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

Full worked solutions for the 20 problems on equations of proportional relationships. Each solution shows how the constant $k$ is found, how the equation $y=kx$ is written, and how it is used. Compare each step with the work on the worksheet and fix any difference.

Grade 7Common Core: 7.RP.A.2c

Problem 1: Constant from a table

Divide each $y$ by its $x$: $\frac{12}{3}=4$, $\frac{24}{6}=4$, $\frac{36}{9}=4$, $\frac{48}{12}=4$. All ratios are equal, so the relationship is proportional with $k=4$.

All four ratios equal 4.

Equation: $y=4x$, with $k=4$.

Problem 2: Potato prices

Divide each cost by its weight: $\frac{6}{4}=1.5$, $\frac{12}{8}=1.5$, $\frac{18}{12}=1.5$, $\frac{24}{16}=1.5$. So $k=1.5$.
$c=1.5p$.
$c=1.5\times 10=15$.

Answers: (a) $k=1.5$ dollars per pound; (b) $c=1.5p$; (c) $15.00.

Problem 3: Equation from one pair

$k=\frac{35}{5}=7$, so the equation is $y=7x$. When $x=9$: $y=7\times 9=63$.

Answer: $y=7x$; $y=63$.

Problem 4: Equation from a pair

$k=\frac{28}{8}=3.5$, so the equation is $y=3.5x$.

(a) $y=3.5\times 12=42$.

(b) $49=3.5x$, so $x=49: 3.5=14$. Check: $3.5\times 14=49$.

From one pair to the equation.

Answers: $y=3.5x$; (a) 42; (b) 14.

Problem 5: Equation from a price

Each muffin costs $2.75, so $k=2.75$ and $c=2.75m$. For 8 muffins: $2.75\times 8=22$.

Answer: $c=2.75m$; the cost is $22.00.

Problem 6: Trail mix price

The price per pound is the constant, so $c=6.4p$.

(a) $c=6.4\times 2.5=16$. The cost is $16.00.

(b) $32=6.4p$, so $p=32: 6.4=5$. Check: $6.4\times 5=32$.

Answer: $c=6.4p$; (a) $16.00; (b) 5 pounds.

Problem 7: Equation from a graph

Use the point $(4,10)$: $k=\frac{10}{4}=2.5$. Check with $(2,5)$: $\frac{5}{2}=2.5$, and with $(8,20)$: $\frac{20}{8}=2.5$. The equation is $y=2.5x$. When $x=10$: $y=2.5\times 10=25$.

The line is y = 2.5x, so every point has y / x = 2.5.

Answer: $y=2.5x$; $y=25$.

Problem 8: Reading a line

Use the point $(10,8)$: $k=\frac{8}{10}=0.8$. Check with the other points: $\frac{4}{5}=0.8$ and $\frac{12}{15}=0.8$. The equation is $y=0.8x$.

When $x=25$: $y=0.8\times 25=20$.

Every ratio y / x equals 0.8.

Answer: $y=0.8x$; $y=20$.

Problem 9: Proportional or not?

Compute the ratios: $\frac{5}{2}=2.5$, $\frac{9}{4}=2.25$, $\frac{13}{6}\approx 2.17$. The ratios are not all equal.

The ratios are different.

Answer: not proportional, because there is no single constant $k$. (The table follows $y=2x+1$.)

Problem 10: Which equations are proportional?

A proportional relationship has the form $y=kx$, with no added or subtracted number.

$y=6x$: proportional, $k=6$.
$y=x+6$: not proportional, because 6 is added. When $x=0$, $y=6$, not 0.
$y=\frac{x}{5}=\frac{1}{5}x$: proportional, $k=\frac{1}{5}=0.2$.
$y=2x+3$: not proportional, because 3 is added.
$y=0.5x$: proportional, $k=0.5$.

Answer: (a), (c) and (e) are proportional, with k equal to 6, $\frac{1}{5}$ and 0.5.

