SAT Math — density, derived units and area and volume conversions

The SAT Math skill on ratios, rates, proportional relationships and units goes beyond recipes and map distances. College Board's description includes problems with derived units, both products such as kilowatt-hours and quotients such as population per square kilometer, and multistep or multidimensional unit conversions. These are the questions where a correct method still produces a wrong answer because a square yard was treated as 3 square feet. This material trains exactly that part of the skill.

It starts with the two classic quotient units. You find the mass of an aluminum block from its density and volume, and the population density of a county from its population and area. A third question converts a population density from people per square mile to people per square kilometer, where the length factor 1.609 has to be squared, and where the right answer must be smaller, since each square kilometer is smaller than a square mile.

The middle of the set is about area and volume conversions, the heart of the gap. Carpet priced by the square yard for a room measured in feet needs 9 square feet per square yard; a concrete patio 6 inches thick needs its thickness in feet and then 27 cubic feet per cubic yard; a water tank measured in meters holds 1,000 liters per cubic meter; and tiles measured in inches cover a floor measured in feet only after both are in the same unit. A map question asks for a real area from a scale of 1 inch to 2 miles, where areas grow by the square of the scale.

The last questions chain rates. A space heater's watts, hours and days become kilowatt-hours and then a monthly cost; the density of cooking oil in grams per milliliter gives the mass of 5 liters in kilograms; a flow of cubic feet per second becomes gallons per minute; and a fuel rating in liters per 100 kilometers gives the fuel for a 340-kilometer trip. Money is written in words, as the test's context questions do.

Each explanation carries the units through the calculation and names the conversion slip behind each wrong answer. The flashcards collect the conversion factors worth knowing by heart (9 square feet, 27 cubic feet, 144 square inches, 1,000 liters), the rule that areas scale by the square and volumes by the cube, and the habit of writing conversion factors as fractions equal to 1 so that units cancel. One question shows the room it describes as a drawing with its two dimensions.

The material offers a quiz and flashcards. It is independent practice; the situations are invented and no question is taken from any published test.

• Use density and population density as quotient units to find mass, volume, population or area
• Convert areas and volumes between units by squaring or cubing the length factor
• Convert a density from square miles to square kilometers
• Compute energy in kilowatt-hours and its cost
• Find a real area from a scale drawing or map
• Chain several unit conversions, including rates per 100 units, so that units cancel

Practice material written by Zestly, based on the SAT Math skill "Ratios, rates, proportional relationships, and units" in the Problem-Solving and Data Analysis domain, as described in College Board's Assessment Framework for the Digital SAT Suite (version 3.01, August 2024, Appendix B). All data sets and situations are invented. Zestly is not affiliated with College Board.