Ratios and Proportional Relationships: Recognizing and Representing Proportional Relationships (CCSS.7.RP.2)
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Common Core 7th Grade Math › Ratios and Proportional Relationships: Recognizing and Representing Proportional Relationships (CCSS.7.RP.2)
On a map, 1 centimeter represents 2.5 kilometers in the real world. Let $c$ be the length on the map (cm) and $d$ be the actual distance (km). Which equation represents this proportional relationship?
$c = 2.5d$
$d = 2.5c$
$d = c + 2.5$
$d = \frac{c}{2.5}$
Explanation
Each centimeter corresponds to 2.5 kilometers, so multiply the map length by 2.5: $d=2.5c$. Choice A inverts the variables, C uses addition instead of multiplication, and D divides instead of multiplying.
A craft store sells ribbon for 40 cents per foot. Let $f$ be the number of feet of ribbon and $c$ be the total cost in dollars. Which equation represents this proportional relationship?
$c=0.40+f$
$c=0.40f+2$
$c=0.40f$
$f=0.40c$
Explanation
The unit rate is \$0.40$ dollars per foot, so $c=0.40f$ (a $y=kx$ form). $c=0.40+f$ and $c=0.40f+2$ add amounts (not proportional), and $f=0.40c$ inverts the variables.
Bags of apples cost 2.50 per bag. Let $C$ be the total cost in dollars and $b$ be the number of bags. Which equation represents this proportional relationship?
$C=b+2.50$
$b=2.50C$
$C=\tfrac{b}{2.50}$
$C=2.50b$
Explanation
A constant price per bag means $C=pb$, so $C=2.50b$. The other choices add, invert, or divide, which are not proportional of the form $y=kx$.
A recipe uses 3 cups of flour for every 2 cups of sugar. Let $f$ be cups of flour and $s$ be cups of sugar. Which equation represents this proportional relationship?
$f=s+1.5$
$s=\tfrac{3}{2}f$
$f=\tfrac{2}{3}s$
$f=\tfrac{3}{2}s$
Explanation
The ratio flour:sugar is \$3:2$, so $\frac{f}{s}=\frac{3}{2}$ and $f=\tfrac{3}{2}s$. Choice C inverts the ratio, B solves for $s$ instead of $f$, and A is additive (not proportional).
One gallon of paint covers 350 square feet. Let $g$ be the gallons of paint needed and $a$ be the area (square feet). Which equation gives $g$ in terms of $a$?
$g = \frac{1}{350}a$
$g = 350a$
$g = a + 350$
$a = 350g$
Explanation
Coverage means $a = 350g$. Solving for gallons gives $g = \frac{a}{350} = \frac{1}{350}a$. Option B flips the rate, C uses addition instead of multiplication, and D gives $a$ in terms of $g$, not $g$ in terms of $a$.
A map uses a scale of 1 inch to 8 miles. Let $m$ be the distance on the map (in inches) and $d$ be the actual distance (in miles). What equation represents this proportional relationship?
$m = 8d$
$d = 8m$
$d = m + 8$
$d = \dfrac{m}{8}$
Explanation
Each inch represents 8 miles, so miles are 8 times the inches: $d = 8m$. Choice A inverts the relationship; C uses addition (not proportional); D uses the reciprocal (dividing by 8).
At a school store, notebooks cost 2.50 each. Let $t$ be the total cost in dollars and $n$ be the number of notebooks. Which equation represents this proportional relationship?
$t = n + 2.5$
$t = \frac{n}{2.5}$
$t = 2.5n$
$t = 2.5n + 1$
Explanation
A proportional relationship has the form $y = kx$. The unit rate is 2.50 per notebook, so $t = 2.5n$. The other choices add a constant, invert the relationship, or include a fixed fee, which are not proportional.
Notebooks all cost the same price: 1.25 dollars each.
Let $t$ be the total cost in dollars and $n$ be the number of notebooks. Which equation represents this proportional relationship?
$t=n+1.25$
$n=1.25t$
$t=\dfrac{n}{1.25}$
$t=1.25n$
Explanation
With a constant price per notebook, $t=pn$ where $p=1.25$, so $t=1.25n$. The other choices use addition, invert the relationship, or divide instead of multiply.
Five stickers cost \$1.25$. Let $x$ be the number of stickers and $y$ be the total cost in dollars. The relationship is proportional.
Which equation represents this situation?
$y = 5x + 1.25$
$y = 1.25x$
$x = 0.25y$
$y = 0.25x$
Explanation
The unit rate is $1.25\div 5=0.25$ dollars per sticker, so $y=0.25x$. A adds a fee, B treats \$1.25$ as the per-sticker price, and C inverts $x$ and $y$.
A bike rental shop charges a constant rate per hour. The relationship between hours $x$ and cost $y$ is shown by the pairs: $(1,8)$, $(2,16)$, $(3,24)$, $(4,32)$.
Which equation represents this proportional relationship?
$y = x + 8$
$y = 8x$
$y = 32x$
$x = 8y$
Explanation
The ratios $y/x$ are all $8$, so $k=8$ and the equation is $y=8x$. Choice A adds instead of multiplying, C uses the wrong constant, and D inverts the variables.