Interpret and compare rates/ratios in context (including constant of proportionality). | Ratios, rates, proportional relationships, and units | SAT Math

Core Idea

The constant of proportionality $𝑘$ in $𝑦 = 𝑘 𝑥$ is the unit rate. To compare two rates, convert both to the same unit and compare the values of $𝑘$ .

Understanding

Constant of proportionality $𝑘$ is the unit rate in $𝑦 = 𝑘 𝑥$ .

From a table, compute $𝑦 𝑥$ for any row. A proportional table gives the same value every time.
From a graph, use any point on the line through the origin and compute $𝑦 𝑥$ .
To compare rates, put them in the same units first, then compare the $𝑘$ values.
Trap: if the ratio changes from row to row, the relationship is not proportional.

Step by Step

Identify the two quantities and which is dependent on which.
Compute $𝑘 = 𝑦 𝑥$ using given values.
Interpret $𝑘$ in context: it's the amount of $𝑦$ per one unit of $𝑥$ .
To compare, express both rates in the same units and compare $𝑘$ values.

Misconceptions

Computing $𝑥 𝑦$ instead of $𝑦 𝑥$ — direction matters.
Assuming a linear relationship is proportional. Proportional means the line goes through the origin; $𝑦 = 2 𝑥 + 3$ is linear but not proportional.
Comparing rates that use different units without converting first.

Question

Worked Example

The table below shows the cost $𝐶$ , in dollars, of $𝑛$ pounds of almonds at a grocery store.

$𝑛$ (pounds)	$𝐶$ (dollars)
2	13.50
5	33.75
8	54.00

What is the cost per pound of almonds?

Select an answer to see the explanation