Middle School Math Quiz: Identify Constant Of Proportionality

20 questions · exam conditions

0:00

Identify Constant Of ProportionalityQuestion 1 of 20

Two proportional relationships are described below.

Relationship 1: $y=6x$
Relationship 2: A table shows $x=1,2,3$ and $y=4,8,12$ .

Which relationship has the greater constant of proportionality $k$ ?

Not enough information to compare

Relationship 2

They have the same

k

Relationship 1

← Back to quizzes

Middle School Math Quiz

Middle School Math Quiz: Identify Constant Of Proportionality

Practice Identify Constant Of Proportionality in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Identify Constant Of Proportionality, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Two proportional relationships are described below.

Relationship 1: $y=6x$
Relationship 2: A table shows $x=1,2,3$ and $y=4,8,12$ .

Which relationship has the greater constant of proportionality $k$ ?

Not enough information to compare
Relationship 2
They have the same $k$
Relationship 1 (correct answer)

Explanation: This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. A proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7); the point (1,k) is special because when x=1, y=k (so a graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, a table showing x:2,4,6 y:14,28,42 allows calculating ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or a graph through (0,0) and (1,7) has k=7 from the point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. Relationship 1 has k=6 from y=6x; Relationship 2 has k=4 from table ratios (4/1=4, 8/2=4, 12/3=4), so Relationship 1 has greater k since 6>4. Common errors include miscalculating table ratios (e.g., x/y=1/4), thinking they are equal, or inverting. To find k, from equation it's the coefficient, from table calculate y/x and verify; special point (1,k) for graphs. Mistakes: assuming not enough info when data is given, or confusing with non-proportional cases.

Question 2

A car uses gasoline at a constant rate: it travels 65 miles per gallon.

If $y$ is miles traveled and $x$ is gallons of gas used, what is the constant of proportionality $k$ in $y=kx$ ?

$k=650$
$k=0.65$
$k=65$ (correct answer)
$k=\dfrac{1}{65}$

y=kx

has constant k equal to: (1) ratio y/x for any point (

14/2=7

28/4=7

k=7

), (2) slope of graph (

\frac{\text{rise}}{\text{run}}

through origin), (3) coefficient of x in equation (

y=7x \to k=7

), (4) unit rate stated ("7 dollars per item"

\to k=7

). Point (1,k) special: when x=1, y=k (so graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). In this verbal description, the car travels 65 miles per gallon, so with y miles and x gallons, k=65 as the stated unit rate in

y=65x

. Common mistakes include inverting to 1/65 (A), using decimals like 0.65 (C) or multiplying unnecessarily to 650 (D). Finding k: from table (pick any (x,y) pair, calculate k=y/x, verify with other pairs—should all equal), from graph (slope=\frac{\text{rise}}{\text{run}}

through origin, or read y at x=1 giving (1,k), k is that y-value), from equation y=kx (k is coefficient:

y=7x \to k=7

), from verbal (stated rate is k: "3 meters per second"

\to k=3

m/s). Not proportional: if y-intercept≠0 (line misses origin), no constant of proportionality exists (

y=mx+b$ with b≠0 is linear but not proportional, no k value).

Question 3

A diagram shows a gear system where Gear A and Gear B rotate proportionally. When Gear A completes 15 rotations, Gear B completes 25 rotations. Based on this proportional relationship, what is the constant of proportionality representing rotations of Gear B per rotation of Gear A?

$\frac{3}{5}$ rotations of Gear B per rotation of Gear A
$\frac{5}{3}$ rotations of Gear B per rotation of Gear A (correct answer)
$10$ rotations of Gear B per rotation of Gear A
$40$ rotations of Gear B per rotation of Gear A

Explanation: The constant of proportionality is

\frac{25}{15} = \frac{5}{3} \approx 1.67

rotations of Gear B per rotation of Gear A. Choice A gives the reciprocal ratio (Gear A per Gear B). Choice C represents the difference

25 - 15 = 10

. Choice D represents the sum

25 + 15 = 40

Question 4

A spring stretches proportionally to the weight attached to it. The relationship can be described as: "For every 3 pounds of weight added, the spring stretches an additional 2.5 inches." What is the constant of proportionality for inches of stretch per pound of weight?

