Translate a context into a proportional model (table/equation/graph) and solve. | Ratios, rates, proportional relationships, and units | SAT Math

Core Idea

A proportional relationship always takes the form $𝑦 = 𝑘 𝑥$ — a straight line through the origin with slope $𝑘$ . Recognize it in any representation: table (constant $𝑦 𝑥$ ), equation (no added constant), or graph (line through $(0, 0)$ ).

Understanding

The SAT may give you a word problem and ask you to write an equation, fill a table, or identify the correct graph. All three represent the same relationship.

Table to Equation: Compute $𝑦 𝑥$ for each row. If it's constant, that constant is $𝑘$ , and the equation is $𝑦 = 𝑘 𝑥$ .

Equation to Graph: $𝑦 = 𝑘 𝑥$ is a line through the origin. Positive $𝑘$ slopes up; negative $𝑘$ slopes down. The steeper the line, the larger $| 𝑘 |$ .

Graph to Equation: Read any point $(𝑥, 𝑦)$ off the line and compute $𝑘 = 𝑦 𝑥$ .

A common SAT trap: a graph shows a line that doesn't pass through the origin. That's linear but not proportional — it has a $𝑦$ -intercept, so $𝑦 = 𝑘 𝑥 + 𝑏$ with $𝑏 \neq 0$ .

Step by Step

Read the problem and identify the two quantities.
Determine whether the relationship is proportional (does zero input give zero output?).
Find $𝑘$ from the given information.
Write the equation $𝑦 = 𝑘 𝑥$ and use it to answer the question.

Misconceptions

Confusing linear ( $𝑦 = 𝑚 𝑥 + 𝑏$ ) with proportional ( $𝑦 = 𝑘 𝑥$ ). Proportional is a special case where $𝑏 = 0$ .
Reading the slope from a graph using $Δ 𝑥 Δ 𝑦$ instead of $Δ 𝑦 Δ 𝑥$ .
Assuming that because two quantities increase together, they must be proportional.

Question

Worked Example

A car uses gasoline at a constant rate. The equation $𝑔 = 0.04 𝑚$ gives the number of gallons $𝑔$ of gasoline used for $𝑚$ miles driven. How many gallons does the car use to drive 250 miles?

Select an answer to see the explanation