65.5 Rate Model Verification

Rate Model Verification is the set of checks applied after solving a motion or work-rate problem, confirming that the solved values satisfy the original relationship they were built from and that they fall within the physical limits that make sense for the situation being modeled.

Motion Equation Substitution Check

Substituting the Solved Value Back into the Original Equation

This check substitutes the solved value for the unknown back into the motion equation exactly as it was originally constructed, confirming that both sides of that equation are equal once simplified.

Why This Substitution Is the Basic Confirmation Step

This check mirrors the same substitution verification used for any solved linear equation, confirming that no arithmetic error occurred while isolating the unknown during the solving process.

Motion Distance Agreement

Confirming Individual Distances Combine Correctly

This check recalculates each traveler's individual distance using the solved time and their own rate, then confirms that these individual distances combine, whether by addition or by equality, exactly as the original motion model required.

Why This Check Goes Beyond the Basic Substitution

While the equation substitution check confirms the algebra was solved correctly, this check separately confirms that the underlying physical picture — how the two travelers' distances relate to one another — genuinely holds true for the solved values.

Travel Unit Consistency Check

Confirming Units Remain Aligned in the Final Answer

This check confirms that the units associated with the solved distance, rate, or time value remain consistent with the units established during the original travel unit alignment step performed before the equation was constructed.

Why This Consistency Must Be Confirmed at the End

Because algebraic solving does not track units automatically, this final check ensures that the units assumed during setup are still the correct units to attach to the answer once the solving process is complete.

Work Fraction Sum Check

Confirming the Completed Fractions Sum to the Whole Task

This check recalculates the fraction of the task each agent completed using the solved combined time and their own individual rate, then confirms that these fractions sum exactly to one, representing the whole completed task.

Why This Check Directly Confirms the Model's Core Condition

Since this sum equaling one is exactly the condition the shared work-time equation was built to satisfy, confirming it holds for the solved combined time is a direct verification that the equation was solved correctly.

Combined Rate Magnitude Check

Confirming the Combined Rate Exceeds Each Individual Rate

This check confirms that the combined rate of two agents working together is greater than either individual agent's rate alone, reflecting that working together should speed up, not slow down, completion of the task.

Why This Check Guards against a Specific Kind of Error

Because adding rates should always produce a larger combined rate than either individual rate, a combined rate that fails this comparison signals an error, such as an inverted fraction, made somewhere during the rate combination step.

Positive Time Feasibility Check

Confirming the Solved Time Is a Positive, Feasible Value

This check confirms that any solved value representing a time is a positive number, since a negative or zero time would not correspond to a real, physically meaningful duration within the situation being modeled.

Why Feasibility Must Be Checked Separately from Correctness

An equation can be solved with complete algebraic correctness and still produce a value that makes no sense within the physical situation it represents, particularly when a modeling assumption, such as the direction of two travelers, was set up incorrectly at the start; this check catches that category of error specifically.