Represent Proportional Relationships by Equations - 7th Grade Math

A proportional relationship is shown on a coordinate plane by points on a line through the origin. The line passes through the point $(4, 14)$ . Which equation represents the relationship between $y$ and $x$ ?
Answer: $y=\frac{14}{4}x$ This question tests writing equations y=kx for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying k and defining variables contextually. Proportional equation y=kx: k is constant of proportionality (unit rate, ratio y/x). From table: calculate k from any pair (14/2=7, k=7 gives y=7x), from graph: k=slope (or read y when x=1: if graph through (1,7), k=7), from context: stated rate is k ("$3 per pound" → k=3, equation c=3p where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example: context "apples $3/lb" write c=3p (c=cost dollars, p=pounds), k=3 from $/lb rate; or table x:2,4,6 y:10,20,30 find k=10/2=5, write y=5x; or graph through origin with slope 8 write y=8x. The correct equation is y=(14/4)x, with k=14/4 from the slope through (0,0) and (4,14). A common error is using wrong k like y=14x without dividing, additive form like y=x+(14/4), or reversing like x=(14/4)y. To write the equation: (1) identify proportional relationship (graph through origin), (2) find k (slope=14/4), (3) choose variables (y and x as given), (4) write y=(14/4)x, (5) define variables if needed, (6) verify (for x=4, y=(14/4)×4=14, matches point✓). Multiple representations: equation y=(14/4)x matches graph with slope 14/4, a table with ratio 14/4, and verbal description—all show same k=14/4.
A bus travels 45 miles in 1.5 hours at a constant rate. Let $d$ be distance (miles) and $t$ be time (hours). Which equation models the proportional relationship?
Answer: $d=30t$ This question tests writing equations y=kx for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying k and defining variables contextually. Proportional equation y=kx: k is constant of proportionality (unit rate, ratio y/x). From table: calculate k from any pair (14/2=7, k=7 gives y=7x), from graph: k=slope (or read y when x=1: if graph through (1,7), k=7), from context: stated rate is k ("$3 per pound" → k=3, equation c=3p where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example, bus 45 miles in 1.5 hours, k=45/1.5=30, write d=30t (d=miles, t=hours); or table x:2,4,6 y:10,20,30 find k=10/2=5, write y=5x; or graph through origin with slope 8 write y=8x. The correct equation is d=30t, with k=30 from calculated rate. A common error is wrong k like d=45t using total without dividing, reversing t=30d, or additive d=t+30. To write: (1) identify proportional from constant rate, (2) find k=30, (3) choose d and t, (4) write d=30t, (5) define d as miles and t as hours, (6) verify t=1.5, d=30×1.5=45. Multiple representations: d=30t matches given point, graph slope 30, verbal "30 mph"—all k=30. Mistakes: wrong k calculation, reversed, added terms.
A landscaping company's profit $P$ (in dollars) is proportional to the number of lawns $n$ they service each week. When they service $24$ lawns, their profit is $960. If their goal is to earn $1400 profit next week, which equation should they use to find how many lawns to service?
Answer: $1400 = 40n$ where the constant $40$ represents dollars per lawn serviced When you see that one quantity is "proportional" to another, this means they have a direct relationship where one equals a constant times the other. Here, profit equals some constant times the number of lawns: $P = k \cdot n$ , where $k$ is the constant rate. To find this constant rate, use the given information: when $n = 24$ lawns, $P = \$960$ . Substituting: $960 = k \cdot 24$ , so $k = 960 ÷ 24 = 40$ dollars per lawn. This means the company earns $40 profit for each lawn they service. Now you can set up the equation for their goal: if they want $1400 profit, then $1400 = 40n$ , where $n$ is the unknown number of lawns needed. Choice A is wrong because proportional relationships are multiplicative ( $P = k \cdot n$ ), not additive ( $P = n + \text{constant}$ ). The "+40" suggests adding a fixed amount rather than multiplying by a rate. Choice B incorrectly uses 24 as the multiplier, but 24 was the number of lawns in the given example, not the profit rate. This confuses the input with the constant. Choice C uses 960 as the multiplier, but 960 was the profit amount from the example, not the rate per lawn. This treats the output as the constant rate. Choice D correctly identifies 40 as the dollars earned per lawn serviced, making $1400 = 40n$ the right equation. Study tip: In proportional relationships, always find the constant rate by dividing the given output by the given input, then use that rate in your equation.
