6.6 Missing Proportion Term Determination

Missing Proportion Term Determination is the process of finding the value of an unknown term in a proportion, using the fact that the two ratios forming the proportion must be equal, through methods such as scale factors, equivalent-ratio reduction, unit rates, or cross multiplication.

Core Methods for Finding a Missing Term

Missing Term by Scale Factor

When one ratio is a scaled version of the other, the missing term is found by identifying the scale factor that relates a known pair of corresponding terms, then applying that same factor to the term paired with the unknown.

Missing Term by Equivalent Ratio Reduction

The missing term can also be found by reducing the known ratio to its simplest form and then scaling that simplest form up or down to match the known term of the second ratio, arriving at the unknown term as the corresponding scaled value.

Missing Term by Unit Rate

When the proportion represents a rate relationship, the missing term is found by first computing the unit rate from the known ratio, then multiplying that unit rate by the known term of the second ratio to obtain the unknown value.

Missing Term by Cross Multiplication

The most general method sets the cross products of the proportion equal to each other and solves the resulting equation for the unknown term. This method works regardless of whether the ratios share an obvious scale factor.

Solving for Each Term Position

Unknown First Ratio Term

When the unknown occupies the first position of the proportion, cross multiplication produces an equation in which the unknown is multiplied by the fourth term; dividing both sides by the fourth term isolates the unknown.

Unknown Second Ratio Term

When the unknown occupies the second position, cross multiplication produces an equation in which the unknown is multiplied by the third term; dividing both sides by the third term isolates the unknown.

Unknown Third Ratio Term

When the unknown occupies the third position, the same cross-multiplication procedure applies with the unknown multiplied by the second term, so dividing both sides by the second term isolates it.

Unknown Fourth Ratio Term

When the unknown occupies the fourth position, cross multiplication places the unknown as a factor multiplied by the first term, and dividing both sides by the first term isolates the unknown, mirroring the same structure used for the other three positions.

Handling Non-Integer Missing Terms

Fractional Missing Proportion Term

When solving for the unknown produces a fraction that does not reduce to a whole number, the fractional result is retained and simplified to lowest terms, since a proportion's solution is not required to be a whole number.

Decimal Missing Proportion Term

When the known terms of the proportion are decimals, or when the division required to isolate the unknown does not terminate evenly as a simple fraction, the missing term is expressed as a decimal, computed by direct division after cross multiplication.

Confirming the Solution

Proportion Solution Verification

After solving for the missing term, the value is substituted back into the original proportion and the cross products are recomputed to confirm they are equal; if the cross products fail to match, an arithmetic error occurred during solving and the calculation must be redone rather than accepted.