Back to Recognize and represent proportional relationships between quantities

Proportional Relationships — Tables, Graphs, Equations, and Verbal Descriptions

Grade 7·21 problems·Common Core Math - Grade 7·standard·7-rp-a-2

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Recall / Warm-Up

Which table shows a proportional relationship between $x$ and $y$ ?

$x$	$y$	$y/x$
2	6	3
4	12	3
6	18	3

$x$	$y$	$y/x$
1	5	5
3	9	3
5	15	3

$x$	$y$	$y/x$
2	7	3.5
4	10	2.5
6	13	2.17

$x$	$y$	$y/x$
0	4	—
2	6	3
4	8	2

A map scale is 1 inch = 25 miles. How many miles does 3 inches represent?

$28$

$75$

$8\frac{1}{3}$

$225$

In a proportional relationship, $y = 9$ when $x = 3$ . What is $y$ when $x = 7$ ? Express as a whole number or fraction.

Fluency Practice

A table shows the relationship between $x$ and $y$ :

$x$	$y$
3	15
6	30
9	45
12	60

Which conclusion is correct?

The relationship is proportional because $y/x = 5$ for the first row, so $k = 5$ .

The relationship is proportional because $y/x = 5$ for every row, confirming a constant ratio.

The relationship is not proportional because the $y$ -values are multiples of 15.

The relationship is proportional because all $y$ -values are larger than $x$ -values.

A proportional relationship is shown in the table:

Pounds of apples ( $x$ )	Total cost in dollars ( $y$ )
2	5
5	12.5
8	20

What is the constant of proportionality $k$ ? Express as a decimal.

A baker uses flour and sugar in a proportional relationship with constant of proportionality $k = \frac{2}{3}$ (cups of sugar per cup of flour). How many cups of sugar does the baker need for 12 cups of flour?

A factory produces widgets at a constant rate: 240 widgets in 4 hours. Write the equation relating widgets $w$ and hours $h$ , then find $w$ when $h = 7$ .

Gas costs $3.50$ per gallon. Let $c$ be the total cost in dollars and $g$ be the number of gallons. Which equation correctly represents this proportional relationship?

$c = \dfrac{g}{3.50}$

$c = g + 3.50$

$c = 3.50g$

$c = 3.50 + g$

Varied Practice

A graph shows a straight line passing through $(0, 4)$ and $(2, 10)$ . Does this graph represent a proportional relationship?

Yes — the graph is a straight line, so the relationship is proportional.

No — the line does not pass through the origin $(0, 0)$ , so the relationship is not proportional.

Yes — the ratio of the two points given is $10/2 = 5$ , confirming proportionality.

Cannot be determined without a full table of values.

A proportional relationship is shown in the table:

$x$	$y$
2	5
4	10
6	15
8	20

The constant of proportionality $k$ = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ . The equation relating $y$ and $x$ is $y$ = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $\times$ $x$ .

coefficient in equation:

$Coordinate plane showing a proportional relationship as a line through the origin passing through labeled points (0,0), (1,7.5), and (4,30). The point (1,7.5) is highlighted to indicate the unit rate.$

The graph of a proportional relationship passes through the origin, $(1, 7.5)$ , and $(4, 30)$ . What does the point $(1, 7.5)$ tell you about this relationship?

The graph starts at $y = 7.5$ when $x = 0$ .

The unit rate (constant of proportionality) is $7.5$ .

When $y = 1$ , the value of $x$ is $7.5$ .

The graph is not proportional because $7.5$ is not a whole number.

A car travels at a constant speed of $55$ miles per hour, where $d$ is distance in miles and $t$ is time in hours. Write an equation of the form $y = kx$ to represent this proportional relationship, then find how far the car travels in $3.5$ hours.

Word Problems

A grocery store sells peaches at a constant price per pound. The table below shows the cost for two purchase sizes: 2 lb costs $3.10$ and 5 lb costs $7.75$ .

What is the price per pound (the constant of proportionality $k$ )? Express as a decimal.

A solar panel generates electricity at a constant rate. After 3 hours of sunlight it has generated 7.5 kilowatt-hours (kWh) of energy.

Write an equation of the form $y = kx$ where $y$ is energy generated (kWh) and $x$ is hours of sunlight. Then find how many kWh the panel generates in 8 hours.

A student measures the amount of ink used ( $y$ mL) by a printer for different numbers of pages ( $x$ ):

Pages ( $x$ )	Ink ( $y$ mL)	$y/x$
10	2.5	0.25
20	5	0.25
40	10	0.25
50	12	0.24

Is the relationship between pages and ink exactly proportional according to all four data rows? Justify using the ratio test.

Yes — all four rows have $y/x \approx 0.25$ , so the relationship is proportional.

No — the ratio in the last row ( $12/50 = 0.24$ ) differs from the others ( $0.25$ ), so not all ratios are equal.

Yes — the first three rows confirm proportionality; the fourth row is close enough.

Cannot be determined without a graph.

Assuming the first three rows represent the true proportional rule, write the equation for ink $y$ in terms of pages $x$ . Use your equation to predict $y$ when $x = 80$ .

A recipe uses $\frac{2}{3}$ cup of butter for every $1$ cup of sugar. The relationship between butter $b$ (cups) and sugar $s$ (cups) is proportional.

Write the equation relating $b$ and $s$ , then find the amount of butter needed for $4.5$ cups of sugar. Express your answer as a whole number or fraction.

Error Analysis

$Two-column contrast card. Left (yellow): a line crossing the y-axis above the origin at (0,3), labeled Not Proportional with a red X. Right (teal): a line through the origin at (0,0), labeled Proportional with a checkmark.$

A graph shows a straight line passing through (0, 3) and (2, 9).

Jaylen claims: "This graph represents a proportional relationship because the line is straight."

What error did Jaylen make?

Jaylen is correct — a straight line always indicates a proportional relationship.

Jaylen confused the slope with the constant of proportionality — the slope is $(9-3)/(2-0) = 3$ , but that does not confirm proportionality.

Jaylen forgot to check whether the line passes through the origin — a proportional relationship requires a straight line through $(0, 0)$ , but this line crosses the $y$ -axis at $(0, 3)$ , not the origin.

Jaylen should have computed $y/x$ for only the point $(2, 9)$ to confirm proportionality.

Sonia finds that the constant of proportionality in a table is $k = 4$ .

She writes the equation: $y = \frac{x}{4}$

What error did Sonia make?

Sonia computed $k$ incorrectly — the constant of proportionality from a table should be $k = x/y$ , not $y/x$ .

Sonia wrote the equation incorrectly — the equation should be $y = 4x$ , not $y = x/4$ . She divided by $k$ instead of multiplying.

Sonia should have written $y = x + 4$ instead of $y = x/4$ .

Sonia's equation is correct — dividing by $k$ gives the same proportional relationship as multiplying by $k$ .

Challenge

A proportional relationship has the equation $y = \frac{5}{4}x$ . Find $x$ when $y = 35$ .

Two graphs are described:

Graph A: A straight line passing through $(0, 0)$ , $(2, 6)$ , and $(5, 15)$ .
Graph B: A straight line passing through $(0, 3)$ , $(2, 9)$ , and $(5, 18)$ .

Explain which graph represents a proportional relationship and why. For the proportional graph, identify the constant of proportionality $k$ , write the equation, and explain what the point $(0, 0)$ tells you about the situation.

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