A gym charges $12 per month for a membership with no starting fee. Let $T$ be the total cost (dollars) and $m$ be the number of months. Which equation represents the relationship, and what is the cost for 7 months?
Answer: $T=12m$ ; $T=84$ This question tests writing equations $y=kx$ for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying $k$ and defining variables contextually. Proportional equation $y=kx$ : $k$ is constant of proportionality (unit rate, ratio $y/x$ ). From table: calculate $k$ from any pair ( $14/2=7$ , $k=7$ gives $y=7x$ ), from graph: $k=$ slope (or read $y$ when $x=1$ : if graph through $(1,7)$ , $k=7$ ), from context: stated rate is $k$ ("$3 per pound" → $k=3$ , equation $c=3p$ where $c=$ cost, $p=$ pounds). Variables: choose meaningful ( $c$ for cost, $n$ for number, $d$ for distance) and define in context. For example, gym $12 per month no fee, write $T=12m$ ( $T=$ total cost in dollars, $m=$ months), $k=12$ from dollars per month; or table $x$ :2,4,6 $y$ :10,20,30 find $k=10/2=5$ , write $y=5x$ ; or graph through origin with slope 8 write $y=8x$ . The correct equation is $T=12m$ with $T=84$ for 7 months, $k=12$ and variables $T$ for total and $m$ for months. A common error is adding fee like $T=12m+12$ , reversing $m=12T$ , or additive $T=m+12$ . To write: (1) identify from "$12 per month no fee," (2) find $k=12$ , (3) choose $T$ and $m$ , (4) write $T=12m$ , (5) define $T$ as dollars and $m$ as months, (6) verify $m=7$ , $T=84$ . Multiple representations: $T=12m$ matches table multiples of 12, graph slope 12, verbal "12 per month"—all $k=12$ . Mistakes: adding intercepts, reversed, wrong form.
A school store sells pencils for $0.50 each. Let $m$ be the total cost (in dollars) and $p$ be the number of pencils. Which equation represents the relationship?
Answer: $m=0.5p$ This question tests writing equations y=kx for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying k and defining variables contextually. Proportional equation y=kx: k is constant of proportionality (unit rate, ratio y/x). From table: calculate k from any pair (14/2=7, k=7 gives y=7x), from graph: k=slope (or read y when x=1: if graph through (1,7), k=7), from context: stated rate is k ("$3 per pound" → k=3, equation c=3p where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example, pencils $0.50 each, write m=0.5p (m=cost in dollars, p=pencils), k=0.5 from dollars per pencil; or table x:2,4,6 y:10,20,30 find k=10/2=5, write y=5x; or graph through origin with slope 8 write y=8x. The correct equation is m=0.5p, with k=0.5 and variables m for money and p for pencils. A common error is wrong form like m=p+0.5, reversing p=0.5m, or intercept m=0.5p+0.5. To write: (1) identify from "$0.50 each," (2) find k=0.5, (3) choose m and p, (4) write m=0.5p, (5) define m as dollars and p as pencils, (6) verify p=2, m=1. Multiple representations: m=0.5p matches table like p=1,m=0.5, graph slope 0.5, verbal "half dollar per pencil"—all k=0.5. Mistakes: additive, reversed, extra terms.
The cost $C$ (in dollars) of buying pencils is proportional to the number of pencils $n$ purchased. A student writes the equation $C = 0.75n$ to model this relationship. If the student later discovers that $30$ pencils actually cost $18, what should the correct equation be?