$\frac{3}{2}$ inches of stretch per pound of weight applied
$\frac{6}{5}$ inches of stretch per pound of weight applied
$\frac{2}{3}$ inches of stretch per pound of weight applied
$\frac{5}{6}$ inches of stretch per pound of weight applied (correct answer)

Explanation: When you encounter problems about proportional relationships, you're looking for a constant rate that describes how one quantity changes with respect to another. Here, you need to find how many inches the spring stretches per pound of weight. The problem tells you that for every 3 pounds added, the spring stretches 2.5 inches. To find the constant of proportionality (inches per pound), you need to create a ratio and simplify it:

\frac{2.5 \text{ inches}}{3 \text{ pounds}} = \frac{2.5}{3}

Convert 2.5 to a fraction:

2.5 = \frac{5}{2}

So you have:

\frac{\frac{5}{2}}{3} = \frac{5}{2} \times \frac{1}{3} = \frac{5}{6}

inches per pound This matches answer choice D. Let's examine why the other answers are wrong: Choice A gives

\frac{3}{2}

, which incorrectly puts pounds in the numerator instead of inches. Choice B gives

\frac{6}{5}

, which appears to flip the correct fraction. Choice C gives

\frac{2}{3}

, which seems to use 2 instead of 2.5 in the calculation, possibly from converting 2.5 incorrectly or misreading the problem. Study tip: For proportional relationships, always set up your ratio with the units you want in the answer. If you need "inches per pound," put inches in the numerator and pounds in the denominator. Then simplify the fraction completely. Double-check by asking: "Does this unit rate make sense given the original relationship?"

Question 5

A proportional relationship is given by the equation $y=\frac{3}{2}x$ .

What is the constant of proportionality $k$ ?

$k=\frac{2}{3}$
$k=2$
$k=3$
$k=\frac{3}{2}$ (correct answer)

Explanation: This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. A proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7); the point (1,k) is special because when x=1, y=k (so a graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, a table showing x:2,4,6 y:14,28,42 allows calculating ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or a graph through (0,0) and (1,7) has k=7 from the point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. In the equation y=(3/2)x, k=3/2 is the coefficient. Common errors include simplifying incorrectly to 2/3, using numerator 3 or denominator 2 alone, or inverting. From an equation y=kx, k is the coefficient; special point (1,k): graphs pass through (1,k) where k=3/2 here. Mistakes: confusing with non-proportional (y=mx+b, b≠0 has no k), or calculating ratios wrong.

Question 6

A car gets 65 miles per gallon of gas. Let $x$ be gallons of gas and $y$ be miles traveled, with a proportional relationship $y=kx$ . What is the constant of proportionality $k$ ?

$k=\dfrac{65}{2}$
$k=\dfrac{1}{65}$
$k=130$
$k=65$ (correct answer)

Explanation: In the proportional relationship y = kx, the constant k is the rate relating miles to gallons, which here is 65 miles per gallon. So k = 65. Choice A comes from dividing the rate by 2 for no clear reason based on the problem. Choice B comes from inverting the rate, mixing up miles per gallon with gallons per mile. Choice C comes from doubling the rate instead of using it directly.

Question 7

The graph shows the relationship between time and the total number of pages printed by a copy machine. What is the constant of proportionality representing pages printed per minute?

$8$ pages per minute at a constant printing speed
$12$ pages per minute at a constant printing speed
$15$ pages per minute at a constant printing speed (correct answer)
$20$ pages per minute at a constant printing speed

Explanation: Using any point on the line to calculate the slope: at (4, 60), the rate is

\frac{60}{4} = 15

pages per minute. This can be verified with (2, 30):

\frac{30}{2} = 15

pages per minute. Choice A uses

\frac{32}{4}

with incorrect y-value. Choice B uses

\frac{60}{5}

with incorrect x-value. Choice D uses

\frac{60}{3}

with incorrect x-value.

Question 8

A runner's distance is proportional to time. The graph shows a line through $(0,0)$ and $(2,10)$ .

What is the constant of proportionality $k$ (miles per minute) in $y=kx$ ?

