64.3 Ratio and Proportion Models

Ratio and Proportion Models are algebraic representations of situations involving a comparison between two or more quantities, used to find an unknown quantity by setting up an equation between two equivalent ratios and solving it, or to divide a total quantity proportionally according to a specified ratio.

Ordered Ratio Reading

Reading a Ratio in Its Stated Order

A ratio is read and recorded in the exact order its quantities are presented, since the position of each quantity within the ratio determines which value it represents.

Why Order Cannot Be Rearranged Freely

Because a ratio compares two specific, distinct quantities, reversing the order of a ratio changes which quantity is being compared to which, producing a completely different relationship rather than an equivalent one.

Ratio Unit Compatibility

Confirming the Quantities Share Compatible Units

Before a ratio is used in any calculation, the quantities being compared are checked to confirm they are expressed in compatible units, converting one of them if necessary.

Why This Check Precedes Further Work

A ratio compares the relative sizes of two quantities, and this comparison is only meaningful once both quantities are expressed using the same unit; skipping this check risks comparing numbers that do not actually represent a valid proportional relationship.

Equivalent Ratio Equation

Setting Two Ratios Equal

An equivalent ratio equation is formed by setting a known ratio equal to a second ratio that contains an unknown quantity, based on the assumption that both ratios describe the same underlying proportional relationship.

Why This Equation Structure Is Used

Writing both ratios as fractions set equal to each other directly expresses the idea that they represent the same proportional relationship, providing a single algebraic equation that can be solved for the unknown quantity.

Ratio Scale-Factor Resolution

Solving Using a Consistent Scaling Factor

When one ratio is a direct multiple of the other, the unknown is found by identifying the factor that scales the known quantities into the given ratio and applying that same factor to find the missing quantity.

Why This Method Is Often the Quickest

Recognizing a whole-number scale factor between the two ratios allows the unknown to be found through direct multiplication, avoiding the more general algebraic manipulation needed when no such simple factor is apparent.

Ratio Model Cross-Product Resolution

Solving Using Cross Multiplication

The unknown in an equivalent ratio equation is found generally by cross-multiplying, setting the product of the numerator of one ratio and the denominator of the other equal to the product of the remaining two values.

Why Cross Multiplication Works in Every Case

Because cross multiplication is derived directly from multiplying both sides of the equivalent ratio equation by both denominators, it applies universally to any equivalent ratio equation, regardless of whether a simple scale factor happens to be visible.

Ratio-Based Total Sharing

Dividing a Total According to a Ratio

When a total quantity is divided among parts using a specified ratio, the total is first divided by the sum of the ratio's terms to find the value of one part, and that value is then multiplied by each individual term to find each share.

Why the Sum of Terms Is the Key Divisor

The sum of the ratio's terms represents the total number of equal parts the ratio divides the whole into, making it the correct divisor for finding the value of a single one of those parts before scaling up to each individual share.

Ratio Model Verification

Confirming the Solved Value Fits the Original Ratio

The value found for the unknown is substituted back into the original ratio, and the resulting ratio is simplified to confirm it matches the known ratio it was set equal to.

Why This Verification Matters

Because setting up the correct proportional relationship depends on correctly matching corresponding quantities between the two ratios, this final check confirms that the equation was constructed correctly and that no quantities were mismatched during setup.