Answer: $C = 0.60n$ because $\frac{18}{30} = 0.60$ dollars per pencil When you encounter proportional relationships, remember that the equation $C = kn$ means the constant $k$ represents the unit rate - in this case, the cost per pencil. To find the correct equation, you need to determine this unit rate from the given data. The student's original equation $C = 0.75n$ suggests each pencil costs $0.75. However, the new information tells us that 30 pencils cost $18 total. To find the actual cost per pencil, divide the total cost by the number of pencils: $\frac{18}{30} = 0.60$ dollars per pencil. Therefore, the correct equation is $C = 0.60n$ . Let's examine why the other answers are incorrect. Answer A ( $C = 30n$ ) incorrectly uses the number of pencils as the rate, which would mean each pencil costs $30 - clearly unreasonable. Answer B claims the original equation was correct, but if you substitute: $C = 0.75(30) = 22.50$ , not $18 as given. Answer C ($ $C = 18n$ $) mistakenly uses the total cost as the rate, meaning each pencil would cost $18. Answer D correctly calculates $\frac{18}{30} = 0.60$ and recognizes this as the cost per individual pencil. Study tip: In proportional relationships, always identify what the constant represents. When given a specific data point, use it to calculate the actual unit rate by dividing the total amount by the number of units. This will help you catch errors in proposed equations and write correct ones.
A recipe uses 2 cups of flour for each batch of muffins. Let $f$ be the number of cups of flour and $b$ be the number of batches. Which equation represents this proportional relationship?
Answer: $f=2b$ This question tests writing equations y=kx for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying k and defining variables contextually. Proportional equation y=kx: k is constant of proportionality (unit rate, ratio y/x). From table: calculate k from any pair (14/2=7, k=7 gives y=7x), from graph: k=slope (or read y when x=1: if graph through (1,7), k=7), from context: stated rate is k ("$3 per pound" → k=3, equation c=3p where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example, recipe 2 cups flour per batch, write f=2b (f=cups of flour, b=batches), k=2 from cups per batch; or table x:2,4,6 y:10,20,30 find k=10/2=5, write y=5x; or graph through origin with slope 8 write y=8x. The correct equation is f=2b, with k=2 and variables f for flour and b for batches. A common error is reversing like b=2f, using additive f=b+2, or including intercept f=2b+2. To write: (1) identify proportional from "2 cups for each batch," (2) find k=2 as rate, (3) choose f and b, (4) write f=2b, (5) define f as cups and b as batches, (6) verify b=1, f=2. Multiple representations: f=2b matches table multiples of 2, graph slope 2, verbal "2 per batch"—all k=2. Mistakes: reversed variables, wrong form, added constants.
A teacher buys markers in bulk. The total cost $c$ (in dollars) is proportional to the number of marker packs $p$ . If 7 packs cost $28, which equation represents the relationship?
Answer: $c=4p$ This question tests writing equations y=kx for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying k and defining variables contextually. Proportional equation y=kx: k is constant of proportionality (unit rate, ratio y/x). From table: calculate k from any pair (14/2=7, k=7 gives y=7x), from graph: k=slope (or read y when x=1: if graph through (1,7), k=7), from context: stated rate is k ("$3 per pound" → k=3, equation c=3p where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example: context "apples $3/lb" write c=3p (c=cost dollars, p=pounds), k=3 from $/lb rate; or table x:2,4,6 y:10,20,30 find k=10/2=5, write y=5x; or graph through origin with slope 8 write y=8x. The correct equation is c=4p, where c is total cost in dollars and p is number of packs, with k=4 from 28/7=4. A common error is using total like c=28p without dividing, additive c=p+28, or reversing p=4c. To write the equation: (1) identify proportional relationship (context says "proportional to the number"), (2) find k (ratio 28/7=4), (3) choose variables (c for cost, p for packs), (4) write c=4p, (5) define variables (c=total cost in dollars, p=number of marker packs), (6) verify (for p=7, c=4×7=28, matches✓). Multiple representations: equation c=4p matches a table with ratio 4, a graph through origin with slope 4, and verbal "$4 per pack"—all show same k=4.
A runner travels at a constant speed of 6 miles per hour. Let $d$ be the distance (in miles) and $h$ be the time (in hours). Which equation models the relationship?