$k=2$
$k=5$ (correct answer)
$k=10$
$k=8$

Explanation: This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. A proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7); the point (1,k) is special because when x=1, y=k (so a graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, a table showing x:2,4,6 y:14,28,42 allows calculating ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or a graph through (0,0) and (1,7) has k=7 from the point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. The graph passes through (0,0) and (2,10), so slope=10/2=5, giving k=5 miles per minute. Common errors include using y=10 or x=2 as k without dividing, inverting to 2/10=1/5 (not an option), or miscalculating slope as 8. From a graph, use slope=rise/run through the origin, or read y at x=1 giving (1,k); proportional graphs pass through (1,k)—here, it would pass through (1,5). Not proportional if y-intercept ≠0, no k; mistakes include reading graph at wrong point.

Question 9

A proportional relationship includes the point $(1,8)$ on its graph.

What is the constant of proportionality $k$ in $y=kx$ ?

$k=9$
$k=8$ (correct answer)
$k=1$
$k=\dfrac{1}{8}$

Explanation: This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. Proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7). Point (1,k) special: when x=1, y=k (so graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, table showing x:2,4,6 y:14,28,42, calculate ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or graph through (0,0) and (1,7) has k=7 from point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. The graph includes (1,8), so k=8 directly from y at x=1. Mistakes: thinking it's 1/8 (inverted), or confusing with other points. Graphs of y=kx pass through (1,k), making k readable; if no (1,k), calculate slope from any point.

Question 10

A recipe uses 3 cups of flour for each batch of muffins. The relationship is proportional.

What is the constant of proportionality $k$ (cups of flour per batch) in $y=kx$ , where $x$ is batches and $y$ is cups of flour?

$k=6$
$k=3$ (correct answer)
$k=9$
$k=\dfrac{1}{3}$

Explanation: This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. Proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7). Point (1,k) special: when x=1, y=k (so graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, table showing x:2,4,6 y:14,28,42, calculate ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or graph through (0,0) and (1,7) has k=7 from point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. The verbal description states 3 cups per batch, so k=3 (cups of flour per batch). A mistake could be inverting to 1/3 (batches per cup) or misreading the units. From a verbal description, the stated unit rate is k, like "3 cups per batch" → k=3; verify if it's proportional (direct variation without added constants).

Question 11

A proportional relationship is graphed as a line that passes through the origin and the point $(1,3)$ .

What is the constant of proportionality $k$ in $y=kx$ ?

$k=3$ (correct answer)
$k=\dfrac{1}{3}$
$k=4$
$k=1$

Explanation: This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. Proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7). Point (1,k) special: when x=1, y=k (so graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, table showing x:2,4,6 y:14,28,42, calculate ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or graph through (0,0) and (1,7) has k=7 from point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. Here, the graph passes through (1,3), so k=3 directly from the y-coordinate at x=1. A common mistake is inverting to 1/3 (x/y instead of y/x), or confusing with other points without calculating slope properly. To find k from a graph, calculate slope=rise/run through origin, or read y at x=1 giving (1,k), where k is that y-value; special point (1,k) makes k directly readable without calculation. Not proportional if y-intercept ≠0, but here it passes through origin, so k=3.

Question 12

The relationship between time $x$ (in hours) and distance $y$ (in miles) is given by the equation $y=4.5x$ . What is the constant of proportionality $k$ ?

$k=5$
$k=0.45$
$k=\dfrac{1}{4.5}$
$k=4.5$ (correct answer)

Explanation: This skill tests identifying the constant of proportionality k (unit rate, slope) from tables (

y/x

ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. Proportional relationship

y=kx

has constant k equal to: (1) ratio

y/x

for any point (

14/2=7

28/4=7

k=7

), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (

y=7x \to k=7

), (4) unit rate stated ("7 dollars per item"

\to k=7

). Point (1,k) special: when

x=1

y=k

(so graph passing through (1,7) has

k=7

directly readable—unit rate at one unit of x). For this equation

y=4.5x

relating distance y to time x, k is the coefficient 4.5, representing miles per hour as the constant rate. Errors might include inverting to

1/4.5

(B), confusing with other numbers like 5 (C) or 0.45 (D), or misreading the equation. Finding k: from table (pick any (x,y) pair, calculate

k=y/x

, verify with other pairs—should all equal), from graph (slope=rise/run through origin, or read y at x=1 giving (1,k), k is that y-value), from equation

y=kx

(k is coefficient:

y=7x \to k=7

), from verbal (stated rate is k: "3 meters per second"

\to k=3

m/s). Mistakes: confusing slope with y-intercept (using b as k), calculating ratios wrong (

x/y

not

y/x

), reading graph at wrong point, non-proportional relationships claimed to have k.

