Identify Constant of Proportionality - 7th Grade Math

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?
Answer: $\frac{3}{2}$ units of $y$ per unit of $x$ 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.
A recipe calls for mixing paint colors in a specific ratio. When 6 ounces of blue paint are mixed with 15 ounces of white paint, the mixture creates the desired shade. If this relationship is proportional, what is the constant of proportionality that represents ounces of white paint per ounce of blue paint?
Answer: $\frac{5}{2}$ ounces of white paint per ounce of blue paint 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).
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 $y=kx$ ?
Answer: $k=65$ 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 ( $\frac{\text{rise}}{\text{run}}$ 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).
A proportional relationship is shown by a line that goes through the origin and the point $(1,3)$ . What is the constant of proportionality $k$ in $y=kx$ ?
Answer: $k=3$ 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).
A proportional relationship has a graph that includes the point $(1,9)$ and passes through the origin. What is the constant of proportionality $k$ in $y=kx$ ?
Answer: $k=9$ 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.
Based on the table shown, which value represents the constant of proportionality for the relationship between time and distance?
Answer: $15$ miles per hour with consistent rate throughout 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.
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?
Answer: $4.5$ gallons per minute with steady flow rate 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.
A bacterial culture grows over time, following the equation $P=150t$ , where $P$ is the population and $t$ is the time in hours. What is the constant of proportionality?
Answer: $150$ bacteria per hour In the equation P=150t, the number multiplying t is the constant of proportionality: 150 bacteria per hour, matching choice C. Choices A, B, and D do not match the coefficient shown in the equation and would only result from misreading or guessing at a different number.
The relationship between distance $y$ (in miles) and time $x$ (in hours) for a bike ride is given by the equation $y=4.5x$ . What is the constant of proportionality $k$ ?
Answer: $k=4.5$ 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 this equation y=4.5x, k=4.5 is the coefficient of x, representing miles per hour. Common errors include inverting to 1/4.5, confusing with nearby numbers like 5 or 4, or thinking it's not proportional if misreading the equation. From an equation y=kx, k is the coefficient (y=7x → k=7); from verbal descriptions, the stated rate is k ("3 meters per second" → k=3 m/s). Not proportional if y-intercept ≠0 (line misses origin), no k exists (y=mx+b with b≠0 is linear but not proportional); mistakes include confusing slope with y-intercept (using b as k).
A proportional relationship is represented by a line on a coordinate plane that passes through $(0,0)$ and $(2,10)$ . What is the constant of proportionality $k$ in $y=kx$ ?
Answer: $k=5$ 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).
The relationship between distance $y$ (miles) and time $x$ (hours) for a bike ride is given by the equation $y = 4.5x$ . What is the constant of proportionality $k$ ?
Answer: $k=4.5$ 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.
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 $k$ ?
Answer: $k=65$ 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.
A line on a coordinate plane represents a proportional relationship and includes the point $(1,9)$ . What is the constant of proportionality $k$ in $y=kx$ ?
Answer: $k=9$ In a proportional relationship written as y equals k times x, the constant of proportionality k can be found directly from any point on the line by dividing the y coordinate by the x coordinate. Since the graph passes through the point (1, 9), dividing 9 by 1 gives k equals 9, matching choice B. Choice A inverts the relationship, treating x divided by y as the ratio instead of y divided by x. Choice C mistakes the x coordinate of the point for the value of k. Choice D does not match the given point at all and appears to come from an unrelated calculation.
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?
Answer: $8.50 per hour before considering any additional expenses 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.
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 $y=kx$ ?
Answer: $k=3$ 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.
A proportional relationship is given by the equation $y=\frac{3}{2}x$ . What is the constant of proportionality $k$ ?
Answer: $k=\frac{3}{2}$ 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.
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?
Answer: $k=3$ 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).
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?
Answer: $\frac{5}{6}$ inches of stretch per pound of weight applied 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?"
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 $k$ ?
Answer: $k=4.5$ 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.
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?
Answer: k=3 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.

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