Problem 11: Using the equation

$c=4.8\times 3.5=16.8$. The cost is $16.80.
$33.6=4.8p$, so $p=33.6: 4.8=7$. Check: $4.8\times 7=33.6$.

Answer: (a) $16.80; (b) 7 pounds.

Problem 12: Hybrid car mileage

$d=48\times 6.5=312$ miles.
$624=48g$, so $g=624: 48=13$ gallons. Check: $48\times 13=624$.

Answers: (a) 312 miles; (b) 13 gallons.

Problem 13: Double number line

$k=\frac{0.5}{10}=0.05$ mile per minute. Check: $\frac{1}{20}=0.05$ and $\frac{1.5}{30}=0.05$. The equation is $d=0.05m$.
$d=0.05\times 45=2.25$.

Answer: (a) $d=0.05m$; (b) 2.25 miles.

Problem 14: Water flow number line

$k=\frac{6}{5}=1.2$ gallons per minute. Check: $\frac{12}{10}=1.2$ and $\frac{18}{15}=1.2$. The equation is $g=1.2m$.
$g=1.2\times 22=26.4$ gallons.
$45=1.2m$, so $m=45: 1.2=37.5$ minutes.

Answers: (a) $g=1.2m$; (b) 26.4 gallons; (c) 37.5 minutes.

Problem 15: Road trip

$k=\frac{195}{3}=65$ miles per hour, so $d=65h$.
$d=65\times 5.5=357.5$ miles.
$520=65h$, so $h=520: 65=8$ hours.

Answer: (a) $d=65h$; (b) 357.5 miles; (c) 8 hours.

Problem 16: Smoothie recipe

$k=\frac{3}{2}=1.5$ cups of fruit per cup of yogurt, so $f=1.5y$.
$f=1.5\times 7=10.5$ cups.
$12=1.5y$, so $y=12: 1.5=8$ cups. Check: $1.5\times 8=12$.

Answers: (a) $f=1.5y$; (b) 10.5 cups of fruit; (c) 8 cups of yogurt.

Problem 17: Comparing two jobs

$\frac{33}{3}=11$, $\frac{66}{6}=11$, $\frac{88}{8}=11$, so $y=11x$.
Maya earns $11.50 per hour and Jamal earns $11 per hour. Maya earns $11.5-11=0.5$ dollars more per hour.
In 40 hours: Maya $11.5\times 40=460$, Jamal $11\times 40=440$, a difference of $20.

Jamal’s constant is 11.

Answer: (a) $y=11x$; (b) Maya, by $0.50 per hour; (c) $20 more.

Problem 18: Two bike riders

$\frac{27}{2}=13.5$, $\frac{67.5}{5}=13.5$, $\frac{81}{6}=13.5$, so $d=13.5t$.
Rider A rides 12 miles per hour and Rider B rides 13.5 miles per hour. Rider B is faster by $13.5-12=1.5$ miles per hour.
In 4 hours: Rider A $12\times 4=48$ miles, Rider B $13.5\times 4=54$ miles. Difference: $54-48=6$ miles.

Distance after 4 hours in miles.

Answers: (a) $d=13.5t$; (b) Rider B, by 1.5 miles per hour; (c) 6 miles farther.

Problem 19: Peaches at two stands

$k=\frac{7.5}{3}=2.5$, so $c=2.5p$.
Stand A: $2.5\times 12=30$. Stand B: $2.4\times 12=28.8$. Costs: $30.00 and $28.80.
Stand B is cheaper. Savings: $30-28.8=1.2$.

Answer: (a) $c=2.5p$; (b) $30.00 at A and $28.80 at B; (c) $1.20.

Problem 20: Copy shop prices

$k=\frac{12}{150}=0.08$, so $c=0.08n$.
Shop X charges $0.08 per copy and Shop Y charges $0.075 per copy, so Shop Y is cheaper.
Shop X: $0.08\times 1{,}200=96$. Shop Y: $0.075\times 1{,}200=90$. Savings: $96-90=6$.

Answers: (a) $c=0.08n$; (b) Shop Y; (c) $6.00.

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