Concept 6

Interpret units and scales; ensure dimensional consistency in solutions.

Core Idea

Units are part of the math. If the units do not line up from one step to the next, the calculation is probably tracking the wrong quantity.

Understanding

Unit check first: the units tell you what should cancel and what the final answer should be.

A rate such as miles per hour or dollars per ticket shows the path for the calculation. Scale on a graph or map can also change the meaning of the printed number. If the final unit is wrong, the setup is wrong.

  • Keep track of the starting unit, the rate unit, and the requested unit.
  • Convert only after the arithmetic if the prompt still asks for a different unit.
  • Do a quick size check before you stop.

Step by Step

  1. Identify the starting unit, the rate unit, and the unit requested in the question.
  2. Multiply or divide so unwanted units cancel in a sensible way.
  3. Convert the result if the final unit still does not match the prompt.
  4. Do a quick size check to see whether the converted answer is reasonable.

Misconceptions

  • Stopping after the arithmetic even though the answer is still in the wrong unit.
  • Multiplying when the unit relationship requires division, or the reverse.
  • Reading the number correctly but ignoring the graph scale or conversion factor.
Question

Worked Example

A water tank fills at a rate of 18 liters per minute. At this rate, how many cubic meters of water enter the tank in 50 minutes? (1 cubic meter = 1,000 liters)

Select an answer to see the explanation