Proportional Relationships: Grade 7 Math Worksheet Answer Key – PDF Download

Use this answer key to check each solution. Every answer shows the ratios of y to x, the conclusion about proportionality, and the constant of proportionality with its units. Compare your own steps with these to find any slips.

Grade 7Common Core: 7.RP.A.2a · 7.RP.A.2b · 7.RP.A.2d

Problem 1: Table check

$\frac{6}{2}=3$, $\frac{12}{4}=3$, $\frac{18}{6}=3$, $\frac{24}{8}=3$. All ratios are equal.

Proportional, with $k=3$.

All four ratios equal 3.

Problem 2: Table of wages

$\frac{31}{2}=15.5$, $\frac{62}{4}=15.5$, $\frac{93}{6}=15.5$, $\frac{124}{8}=15.5$. All the ratios are equal.

Proportional, with $k=15.50$ dollars per hour.

Problem 3: Not constant

$\frac{4}{1}=4$, $\frac{7}{2}=3.5$, $\frac{10}{3}\approx 3.33$, $\frac{13}{4}=3.25$. The ratios are different.

Not proportional.

The ratios keep changing.

Problem 4: Ratios that differ

$\frac{8}{3}\approx2.67$, $\frac{15}{6}=2.5$, $\frac{22}{9}\approx2.44$, $\frac{29}{12}\approx2.42$. The ratios are different.

Not proportional. The $y$-values rise by 7 each time, but a constant difference does not make a relationship proportional.

Problem 5: Cost of gasoline

$\frac{7.00}{2}=3.50$, $\frac{14.00}{4}=3.50$, $\frac{21.00}{6}=3.50$. All ratios are equal, so the cost is proportional to the gallons.

$k=\$3.50$ per gallon. It is the price of 1 gallon of gasoline.

Problem 6: Reading speed graph

The graph is a straight line through the origin, and $\frac{6}{4}=\frac{12}{8}=\frac{18}{12}=1.5$.

Every ratio equals 1.5.

Proportional, with $k=1.5$ pages per minute.

Problem 7: Graph through the origin

The line passes through the origin, and $\frac{2}{1}=\frac{4}{2}=\frac{6}{3}=\frac{8}{4}=2$.

Proportional, with $k=2$. The point $(1,2)$ shows the unit rate.

The point (1, 2) shows k = 2.

Problem 8: Graph that misses the origin

The line crosses the $y$-axis at $(0,3)$, not at the origin $(0,0)$. The ratios are also different: $\frac{5}{1}=5$, $\frac{7}{2}=3.5$, $\frac{9}{3}=3$.

It is not proportional because the line does not pass through $(0,0)$.

Problem 9: Miles per gallon

$\frac{150}{6}=25$ and $\frac{225}{9}=25$. The ratios are equal, so the distance is proportional to the gas used.

$k=25$ miles per gallon.

Problem 10: Earnings number line

$\frac{26}{2}=13$, $\frac{52}{4}=13$, $\frac{78}{6}=13$. The ratios are equal, so the earnings are proportional to the hours, with $k=13$ dollars per hour.

For 10 hours: $13\times 10=130$.

Proportional; $k=13$ dollars per hour; $130 for 10 hours.

Problem 11: Notebook bar model

$k=\frac{24}{4}=6$. One notebook costs $6.

For 7 notebooks: $6\times 7=42$. $k=\$6$ per notebook; 7 notebooks cost $42.

Problem 12: Meaning of the points

(a) $k=\frac{5}{2}=2.5$. $k=2.50$ dollars per pound

(b) $(1,2.50)$: 1 pound of grapes costs $2.50.

The three important points.

Problem 13: Points on a graph

(a) $k=\frac{12}{3}=4$. $k=4$ dollars per bag.

(b) $(1,4)$: 1 bag of oranges costs $4. This is the unit rate.

Three important points on the line.

Problem 14: Missing table values

$k=\frac{9}{2}=4.5$. Check with the other known pair: $\frac{27}{6}=4.5$.

For $x=4$: $y=4.5\times 4=18$. For $x=10$: $y=4.5\times 10=45$.

$k=4.5$; the missing values are 18 and 45.

Problem 15: Two gym plans

Gym A: $\frac{60}{1}=60$, $\frac{120}{2}=60$, $\frac{180}{3}=60$, $\frac{240}{4}=60$. The ratios are equal, so Gym A is proportional with $k=60$ dollars per month.

Gym B: $\frac{85}{1}=85$, $\frac{110}{2}=55$, $\frac{135}{3}=45$, $\frac{160}{4}=40$. The ratios are different, so Gym B is not proportional.

Gym B costs $25 per month plus a $60 sign-up fee, so zero months would still cost $60, not $0.

Gym A is proportional; Gym B is not.

Problem 16: Three riders

Liam: $\frac{5}{1}=\frac{10}{2}=\frac{15}{3}=5$, proportional. Sofia: $\frac{6}{1}=6$, $\frac{11}{2}=5.5$, $\frac{16}{3}\approx5.33$, not proportional. Noah: $\frac{4.5}{1}=\frac{9}{2}=\frac{13.5}{3}=4.5$, proportional.
Liam: $k=5$ miles per hour. Noah: $k=4.5$ miles per hour.
Liam is faster by $5-4.5=0.5$ miles per hour.

Answers: (a) Liam and Noah; (b) 5 and 4.5 miles per hour; (c) Liam, by 0.5 miles per hour.

Problem 17: Bake sale table

(a) $\frac{7.50}{3}=2.50$, $\frac{12.50}{5}=2.50$, $\frac{20.00}{8}=2.50$, $\frac{30.00}{12}=2.50$. All ratios equal, so it is proportional. $k=2.50$

(b) $2.50\times 20=50$. $50.00

From the ratio to the answer.

Problem 18: Straight line error

A proportional relationship needs equal ratios and a graph that passes through the origin. Compare the ratios: $\frac{6}{2}=3$ and $\frac{10}{4}=2.5$. They are not equal.

The line through the two points rises 4 for every 2 across, so each step of 1 in $x$ adds 2 to $y$. Going back from $(2,6)$ by 2 steps gives $(0,2)$, so the line crosses the $y$-axis at 2, not at the origin.

The pattern reaches (0, 2), not (0, 0).

Ethan is incorrect: a straight line is proportional only when it passes through (0, 0).

Problem 19: Find the error

Liam is not correct. A constant difference does not make a relationship proportional. The ratios are: for $x=2$, $\frac{9}{2}=4.5$; for $x=4$, $\frac{13}{4}=3.25$; for $x=6$, $\frac{17}{6}\approx 2.83$. They are not equal.

Also, $x=0$ goes with $y=5$, not $y=0$. The relationship is not proportional.

Problem 20: Cheese at two stores

Store A: $k=\frac{26}{4}=6.5$ dollars per pound. Store B: $k=\frac{31.25}{5}=6.25$ dollars per pound.
Store B is cheaper by $6.5-6.25=0.25$ dollars per pound.
$0.25\times 12=3$, or check: Store A $6.5\times 12=78$ and Store B $6.25\times 12=75$, a difference of $3.

Cost of 12 pounds in dollars.

Answers: (a) $6.50 and $6.25 per pound; (b) Store B, by $0.25 per pound; (c) $3.00.

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