10.6 Formula Evaluation

Formula Evaluation is the process of substituting values into algebraic expressions to compute their numerical results.

Formula Evaluation applies expression substitution to real-world formulas, requiring that given quantities be correctly matched to formula symbols, carried through with appropriate units, computed either directly or through intermediate steps, and checked for a result that is both correctly rounded and physically reasonable.

Preparing to Evaluate a Formula

Formula Variable Identification

Before substituting into a formula, every symbol in the formula must be identified along with the quantity it represents, since formulas often use letters chosen to suggest their meaning, such as a symbol representing radius or time, rather than the generic letters used in an abstract expression.

A = π r^{2} \to A is area, r is radius

Given Quantity and Formula Symbol Matching

Once the formula's symbols are identified, each quantity given in the problem must be matched to the correct symbol, based on what that quantity represents rather than the order in which it is mentioned, since mismatching a given value to the wrong symbol produces an incorrect substitution even if the arithmetic afterward is performed correctly.

Complete Formula Input Assignment

Every symbol in the formula that is not itself the quantity being solved for must receive an assigned value before evaluation can proceed; a formula with any required input left unassigned cannot be reduced to a single numerical result.

Carrying Units Through the Substitution

Formula Substitution with Units

When a formula's variables represent physical quantities, the units attached to each given value should be carried alongside the number through the substitution, and converted to a consistent system of units beforehand if the formula's variables are not already expressed in matching units.

Carrying Out the Calculation

Formula Evaluation in a Single Step

Some formulas allow every substituted value to be combined in one continuous calculation, following the order of operations directly to the final result without needing to record any separate intermediate quantity.

A = π 5^{2} = 25 π

Formula Evaluation through Intermediate Values

Other formulas require computing one or more intermediate quantities first, recording each one clearly, before those intermediate results are substituted into a further step of the same formula or into another related formula to reach the final result.

Precision of the Result

Exact Formula Result

A formula's result is exact when every input is exact and no term in the formula requires an irrational approximation, such as when a named constant is left unevaluated in symbolic form or when all operations involve only whole numbers or exact fractions.

Rounded Formula Result

A formula's result must be rounded when it involves an irrational quantity, such as a named constant expressed as a decimal approximation, or when the given inputs themselves were already rounded measurements, and the appropriate number of decimal places for the rounded result should match the precision of the least precise input.

Interpreting the Result

Result Unit Determination

The unit of a formula's result is determined by combining the units of every input according to the operations of the formula itself, such as squaring a length unit when computing an area, or dividing a distance unit by a time unit when computing a speed.

A = π {(5 cm)}^{2} \to result in {cm}^{2}

Formula Value Reasonableness Check

After a formula is evaluated, the resulting value and its unit should be checked against the physical context of the problem: a computed area, distance, or rate that is far too large or too small, or that carries the wrong type of unit for what was being calculated, signals an error in matching quantities to symbols, in unit conversion, or in the arithmetic itself.