Answer: $d=6h$ This question tests writing equations y=kx for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying k and defining variables contextually. Proportional equation y=kx: k is constant of proportionality (unit rate, ratio y/x). From table: calculate k from any pair (14/2=7, k=7 gives y=7x), from graph: k=slope (or read y when x=1: if graph through (1,7), k=7), from context: stated rate is k ("$3 per pound" → k=3, equation c=3p where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example, in context of speed at 6 mph, write d=6h (d=distance in miles, h=time in hours), k=6 from miles per hour; or table x:2,4,6 y:10,20,30 find k=10/2=5, write y=5x; or graph through origin with slope 8 write y=8x. The correct equation is d=6h, with k=6 and variables d for distance and h for hours. A common error is wrong form like d=h+6 not proportional, reversing variables like h=6d, or including intercept like d=6h+6. To write the equation: (1) identify proportional from "constant speed," (2) find k=6 as rate, (3) choose d and h, (4) write d=6h, (5) define d as miles and h as hours, (6) verify with h=1, d=6. Multiple representations: d=6h matches table of multiples of 6, graph with slope 6 through origin, verbal "6 miles per hour"—all k=6. Mistakes: additive form, reversed variables, added constants.
A movie theater charges $9 for each ticket. Let $c$ be the total cost (in dollars) and let $t$ be the number of tickets. Which equation represents this proportional relationship?
Answer: $c=9t$ This question tests writing equations $y=kx$ for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying $k$ and defining variables contextually. Proportional equation $y=kx$ : $k$ is constant of proportionality (unit rate, ratio $y/x$ ). From table: calculate $k$ from any pair ( $14/2=7$ , $k=7$ gives $y=7x$ ), from graph: $k=$ slope (or read y when x=1: if graph through (1,7), $k=7$ ), from context: stated rate is $k$ ("$3 per pound" → $k=3$ , equation $c=3p$ where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example: context "apples $3/lb" write $c=3p$ (c=cost dollars, p=pounds), $k=3$ from $/lb rate; or table x:2,4,6 y:10,20,30 find $k=10/2=5$ , write $y=5x$ ; or graph through origin with slope 8 write $y=8x$ . The correct equation is $c=9t$ , where c is the total cost in dollars and t is the number of tickets, with $k=9$ from the $9 per ticket rate. A common error is reversing variables like $t=9c$ instead of $c=9t$ , using a non-proportional form like $c=t+9$ , or adding constants like $c=9+t$ when the relationship passes through the origin. To write the equation: (1) identify proportional relationship (context says "$9 for each ticket"), (2) find $k$ (stated rate of 9), (3) choose variables (c for cost, t for tickets), (4) write $c=9t$ , (5) define variables (c=total cost in dollars, t=number of tickets), (6) verify (for $t=2$ , $c=9×2=18$ , reasonable? yes✓). Multiple representations: equation $c=9t$ matches a table where costs are multiples of 9, a graph through origin with slope 9, and verbal "$9 per ticket"—all show same $k=9$ .
A recipe calls for ingredients in the following proportional relationship: the amount of flour $f$ (in cups) needed is always $1.5$ times the amount of sugar $s$ (in cups). Which equation represents this relationship, and what would be the flour requirement if $\frac{2}{3}$ cup of sugar is used?
Answer: $f = 1.5s$ ; flour needed is $1$ cup exactly The correct answer is A. Since flour is $1.5$ times the sugar, $f = 1.5s$ . With $s = \frac{2}{3}$ : $f = 1.5 × \frac{2}{3} = \frac{3}{2} × \frac{2}{3} = 1$ cup. Choice B reverses the relationship. Choice C has the right equation but wrong calculation ( $\frac{4}{9}$ instead of $1$ ). Choice D uses addition instead of multiplication and gets $\frac{2}{3} + 1.5 = \frac{13}{6}$ .
A gym charges a proportional fee based on the number of classes taken. The fee is $9 per class. Let $f$ be the total fee (in dollars) and let $c$ be the number of classes. Using the proportional equation, what is the fee for $7$ classes?
