7th Grade · 7.RP.A.2

Proportional relationships in 7th grade

Proportional relationships are a big part of 7th grade math. Two amounts are in a proportional relationship when one is always the same number times the other, like pay of 12 dollars for every hour worked. Your child learns to spot these relationships in tables, graphs and word problems, find that number and write an equation for it.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

7.RP.A.2 Recognize and represent proportional relationships between quantities.
7.RP.A.2.a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
7.RP.A.2.b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
7.RP.A.2.c Represent proportional relationships by equations.
7.RP.A.2.d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

What it means, in plain words

There are two quick checks. In a table, divide each $y$ value by the $x$ value in the same row. If every row gives the same answer, the relationship is proportional. On a graph, a proportional relationship is a straight line that passes through $(0, 0)$ , the point where both amounts are zero.

The answer that every row shares is called the constant of proportionality, written $k$ . It is the unit rate: how much of $y$ goes with one unit of $x$ . Once your child knows $k$ , the equation is $y = kx$ . For pay of 12 dollars an hour, $k = 12$ and the equation is $y = 12x$ , with $x$ in hours and $y$ in dollars.

Proportional relationships graphs also tell a story point by point. The point $(0, 0)$ means zero hours earn zero dollars. The point $(1, 12)$ means one hour earns 12 dollars, so the $y$ value at $x = 1$ is always the constant of proportionality. Some relationships have a starting charge, such as a fee to join plus a price per visit. Their graphs start above zero, so they are not proportional.

Your child should be able to:

Decide whether a table shows a proportional relationship by dividing $y$ by $x$ in each row
Tell from a graph whether a relationship is proportional: a straight line through $(0, 0)$
Find the constant of proportionality in a table, a graph, an equation or a word problem
Write the equation $y = kx$ and explain what the points $(0, 0)$ and $(1, k)$ mean in the situation

Example

A table shows hours worked and pay: 2 hours earn 24 dollars, 5 hours earn 60 dollars and 8 hours earn 96 dollars. Is the pay proportional to the hours, and what is the equation?

Divide pay by hours in each row: $24 \div 2 = 12$ , $60 \div 5 = 12$ and $96 \div 8 = 12$ .
Every row gives 12, so the relationship is proportional and $k = 12$ .
The equation is $y = 12x$ , with $x$ in hours and $y$ in dollars.
Yes, the pay is proportional to the hours, at 12 dollars per hour.

Example

A table shows tickets and total cost: 1 ticket costs 9 dollars, 2 tickets cost 14 dollars and 3 tickets cost 19 dollars. Is the cost proportional to the number of tickets?

Divide cost by tickets in each row: $9 \div 1 = 9$ , $14 \div 2 = 7$ and $19 \div 3 \approx 6.33$ .
The answers are not the same, so there is no single constant of proportionality.
Each ticket adds 5 dollars, so zero tickets would still cost 4 dollars, a booking fee. The graph would not pass through $(0, 0)$ .
No, the cost is not proportional to the number of tickets.

Where children usually get stuck

Your child sees costs of 9, 14 and 19 dollars for 1, 2 and 3 tickets and calls the relationship proportional, because the cost goes up by 5 each time.
A steady pattern feels like a steady rate. Going up by the same amount each time does give a straight line, but a proportional line must also start at zero.
What helps. Ask what zero tickets would cost. Work backward from 9 dollars: the answer is 4 dollars, not zero, so the relationship is not proportional.
From a graph that passes through $(4, 10)$ , your child writes the constant of proportionality as $\frac{4}{10} = 0.4$ instead of 2.5.
The constant is $y$ divided by $x$ . The child divides the two numbers in the order they appear in the point.
What helps. Ask how much $y$ goes with one unit of $x$ . Read the line at $x = 1$ , or divide $y$ by $x$ : $10 \div 4 = 2.5$ .
For pay of 12 dollars an hour, your child writes the equation $y = x + 12$ .
The child knows 12 belongs in the equation but adds it once, instead of counting it again for every hour.
What helps. Have your child check the equation with 3 hours of work. $3 + 12 = 15$ dollars is far too little, while $12 \times 3 = 36$ dollars matches the pay.

