passing through the origin is just as necessary as having a constant ratio. Choice B is wrong because even small differences in the ratio mean it isn't truly constant. Choice C is wrong because meeting only one of the two conditions isn't enough for a relationship to be proportional.
Two classmates make graphs from different situations. Graph 1 has points $(0,0)$ , $(2,6)$ , $(4,12)$ . Graph 2 has points $(0,3)$ , $(2,7)$ , $(4,11)$ . Which graph shows a proportional relationship?
Answer: Graph 1 only, because it forms a line through the origin. Tests identifying proportional relationships by checking equivalent ratios in tables (y/x constant for all pairs) or verifying graphs pass through origin (straight line through (0,0)). Proportional relationship y=kx has constant ratio k: in table, calculate y/x for each pair (10/2=5, 20/4=5, 30/6=5 all equal → k=5 constant → proportional), on graph plots as straight line through origin (0,0) (proportional must have y-intercept=0, form y=kx not y=kx+b). Non-proportional: ratios vary (3/1=3, 5/2=2.5, 7/3≈2.33 different → not constant → not proportional), or graph misses origin (y=2x+1 through (0,1) not (0,0) → not proportional even though linear). Graph 1 with points (0,0), (2,6), (4,12) shows a proportional relationship as it forms a straight line through the origin with constant ratio 3 (6/2=3, 12/4=3), while Graph 2 with (0,3), (2,7), (4,11) does not pass through (0,0) and has varying ratios. A common error is selecting option B, thinking Graph 2 is proportional just because it is straight, without verifying the origin. Graph method: (1) plot points, (2) check if collinear (straight line? if not, definitely not proportional), (3) extend line to y-axis (does it pass (0,0)? yes→proportional; passes (0,b) with b≠0→not proportional, linear but y=mx+b form). Both methods: proportional requires BOTH equivalent ratios (constant k) AND line through origin—they're equivalent tests (if k constant, graph through origin; if through origin, k must be constant).
Use the table to determine which relationship represents a proportional relationship.
Answer: Relationship Q, because the ratios $\frac{y}{x}$ are constant and equal to 3 A proportional relationship requires constant ratios y/x for all data points. Relationship Q has ratios: 6/2=3, 9/3=3, 12/4=3, showing constant ratios. Relationship P has equal differences (linear) but ratios 4/1=4, 7/2=3.5, 10/3≈3.33 (not constant). Relationship R has ratios 3/1=3, 8/2=4, 15/3=5 (not constant). Choice A confuses linear with proportional. Choice C describes correlation, not proportionality. Choice D incorrectly assumes different rates mean no proportional relationships exist.
A recipe calls for ingredients in these amounts: 2 cups flour with 3 cups milk, 4 cups flour with 6 cups milk, and 6 cups flour with 9 cups milk. A student claims this shows a proportional relationship between flour and milk. However, the recipe also requires a constant 1 tablespoon of vanilla regardless of batch size. How does the vanilla requirement affect the analysis?
Answer: It doesn't affect the analysis because vanilla amount stays constant while flour and milk vary proportionally The proportional relationship between flour and milk (ratios 3/2 = 6/4 = 9/6 = 1.5) is independent of the vanilla requirement. Proportionality is about the relationship between two specific variables, not all variables in a system. The vanilla affects the relationship between 'total ingredients' and 'batch size' but not the flour-milk relationship. Choice A incorrectly focuses on units. Choice C correctly identifies that total ingredients aren't proportional to batch size, but this doesn't affect the flour-milk relationship. Choice D incorrectly assumes all variables must be proportional.
Marcus claims that the relationship between the side length of a square and its perimeter is proportional. Sophia argues that the relationship between the side length and the area is proportional. To test their claims, they create data for squares with side lengths 2, 4, and 6 units. Which conclusion is correct?
Answer: Only Marcus is correct because perimeter = 4 × side length, giving constant ratios of 4:1 For proportional relationships, y/x must be constant. Perimeter ratios: 8/2=4, 16/4=4, 24/6=4 (constant). Area ratios: 4/2=2, 16/4=4, 36/6=6 (not constant). The area relationship is quadratic (y = x²), not proportional. Choice A incorrectly assumes patterns indicate proportionality. Choice C confuses the magnitude of change with proportionality. Choice D is incorrect because perimeter is indeed proportional to side length.

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