Answer: $63 This question tests writing equations $y=kx$ for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying $k$ and defining variables contextually, and applying to find values. Proportional equation $y=kx$ : $k$ is constant of proportionality (unit rate, ratio $y/x$ ). From table: calculate $k$ from any pair ( $14/2=7$ , $k=7$ gives $y=7x$ ), from graph: $k=$ slope (or read $y$ when $x=1$ : if graph through $(1,7)$ , $k=7$ ), from context: stated rate is $k$ ("$3 per pound" → $k=3$ , equation $c=3p$ where $c=$ cost, $p=$ pounds). Variables: choose meaningful ( $c$ for cost, $n$ for number, $d$ for distance) and define in context. For example, context "apples $3/lb" write $c=3p$ ( $c=$ cost dollars, $p=$ pounds), $k=3$ from $/lb rate; or table $x$ :2,4,6 $y$ :10,20,30 find $k=10/2=5$ , write $y=5x$ ; or graph through origin with slope 8 write $y=8x$ . The correct equation is $f=9c$ with $k=9$ , and for 7 classes, $f=9\times7=63$ . A common error is wrong calculation like $72 if using 8 instead, or non-proportional forms. To solve: (1) identify proportional ("$9 per class"), (2) find $k=9$ , (3) variables $f$ fee, $c$ classes, (4) write $f=9c$ , (5) define ( $f=$ fee in dollars, $c=$ classes), (6) substitute $c=7$ , $f=63\checkmark$ . Multiple representations: $f=9c$ matches table multiples of 9, graph slope 9, verbal "$9 per"—all $k=9$ . Mistakes: wrong form, miscalculation, wrong $k$ , no verification.
A school store sells water bottles. The total cost c (in dollars) is proportional to the number of bottles n. If 3 bottles cost $18, which equation represents this relationship?
Answer: c=6n Since the cost is proportional to the number of bottles, the relationship has the form c = kn, where k is the cost per bottle. Given that 3 bottles cost 18 dollars, dividing 18 by 3 gives k = 6, so the equation is c = 6n, matching Choice D. Choice A, c = 18n, mistakenly uses the total cost as the rate instead of dividing by the number of bottles. Choice B, n = 6c, reverses the variables, solving for the number of bottles instead of the cost. Choice C, c = n + 6, is not a proportional relationship at all, since it adds a constant instead of multiplying by a rate.
At a school fundraiser, a student earns $2.50 for each box of candy sold. Let $m$ be the money earned (in dollars) and let $b$ be the number of boxes sold. Which equation represents this proportional relationship?
Answer: $m=2.50b$ This question tests writing equations $y=kx$ for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying $k$ and defining variables contextually. Proportional equation $y=kx$ : $k$ is constant of proportionality (unit rate, ratio $y/x$ ). From table: calculate $k$ from any pair ( $14/2=7$ , $k=7$ gives $y=7x$ ), from graph: $k=$ slope (or read $y$ when $x=1$ : if graph through $(1,7)$ , $k=7$ ), from context: stated rate is $k$ ("$3 per pound" → $k=3$ , equation $c=3p$ where $c=$ cost, $p=$ pounds). Variables: choose meaningful ( $c$ for cost, $n$ for number, $d$ for distance) and define in context. For example: context "apples $3/lb" write $c=3p$ ( $c=$ cost dollars, $p=$ pounds), $k=3$ from $/lb rate; or table $x$ :2,4,6 $y$ :10,20,30 find $k=10/2=5$ , write $y=5x$ ; or graph through origin with slope 8 write $y=8x$ . The correct equation is $m=2.50b$ , where $m$ is money earned in dollars and $b$ is boxes sold, with $k=2.50$ from $2.50 per box. A common error is using additive form like $m=b+2.50$ , reversing variables like $b=2.50m$ , or adding extra constants like $m=2.50b+5$ . To write the equation: (1) identify proportional relationship (context says "$2.50 for each box"), (2) find $k$ (stated rate of 2.50), (3) choose variables ( $m$ for money, $b$ for boxes), (4) write $m=2.50b$ , (5) define variables ( $m=$ money earned in dollars, $b=$ number of boxes), (6) verify (for $b=2$ , $m=2.50×2=5$ , reasonable? yes✓). Multiple representations: equation $m=2.50b$ matches a table with multiples of 2.50, a graph through origin with slope 2.50, and verbal "$2.50 per box"—all show same $k=2.50$ .
