Middle School Math Quiz: Represent Proportional Relationships By Equations

20 questions · exam conditions

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Represent Proportional Relationships By EquationsQuestion 1 of 20

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?

36

pages; it shows that doubling the time doubles the output in proportional relationships

45

pages; it shows that the constant rate applies to any time interval in proportional relationships

20.5

pages; it shows that fractional inputs produce fractional outputs in proportional relationships

72

pages; it shows that proportional relationships always involve whole number coefficients and results

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Middle School Math Quiz

Middle School Math Quiz: Represent Proportional Relationships By Equations

Practice Represent Proportional Relationships By Equations 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 Represent Proportional Relationships By Equations, 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

$36$ pages; it shows that doubling the time doubles the output in proportional relationships
$45$ pages; it shows that the constant rate applies to any time interval in proportional relationships (correct answer)
$20.5$ pages; it shows that fractional inputs produce fractional outputs in proportional relationships
$72$ pages; it shows that proportional relationships always involve whole number coefficients and results

Explanation: 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.

Question 2

A movie theater charges $9 per ticket. Let $c $be the total cost (in dollars) and$ t$ be the number of tickets. Which equation represents this proportional relationship?

$c=t+9$
$c=9t$ (correct answer)
$c=9t+9$
$t=9c$

Explanation: 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 the context of movie tickets at $9 each, write c=9t (c=total cost in dollars, t=number of tickets), where k=9 from the dollars per ticket 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, with k=9 and variables c for total cost and t for tickets. A common error is using the wrong form like c=t+9 which is not proportional, or reversing variables like t=9c, or including an intercept like c=9t+9 when it should pass through the origin. To write the equation: (1) identify the proportional relationship from the context "charges

9 per ticket," (2) find k=9 as the stated rate, (3) choose variables c for cost and t for tickets, (4) write c=9t, (5) define c as total cost in dollars and t as number of tickets, (6) verify by substituting t=1, c=9×1=9, which is reasonable. Multiple representations: equation c=9t matches a table where costs are multiples of 9, a graph through origin with slope 9, and the verbal "

9 per ticket"—all show k=9. Mistakes include using additive forms like c=t+9 instead of multiplicative, reversing variables, or adding unnecessary constants.

Question 3

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?

Only Student A is correct; Student B should have written $g = 28d$ for the relationship
Only Student B is correct; Student A confused the independent and dependent variables completely
Both students are correct; they represent the same proportional relationship expressed in different equivalent forms (correct answer)
Neither student is correct; proportional relationships cannot be written with division or fractions in the equations

Explanation: 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.

Question 4

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?

$m=2.50b+5$
$b=2.50m$
$m=2.50b$ (correct answer)
$m=b+2.50$

Explanation: 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

cost,

pounds). Variables: choose meaningful (

for cost,

for number,

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 $k=2.50$.

Question 5

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?

$16
$63 (correct answer)
$9
$72

Explanation: 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

cost,

pounds). Variables: choose meaningful (

for cost,

for number,

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.

Question 6

A movie theater charges $8 per ticket. Let $t$ be the total cost (in dollars) and let $n$ be the number of tickets. Which equation represents this proportional relationship?

$t=8n$ (correct answer)
$t=n+8$
$t=8n+5$
$n=8t$

8 per ticket, write t=8n (t=total cost in dollars, n=number of tickets), where k=8 from the dollars per ticket 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 t=8n with proper k=8 and variables t for total cost and n for number of tickets. A common error is reversing variables like n=8t instead of t=8n, using a wrong form like t=n+8 which is not proportional, or including an intercept like t=8n+5 when it should pass through the origin. To write the equation: (1) identify proportional relationship (context says "

8 per ticket"), (2) find k (stated rate of 8), (3) choose variables (t for cost, n for tickets), (4) write t=8n, (5) define variables (t=total cost in dollars, n=number of tickets), (6) verify (substitute n=1, t=8×1=8, reasonable? yes✓). Multiple representations: equation t=8n matches a table where costs are multiples of 8, a graph through origin with slope 8, and verbal "$8 per ticket"—all show same k=8. Mistakes include wrong form (additive t=n+8 not multiplicative), variables reversed (n=8t), k wrong, or forgetting to define variables in context.

Question 7

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?