Try four practice questions

Real 7th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Look at the equation and Mia's claim below. Which explanation is correct?
$\displaystyle 2y = 4x \qquad \text{Mia says it is proportional because } y = 2x$
A
Mia is wrong because the equation is not already written as y = kx.
B
Mia is wrong because dividing by 2 gives y = 2x + 3.
C
Mia is correct because dividing by 2 gives y = 2x.
D
Mia is wrong because dividing by 2 gives y = x + 3.
Show the answer and how to solve it
Mia is correct because dividing every term by 2 gives exactly y = 2x. With no constant term remaining, the equation is proportional.
1. 1Look at the original equation and Mia's claim that y = 2x.
2. 2To check the claim, divide every term by 2. There is no constant term to divide.
3. 3Simplify each term.
4. 4The equation simplifies to y = 2x with no added or subtracted constant, so the relationship is proportional. Mia's reasoning is correct.
Common mistake. A common mistake is to expect a leftover constant out of habit. Here every term divided by 2 is exact: 2y ÷ 2 = y and 4x ÷ 2 = 2x, with nothing added.
The idea underneath. A relationship is proportional only if isolating y leaves exactly a constant times x with nothing added or subtracted. A constant term is what breaks proportionality.
2On level
Rewrite the equation so y is alone. Then choose proportional or not proportional.
$\displaystyle 3x + y = 8$
Rewrite so y is alone Proportional or not?
Loading...
Show the answer and how to solve it
The answer is y = -3x + 8, and the relationship is not proportional because the rewritten equation has a constant term.
1. 1Look at the equation 3x + y = 8. The y term is not alone yet because 3x is on the same side.
2. 2Subtract 3x from both sides to move the x term to the right.
3. 3The left side simplifies to y, giving $y = 8 - 3x$ . That is the same as y = -3x + 8.
4. 4A proportional relationship must be exactly y = kx. Since y = -3x + 8 still has the constant +8, the relationship is not proportional.
Common mistake. A common mistake is writing y = 3x + 8 by moving 3x across the equal sign without changing its sign. Another mistake is calling y = -3x + 8 proportional even though the +8 means it is not exactly y = kx.
The idea underneath. To decide whether an equation is proportional, first rewrite it so y is alone, then check whether any constant term is left.
3On level
Which descriptions show a proportional relationship? Select all that apply.
A
A cab ride costs $3 to start and $2 per mile.
B
The first 4 songs are free, then 50 cents per song.
C
A baker uses 9 cups of flour for every 3 loaves of bread.
D
A store charges $4 for each bottle of juice.
E
A plant is 6 cm taller than its pot.
Show the answer and how to solve it
The proportional descriptions are 'A store charges $4 for each bottle of juice' and 'A baker uses 9 cups of flour for every 3 loaves of bread' because each can be written as one constant times the other quantity.
1. 1A description is proportional if one quantity is always the same multiple of the other—it can be written as $y = kx$ for one constant $k$ , with no extra added or subtracted amount and no rate change.
2. 2The juice store and the bread recipe are both proportional. Charging $4 per bottle means cost is always 4 times the number of bottles. Using 9 cups for 3 loaves means cups are always $\frac{9}{3} = 3$ times the number of loaves—a constant ratio even though it is not written per 1.
3. 3The cab ride ($3 start plus $2 per mile) and the plant description (6 cm taller than its pot) are not proportional because each adds an extra constant instead of using one multiplier only.
4. 4The streaming description is also not proportional: the rate is $0 per song for the first 4 songs, then changes to $0.50 per song. One constant $k$ cannot describe the whole relationship.
Common mistake. A common mistake is selecting only the juice description and missing the bread recipe because it is not written per 1 loaf. Any constant ratio—such as 9 cups for 3 loaves—describes a proportional relationship, not just unit rates.
The idea underneath. You do not need to see 'per 1' to have a proportional relationship—any situation with one constant multiplier and no extra amount is proportional.
4Stretch
Which graph shows a proportional relationship?
A
B
C
D
Show the answer and how to solve it
This is the only proportional relationship because all four points, (1, 2), (3, 6), (5, 10), and (7, 14), satisfy y = 2x, a straight line through the origin.
1. 1A proportional relationship requires all points to lie on a straight line through the origin, following y = kx for some constant k.
2. 2The proportional graph has the points (1, 2), (3, 6), (5, 10), and (7, 14). Check the y-to-x ratio for each point:
3. 3All four ratios equal 2, so the points follow y = 2x. That line passes through (0, 0), confirming a proportional relationship.
4. 4The graph with points (1,4), (3,8), (5,12), (7,16): the points are collinear on y = 2x + 2, which crosses the y-axis at 2 rather than the origin. The graph with points (1,2), (3,8), (5,10), (7,14): three points fit y = 2x, but (3, 8) sits 2 units above the line; that single interior point breaks the constant ratio. The graph with points (1,2), (3,6), (5,10), (7,12): three points fit y = 2x, but (7, 12) should be 14; it is 2 units below the line, so the fourth point destroys the proportional rule.
Common mistake. A common mistake is choosing the graph with points (1,4), (3,8), (5,12), (7,16) because its points look collinear. But a proportional graph must pass through (0, 0); y = 2x + 2 is shifted 2 units upward and never reaches the origin.
The idea underneath. In a proportional relationship, 0 of one quantity must match 0 of the other, so the graph has to pass through the origin.