Unit Conversions and Dimensional Analysis

Objective: Cancel a starting unit using a conversion ratio.

Prerequisite skills: fraction multiplication and rate units. Multiply by a ratio of equal physical quantities so the old unit cancels. The numerical result changes while the measured quantity does not. Write the units through the calculation to choose the ratio direction. Common error: using the reciprocal conversion factor. Restrictions and interpretation: Use stated conversion factors exactly; squared and cubed units require squared and cubed factors.

Objective: Convert both numerator and denominator units.

For rates, changing the time unit often reverses the apparent multiplication direction. Write the complete dimensional chain instead of relying on a memorized rule. Converting kilometers per hour to meters per second uses both length and time factors. Common error: multiplying by 3600 when the hour is already in the denominator. Restrictions and interpretation: Use stated conversion factors exactly; squared and cubed units require squared and cubed factors.

Objective: Raise conversion factors to match the dimension.

A square unit contains two length factors and a cubic unit contains three. If one length unit equals c smaller units, one square unit equals c² and one cubic unit equals c³ smaller units. Common error: using a linear conversion factor on an area. Restrictions and interpretation: Use stated conversion factors exactly; squared and cubed units require squared and cubed factors. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.