7th Grade Math Quiz: Identify Constant Of Proportionality

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 7th Grade 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.

Question 1

1. $6.0$ gallons per minute with steady flow rate
2. $4.0$ gallons per minute with steady flow rate
3. $5.5$ gallons per minute with steady flow rate
4. $4.5$ 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 2

1. $\frac{3}{5}$ rotations of Gear B per rotation of Gear A
2. $\frac{5}{3}$ rotations of Gear B per rotation of Gear A (correct answer)
3. $10$ rotations of Gear B per rotation of Gear A
4. $40$ 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 3

1. $\frac{2}{5}$ ounces of white paint per ounce of blue paint
2. $\frac{5}{2}$ ounces of white paint per ounce of blue paint (correct answer)
3. $21$ ounces of white paint per ounce of blue paint
4. $\frac{6}{15}$ ounces of white paint per ounce of blue paint

Explanation: The constant of proportionality is the ratio of white paint to blue paint: $\frac{15}{6} = \frac{5}{2} = 2.5$ ounces of white paint per ounce of blue paint. Choice A gives the ratio of blue to white (reciprocal). Choice C adds the quantities instead of finding their ratio. Choice D presents the unreduced fraction in the wrong order (blue to white).

Question 4

1. $\frac{3}{2}$ inches of stretch per pound of weight applied
2. $\frac{6}{5}$ inches of stretch per pound of weight applied
3. $\frac{2}{3}$ inches of stretch per pound of weight applied
4. $\frac{5}{6}$ 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

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

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 6

1. Not enough information to compare
2. Relationship 2
3. They have the same $k$
4. 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 7

1. $k=650$
2. $k=0.65$
3. $k=65$ (correct answer)
4. $k=\dfrac{1}{65}$

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 (rac{\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 8

1. $k=\frac{1}{9}$
2. $k=9$ (correct answer)
3. $k=1$
4. $k=10$

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 the origin and (1,9), so k=9 directly from the y-coordinate at x=1. Common errors include using 1 as k (from x=1), inverting to 1/9, or misreading as 10. From a graph, read y at x=1 giving (1,k); proportional graphs pass through (1,k)—makes k directly readable. Not proportional if misses origin, no k; mistakes include reading at wrong point or confusing with y-intercept.

Question 9

1. $k=4.5$ (correct answer)
2. $k=9$
3. $k=5$
4. $k=\dfrac{1}{4.5}$

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. In the equation y=4.5x, k is the coefficient 4.5. Errors include mistaking it for 1/4.5 (inverting) or confusing with non-proportional forms like y=4.5x+something. From an equation y=kx, k is simply the coefficient of x; if there's a y-intercept (b≠0), it's not proportional and has no k.

Question 10

1. $k=\frac{2}{3}$
2. $k=2$
3. $k=3$
4. $k=\frac{3}{2}$ (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 11

1. $k=\dfrac{65}{2}$
2. $k=\dfrac{1}{65}$
3. $k=130$
4. $k=65$ (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 12

1. $k=\dfrac{2}{10}$
2. $k=8$
3. $k=5$ (correct answer)
4. $k=10$

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). In this graph passing through (0,0) and (2,10), the slope is rise/run=10/2=5, so k=5, or equivalently y/x=10/2=5. Errors include using the y-value 10 directly (A), inverting to 2/10=1/5 (B), or picking unrelated numbers like 8 (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).

Question 13

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

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). In this question, the line passes through the origin and (1,3), so k is directly the y-value at x=1, giving k=3 as the slope or unit rate. Common errors include mistaking k for the x-value (choosing 1 for A), inverting to x/y=1/3 (B), or using another point incorrectly like assuming k=4 without basis (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).

Question 14

1. $k=2$
2. $k=5$ (correct answer)
3. $k=10$
4. $k=8$

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 15

1. $k=9$
2. $k=8$ (correct answer)
3. $k=1$
4. $k=\dfrac{1}{8}$

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 16

1. $k=\dfrac{1}{5}$
2. $k=2$
3. $k=10$
4. $k=5$ (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. 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 line passes through (2,10), so k=10/2=5 (slope from origin). Common errors: inverting to 2/10=1/5, or reading y=10 as k without dividing, or thinking it's 2. From a graph, use slope=rise/run, like 10/2=5; if (1,k) isn't given, scale to find y at x=1 (here, half of 10 at x=2 is 5 at x=1).

Question 17

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

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 18

1. $k=3$ (correct answer)
2. $k=\dfrac{1}{3}$
3. $k=4$
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 19

1. $k=5$
2. $k=0.45$
3. $k=\dfrac{1}{4.5}$
4. $k=4.5$ (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 20

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

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).