$m=12T$ ; $T=7$
$T=12m+12$ ; $T=96$
$T=m+12$ ; $T=19$
$T=12m$ ; $T=84$ (correct answer)

Explanation: 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

cost,

pounds). Variables: choose meaningful (

for cost,

for number,

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

and

, (4) write

T=12m

, (5) define

as dollars and

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.

Question 8

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?

$P = n + 40$ where $40$ represents the additional profit per lawn above fixed costs
$1400 = 24n$ where the constant $24$ represents the previous number of lawns
$1400 = 960n$ where the constant $960$ represents the base profit amount
$1400 = 40n$ where the constant $40$ represents dollars per lawn serviced (correct answer)

Explanation: 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.

Question 9

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?

$C = 30n$ because $30$ pencils were purchased in the verification step
$C = 0.75n$ because the original equation was already correct for this scenario
$C = 18n$ because $18$ dollars is the total cost for the pencils
$C = 0.60n$ because $\frac{18}{30} = 0.60$ dollars per pencil (correct answer)

Explanation: 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.

Question 10

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?

$t=9c$
$c=9t$ (correct answer)
$c=9+t$
$c=t+9$

Explanation: 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 $k=9$.

Question 11

A car rental company charges $0.25 per mile driven. If the total cost $ $T$ $is proportional to the number of miles$ $m$ $driven, and a customer pays$ 37.50 for driving $150$ miles, which equation correctly represents this proportional relationship?

$T = 0.25m$ (correct answer)
$T = 37.50m$
$T = 150m$
$T = 0.25m + 37.50$

Explanation: The correct answer is A. Since the cost is proportional to miles driven at $0.25 per mile, the equation is $

T = 0.25m

. We can verify:

T = 0.25(150) = 37.50

$, which matches the given information. Choice B uses the total payment as the rate. Choice C uses the miles driven as the rate. Choice D adds a constant term, making it non-proportional.

Question 12

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?

$f = 1.5s$ ; flour needed is $1$ cup exactly (correct answer)
$s = 1.5f$ ; flour needed is $\frac{4}{9}$ cup exactly
$f = 1.5s$ ; flour needed is $\frac{4}{9}$ cup exactly
$f = s + 1.5$ ; flour needed is $\frac{13}{6}$ cups exactly

Explanation: 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}

Question 13

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 $x$ ?

$y=x+\frac{14}{4}$
$y=14x$
$x=\frac{14}{4}y$
$y=\frac{14}{4}x$ (correct answer)

Explanation: 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.

Question 14

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?

$f=2.5b+1$
$f=b+2.5$
$f=2.5b$ (correct answer)
$b=2.5f$

Explanation: 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.

Question 15

A runner runs at a constant speed of 6 miles per hour. Let $d$ be the distance (in miles) and let $h$ be the time (in hours). Which equation models this proportional relationship?

$h=6d$
$d=h+6$
$d=6h+2$
$d=6h$ (correct answer)

Explanation: 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=6h, where d is the distance in miles and h is the time in hours, with k=6 from the 6 miles per hour speed. A common error is reversing variables like h=6d instead of d=6h, using a non-proportional form like d=h+6, or adding constants like d=6h+2 when the relationship passes through the origin. To write the equation: (1) identify proportional relationship (context says "constant speed of 6 miles per hour"), (2) find k (stated rate of 6), (3) choose variables (d for distance, h for hours), (4) write d=6h, (5) define variables (d=distance in miles, h=time in hours), (6) verify (for h=2, d=6×2=12, reasonable? yes✓). Multiple representations: equation d=6h matches a table where distances are multiples of 6, a graph through origin with slope 6, and verbal "6 miles per hour"—all show same k=6.

Question 16

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?

$d=30t$ (correct answer)
$d=t+30$
$d=45t$
$t=30d$

Explanation: 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.

Question 17

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?

$c=p+28$
$c=4p$ (correct answer)
$c=28p$
$p=4c$

Explanation: 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.

Question 18

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?

$m=0.5p+0.5$
$m=0.5p$ (correct answer)
$m=p+0.5$
$p=0.5m$

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.

Question 19

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?

The distance increases by $4.5$ miles for each additional hour. (correct answer)
The distance starts at $4.5$ miles when $t=0$ .
The distance increases by $t$ miles for each additional $4.5$ hours.
The relationship is not proportional because $4.5$ is a decimal.

Explanation: 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.

Question 20

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 $x$ ?

$y=7x$ (correct answer)
$y=7x+2$
$x=7y$
$y=x+7$

Explanation: 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.