SAT® · Math: Problem-Solving and Data Analysis
Derived Units from Products and Quotients
A derived unit is built by multiplying or dividing two other units, and the words in its name tell you exactly which operation to do and in what order.
A unit like a kilowatt sounds finished, but it's actually a rate: kilowatts measure energy per unit time, how fast energy is being used. Multiply that rate by an amount of time and the time units don't cancel, they combine, giving you kilowatt-hours, a unit of energy itself. This is the general rule for derived units: when a word problem names a unit as one thing "times" or "per" another, it's telling you the exact operation to perform. Power times time gives energy. Mass divided by volume gives density. Population divided by area gives density of a different kind.
The key habit is reading the unit's name as an instruction. "Kilograms per cubic meter" means kilograms divided by cubic meters, so if you know density and mass, you find volume by dividing mass by density, not by guessing some other combination. "A per B" always means A divided by B, no exceptions, which is why population density is population divided by land area, never divided by the number of towns or cities sitting in that area. The number of towns is irrelevant data, meant to tempt you into dividing by the wrong thing.
This skill connects to ratios and rates elsewhere in data analysis, since a derived unit is just a ratio with a name. It also sets up unit conversions: before combining units you often have to match their scales first, like converting watts to kilowatts before multiplying by hours. Once you see density, speed, or cost-per-kilowatt-hour as multiplication or division of simpler units, every problem of this type turns into picking the right two numbers and the right operation.
Key terms
- Derived unit
- A unit formed by multiplying or dividing two other units, such as watt-hours or kilograms per cubic meter.
- Density
- Mass divided by volume, often expressed in kilograms per cubic meter.
- Kilowatt-hour
- A unit of energy equal to one kilowatt of power sustained for one hour.
- Rate
- A quantity that compares two different units, expressed as one unit per another, like dollars per hour.
Watch it explained
- How it worksHow do you build a new unit by multiplying two old ones, the way power times time makes energy?
- Worked exampleFind the cost of running a 1500 watt heater for 6 hours at 14 cents per kilowatt-hour.
- Common mistakeWhy isn't population per square kilometer just the population divided by the number of cities?
- Worked exampleFind the edge length of a cube of wood with mass 345 kg when the wood's density is 353 kg per cubic meter.
- Going furtherWhat changes about density when you cut a block of wood exactly in half?
Check yourself
End of topic.