Ratios, rates, proportional relationships, and units: solve proportions, work with unit rates and constant of proportionality, and convert between units including compound units.

What this skill is asking

A ratio compares two quantities, a rate is a ratio of quantities with different units (like miles per hour), and a proportional relationship is one where two quantities keep a constant ratio. The Digital SAT (Problem-Solving and Data Analysis domain) tests setting up and solving proportions, finding and using unit rates, and converting units, including compound units such as speed and density. These are word-problem heavy and reward careful setup.

Setting up proportions

The reliable method is to write a ratio with consistent units on both sides.

A worked unit-conversion

Conversions are about cancelling units cleanly.

Unit rates and the constant of proportionality

A unit rate answers "how much per one?" and is found by division: $\frac{150 \text{ miles}}{2.5 \text{ hours}} = 60$ miles per hour. In a proportional relationship $y = kx$, that unit rate is the constant $k$, and it is also the slope of the straight line through the origin that represents the relationship. So a question that gives a proportional table and asks for $k$ is asking for the unit rate: divide any $y$ by its $x$. Recognising that "constant of proportionality", "unit rate", and "slope through the origin" all name the same number lets you answer many PSDA questions with a single division.

Proportional reasoning shortcuts

Many proportion questions yield to a quick scale factor rather than full cross-multiplication. If $12$ cups of flour is $4$ times the $3$ cups in the ratio, then every other ingredient also scales by $4$. Spotting the multiplier between the given value and the ratio value often beats setting up and solving an equation. For inverse relationships (where one quantity goes up as the other goes down, like speed and travel time for a fixed distance), the product stays constant rather than the ratio, so set $x_1 y_1 = x_2 y_2$. Telling a direct proportion (constant ratio) from an inverse one (constant product) is the key reading skill.