64.5 Model Verification

Model Verification ensures mathematical models are accurate by checking their consistency with real-world data and logical rules.

Model Verification is the set of checks applied after solving a percentage, ratio, or mixture problem, confirming that the resulting values make sense within the constraints specific to each type of model, from reasonable percentage magnitudes to the conservation of total amounts within a mixture.

Percentage Magnitude Check

Confirming a Reasonable Percentage Value

This check confirms that a calculated percentage rate falls within an expected range for the situation described, flagging results that fall far outside what the described situation would reasonably allow.

Rate = 250 %  may indicate an error, depending on context

Why This Check Matters

While a percentage rate exceeding one hundred is not always mathematically invalid, an unexpectedly extreme value is often a signal that the part and whole quantities were switched or misidentified during equation construction.

Ratio Order Preservation Check

Confirming the Solved Ratio Matches the Original Order

This check confirms that the two quantities in a solved ratio problem remain in the same order they were originally presented, rather than having been inadvertently swapped during solving.

a : b still matches the original comparison order

Why Order Preservation Matters

Because a ratio's meaning depends entirely on which quantity is compared to which, this check catches the specific error of reversing that comparison, which would otherwise produce a numerically plausible but conceptually incorrect result.

Proportional Equality Check

Confirming Both Ratios Are Truly Equivalent

This check substitutes the solved unknown back into its ratio and simplifies both ratios in the original proportion to confirm they reduce to the identical simplified form.

\frac{a}{b} = \frac{c}{x} confirmed by simplification

Why This Check Is the Direct Test of Correctness

Since the entire ratio and proportion model depends on the assumption that two ratios describe the same relationship, directly confirming that equivalence after solving is the most direct possible test of whether the solution is correct.

Mixture Amount Conservation Check

Confirming the Source Amounts Sum to the Total

This check confirms that the two individual source amounts used in a mixture model add up exactly to the stated total amount of the final mixture.

Why This Conservation Must Hold

Because combining two sources is treated as a purely additive process within this scope, any solved amounts that fail to sum to the given total indicate an arithmetic or setup error somewhere in the construction of the model.

Mixture Component Conservation Check

Confirming the Component Contributions Sum Correctly

This check confirms that the individual component quantities contributed by each source, once calculated using the solved amounts, sum to the total component quantity implied by the final mixture's concentration and total amount.

r_{1} x_{1} + r_{2} x_{2} = r_{mix} (x_{1} + x_{2})

Why This Check Directly Confirms the Balance Equation

Since this relationship is exactly the balance equation the mixture model was built upon, confirming it holds true for the solved values is a direct verification that the equation was solved correctly.

Mixture Concentration Range Check

Confirming the Final Concentration Falls between the Sources

This check confirms that the target concentration of the mixture falls numerically between the concentrations of the two individual sources that were combined to create it.

r_{1} < r_{mix} < r_{2}

Why This Range Must Hold

Because combining two sources through a purely additive process cannot produce a concentration more extreme than either individual source contributed, a target concentration falling outside this range signals an error somewhere in the setup or solving of the model.

Model Unit Agreement Check

Confirming Units Remain Consistent through the Solution

This check confirms that the units associated with every quantity in the completed solution remain consistent with the units established during the original unit compatibility check performed at the start of the modeling process.

Why This Final Consistency Check Matters

Because algebraic manipulation during solving does not track units automatically, this final check ensures that no unit mismatch was silently introduced partway through the solving process, keeping the final answer meaningful within its original context.