Unit Rates, Speed, and Combined Quantities

Objective: Find a per-unit amount and scale it.

Prerequisite skills: division, compatible units, and rate models. A unit rate divides an output quantity by the associated input quantity. Keep the unit order explicit. When the relationship stays proportional, multiplying by another input yields the new output. Common error: inverting the unit rate or using incompatible time units. Restrictions and interpretation: Average speed is total distance divided by total time, not generally the mean of component speeds.

Objective: Combine distances and times before computing an overall rate.

For different trip segments, add all distances and all times. Their quotient is the overall average speed. Averaging speeds directly gives the correct result only under appropriate equal-time conditions. Common error: taking the unweighted mean of speeds for equal-distance legs. Restrictions and interpretation: Average speed is total distance divided by total time, not generally the mean of component speeds.

Objective: Add rates when independent workers contribute to the same total.

If one worker completes one task in a hours, its rate is 1/a task per hour. Independent simultaneous rates add. The combined time is the reciprocal of the total rate, not the sum of the solo times. Common error: adding the completion times instead of their rates. Restrictions and interpretation: Average speed is total distance divided by total time, not generally the mean of component speeds. 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.