Two students are modeling the same proportional relationship between gallons of gas $g$ and total driving distance $d$ in miles. Student A writes $d = 28g$ while Student B writes $g = \frac{d}{28}$ . Which statement best describes these equations?
Answer: Both students are correct; they represent the same proportional relationship expressed in different equivalent forms The correct answer is C. Both equations represent the same proportional relationship. Student A's equation $d = 28g$ shows distance as a function of gallons ( $28$ miles per gallon). Student B's equation $g = \frac{d}{28}$ is the inverse, showing gallons as a function of distance. These are equivalent: solving $d = 28g$ for $g$ gives $g = \frac{d}{28}$ . Choice A and B incorrectly claim only one is right. Choice D makes a false statement about proportional relationships.
A recipe uses $2.5$ cups of flour for each batch of cookies. Let $f$ be the number of cups of flour and let $b$ be the number of batches. Which equation shows the proportional relationship?
Answer: $f=2.5b$ This question tests writing equations y=kx for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying k and defining variables contextually. Proportional equation y=kx: k is constant of proportionality (unit rate, ratio y/x). From table: calculate k from any pair (14/2=7, k=7 gives y=7x), from graph: k=slope (or read y when x=1: if graph through (1,7), k=7), from context: stated rate is k ("$3 per pound" → k=3, equation c=3p where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example, context "apples $3/lb" write c=3p (c=cost dollars, p=pounds), k=3 from $/lb rate; or table x:2,4,6 y:10,20,30 find k=10/2=5, write y=5x; or graph through origin with slope 8 write y=8x. The correct equation is f=2.5b with proper k=2.5 and variables f for flour and b for batches. A common error is reversing variables like b=2.5f instead of f=2.5b, wrong form like f=b+2.5 not proportional, or including intercept like f=2.5b+1. To write the equation: (1) identify proportional relationship (context says "2.5 cups per batch"), (2) find k (stated rate of 2.5), (3) choose variables (f for flour, b for batches), (4) write f=2.5b, (5) define variables (f=cups of flour, b=number of batches), (6) verify (b=1, f=2.5×1=2.5, yes✓). Multiple representations: equation f=2.5b matches table of multiples of 2.5, graph with slope 2.5, verbal "2.5 per batch"—all show k=2.5. Mistakes: wrong form (additive), variables reversed, k wrong, undefined variables.
A bus travels 180 miles in 3 hours at a constant speed. Let $d$ be the distance (in miles) and let $t$ be the time (in hours). Which equation models this proportional relationship?
Answer: $d=60t$ This question tests writing equations y=kx for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying k and defining variables contextually. Proportional equation y=kx: k is constant of proportionality (unit rate, ratio y/x). From table: calculate k from any pair (14/2=7, k=7 gives y=7x), from graph: k=slope (or read y when x=1: if graph through (1,7), k=7), from context: stated rate is k ("$3 per pound" → k=3, equation c=3p where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example: context "apples $3/lb" write c=3p (c=cost dollars, p=pounds), k=3 from $/lb rate; or table x:2,4,6 y:10,20,30 find k=10/2=5, write y=5x; or graph through origin with slope 8 write y=8x. The correct equation is d=60t, where d is distance in miles and t is time in hours, with k=60 from 180 miles / 3 hours. A common error is using total like d=180t without dividing, reversing like t=60d, or additive d=t+60. To write the equation: (1) identify proportional relationship (context says "constant speed"), (2) find k (ratio 180/3=60), (3) choose variables (d for distance, t for time), (4) write d=60t, (5) define variables (d=distance in miles, t=time in hours), (6) verify (for t=3, d=60×3=180, matches✓). Multiple representations: equation d=60t matches a table with ratio 60, a graph through origin with slope 60, and verbal "60 miles per hour"—all show same k=60.