Question 13

Using the data from Sarah's visits shown in the table, determine the constant of proportionality for dollars earned per pound of cans.

$0.65 per pound of aluminum cans collected
$0.75 per pound of aluminum cans collected (correct answer)
$0.85 per pound of aluminum cans collected
$1.25 per pound of aluminum cans collected

Explanation: Calculate the rate for each visit: Visit 1:

\frac{\$3.00}{4} = \$0.75

per pound. Visit 2:

\frac{\$4.50}{6} = \$0.75

per pound. Visit 3:

\frac{\$6.00}{8} = \$0.75

per pound. The constant rate is $0.75 per pound. Choice A results from using $

\frac{\

3.90}{6}

. Choice C results from miscalculating

\frac{$6.00}{7}

. Choice D uses

\frac{$6.00}{$4.50}

incorrectly.

Question 14

Examine the coordinate plane shown. A line passes through the origin and represents a proportional relationship between $x$ and $y$ . What is the constant of proportionality?

$\frac{1}{3}$ units of $y$ per unit of $x$
$\frac{2}{3}$ units of $y$ per unit of $x$
$\frac{3}{2}$ units of $y$ per unit of $x$ (correct answer)
$3$ units of $y$ per unit of $x$

Explanation: The constant of proportionality is the slope of the line through the origin. Using the point (4, 6): slope =

\frac{6-0}{4-0} = \frac{6}{4} = \frac{3}{2}

. This can be verified with (2, 3):

\frac{3}{2} = \frac{3}{2}

. Choice A is the reciprocal. Choice B results from incorrectly calculating

\frac{4}{6}

. Choice D uses only the y-coordinate of one point.

Question 15

Based on the table shown, which value represents the constant of proportionality for the relationship between time and distance?

$12$ miles per hour with consistent rate throughout
$15$ miles per hour with consistent rate throughout (correct answer)
$18$ miles per hour with consistent rate throughout
$20$ miles per hour with consistent rate throughout

Explanation: To find the constant of proportionality, divide distance by time for any row:

\frac{30}{2} = 15

\frac{45}{3} = 15

\frac{60}{4} = 15

. The constant rate is 15 miles per hour. Choice A results from miscalculating

\frac{60}{5}

instead of

\frac{60}{4}

. Choice C comes from adding 3 to the correct answer. Choice D results from using

\frac{60}{3}

incorrectly.

Question 16

A proportional relationship is shown on a coordinate plane by a line that goes through the origin and the point (1,3). What is the constant of proportionality k in y=kx?

k=1
k=3 (correct answer)
k=4
k=1/3

Explanation: In a proportional relationship y = kx, the constant k can be found by looking at any point on the line, since k equals y divided by x. The line passes through (1, 3), so k = 3 divided by 1, which equals 3, matching Choice B. Choice A, k = 1, and Choice C, k = 4, do not match this point on the line. Choice D, k = 1/3, is the reciprocal of the correct value, which would happen if x and y were divided in the wrong order.

Question 17

Marcus earns money by walking dogs. The equation $d = 8.5h$ represents the relationship between the number of hours worked ( $h$ ) and the total dollars earned ( $d$ ). After working for several days, Marcus realizes he needs to account for $12 in transportation costs. What is the constant of proportionality in the original earning relationship?

$8.50 per hour before considering any additional expenses (correct answer)
$12.00 per hour before considering any additional expenses
$20.50 per hour before considering any additional expenses
$96.50 per hour before considering any additional expenses

Explanation: In the equation

d = 8.5h

, the constant of proportionality is the coefficient 8.5, representing $8.50 earned per hour. The $12 transportation cost is additional information that doesn't affect the original proportional relationship. Choice B uses only the transportation cost. Choice C incorrectly adds 8.5 + 12. Choice D multiplies 8.5 × 12 - 6.