A proportional relationship is given by the equation $d=4.5t$ , where $d$ is distance (in miles) and $t$ is time (in hours). Which statement is true?
Answer: The distance increases by $4.5$ miles for each additional hour. This question tests writing equations y=kx for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying k and defining variables contextually, and interpreting meaning. Proportional equation y=kx: k is constant of proportionality (unit rate, ratio y/x). From table: calculate k from any pair (14/2=7, k=7 gives y=7x), from graph: k=slope (or read y when x=1: if graph through (1,7), k=7), from context: stated rate is k ("$3 per pound" → k=3, equation c=3p where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example, context "apples $3/lb" write c=3p (c=cost dollars, p=pounds), k=3 from $/lb rate; or table x:2,4,6 y:10,20,30 find k=10/2=5, write y=5x; or graph through origin with slope 8 write y=8x. The correct interpretation of d=4.5t is distance increases by 4.5 miles per hour, with k=4.5 as rate. A common error is reversing like increases by t per 4.5 hours, assuming intercept like starts at 4.5 when t=0 (but it's 0), or thinking not proportional due to decimal. To interpret: (1) identify proportional (form y=kx), (2) find k=4.5 (miles per hour), (3) variables d distance, t time, (4) equation d=4.5t, (5) define (d=miles, t=hours), (6) verify (t=1, d=4.5, rate matches✓). Multiple representations: d=4.5t matches table multiples of 4.5, graph slope 4.5, verbal "4.5 mph"—all k=4.5. Mistakes: reversed meaning, assuming intercept, wrong form, misinterpreting decimal.
A printer produces pages at a constant rate. The equation $p = 18t$ represents the number of pages $p$ printed after $t$ minutes. How many pages will be printed in the first $2.5$ minutes, and what does this demonstrate about proportional relationships?
Answer: $45$ pages; it shows that the constant rate applies to any time interval in proportional relationships The correct answer is B. Using $p = 18t$ with $t = 2.5$ : $p = 18(2.5) = 45$ pages. This demonstrates that the constant rate of $18$ pages per minute applies to any time interval, including fractional times. Choice A gives the wrong calculation ( $18 × 2 = 36$ ). Choice C gives an incorrect sum ( $18 + 2.5$ ). Choice D uses incorrect multiplication ( $18 × 4$ ) and makes a false claim about whole numbers.
A proportional relationship is graphed on the coordinate plane. The line passes through the points $(0,0)$ and $(1,7)$ . Which equation represents the relationship between $y$ and $x$ ?
Answer: $y=7x$ This question tests writing equations y=kx for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying k and defining variables contextually. Proportional equation y=kx: k is constant of proportionality (unit rate, ratio y/x). From table: calculate k from any pair (14/2=7, k=7 gives y=7x), from graph: k=slope (or read y when x=1: if graph through (1,7), k=7), from context: stated rate is k ("$3 per pound" → k=3, equation c=3p where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example, context "apples $3/lb" write c=3p (c=cost dollars, p=pounds), k=3 from $/lb rate; or table x:2,4,6 y:10,20,30 find k=10/2=5, write y=5x; or graph through origin with slope 8 write y=8x. The correct equation is y=7x with proper k=7 from the slope through (0,0) and (1,7). A common error is including intercept like y=x+7 or y=7x+2 when proportional must pass through origin, reversing variables like x=7y, or wrong form. To write the equation: (1) identify proportional relationship (graph through origin), (2) find k (slope = 7/1=7), (3) choose variables (y and x), (4) write y=7x, (5) define if needed, (6) verify (x=1, y=7×1=7, matches point✓). Multiple representations: y=7x matches table of multiples of 7, graph with slope 7, verbal rate 7—all show k=7. Mistakes: additive form, meaningless variables, wrong k, undefined in context.

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