Question 18

A water tank fills at a constant rate. In the first 4 minutes, 18 gallons flow into the tank. In the next 6 minutes, 27 gallons flow into the tank. What is the constant of proportionality that describes gallons per minute?

$6.0$ gallons per minute with steady flow rate
$4.0$ gallons per minute with steady flow rate
$5.5$ gallons per minute with steady flow rate
$4.5$ gallons per minute with steady flow rate (correct answer)

Explanation: When you encounter a problem about constant rates, you're working with proportional relationships where the rate of change stays the same throughout. The key is finding the rate per unit of time. To find the constant rate in gallons per minute, you need to calculate the total gallons divided by the total time. In the first 4 minutes, 18 gallons flow in. In the next 6 minutes, 27 gallons flow in. This gives you a total of

18 + 27 = 45

gallons over a total time of

4 + 6 = 10

minutes. The constant rate is

\frac{45 \text{ gallons}}{10 \text{ minutes}} = 4.5

gallons per minute. You can verify this works: in 4 minutes at 4.5 gallons/minute, you get

4 \times 4.5 = 18

gallons. In 6 minutes, you get

6 \times 4.5 = 27

gallons. Perfect! Answer choice A (

6.0

gallons per minute) likely comes from incorrectly dividing 27 by 4 or 18 by 3, mixing up the time periods. Answer choice B (

4.0

gallons per minute) might result from rounding 4.5 down or dividing 24 by 6 through some calculation error. Answer choice C (

5.5

gallons per minute) could come from finding the average of individual rates (4.5 and 6.75) rather than using total gallons over total time. Remember: for constant rate problems, always use total amount divided by total time. Don't average separate rates or use individual time periods incorrectly.

Question 19

A recipe uses 3 cups of flour for each batch of muffins. The number of cups of flour $y$ is proportional to the number of batches $x$ .

What is the constant of proportionality $k$ in $y=kx$ (cups per batch)?

$k=3$ (correct answer)
$k=9$
$k=6$
$k=\frac{1}{3}$

Explanation: This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. A proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7); the point (1,k) is special because when x=1, y=k (so a graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, a table showing x:2,4,6 y:14,28,42 allows calculating ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or a graph through (0,0) and (1,7) has k=7 from the point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. Here, the verbal description states 3 cups of flour per batch, so k=3 cups per batch in y=3x. Common errors include inverting to 1/3, using multiples like 6 or 9, or misinterpreting "per batch" as x instead of the rate. From verbal descriptions, the stated rate is k ("3 meters per second" → k=3 m/s); to find k from a table, pick any (x,y) pair, calculate k=y/x, verify with others. Special point (1,k): proportional graphs pass through (1,k) where k is constant—makes k directly readable; not proportional if y-intercept ≠0, no k value.

Question 20

A recipe uses flour in a proportional way: 1 batch needs 3 cups of flour, 2 batches need 6 cups, and 3 batches need 9 cups.

Let $x$ be the number of batches and $y$ be the number of cups of flour. What is the constant of proportionality $k$ in $y=kx$ ?

$k=\dfrac{1}{3}$
$k=9$
$k=3$ (correct answer)
$k=6$

Explanation: This skill tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. Proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7). Point (1,k) special: when x=1, y=k (so graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). Here, the recipe gives pairs like 1 batch with 3 cups, 2 with 6, 3 with 9, so ratios 3/1=3, 6/2=3, 9/3=3 yield k=3 for y=3x. Errors could be using a y-value like 6 (B) or 9 (C) without dividing, or inverting to 1/3 (D). Finding k: from table (pick any (x,y) pair, calculate k=y/x, verify with other pairs—should all equal), from graph (slope=rise/run through origin, or read y at x=1 giving (1,k), k is that y-value), from equation y=kx (k is coefficient: y=7x → k=7), from verbal (stated rate is k: "3 meters per second" → k=3 m/s). Special point (1,k): proportional graphs pass through (1,k) where k is constant—makes k directly readable (no calculation needed, just read y-coordinate at x=1).