33.4 Proportional Pair Scaling

Proportional Pair Scaling is the study of what happens to an output value when its corresponding input value in a direct variation relationship is scaled by some factor, based on the fact that scaling the input by any factor scales the output by that exact same factor, since both values are connected through multiplication by the same fixed constant of proportionality. This behavior follows directly from the output proportional to input requirement established under direct proportionality scope, and understanding it allows new input-output pairs to be generated quickly from a single known pair without needing to first calculate the constant of proportionality or construct the full rule.

Because scaling applies uniformly regardless of whether the scale factor is a whole number, a fraction, or a negative value, this topic works through several specific scaling situations to show that the same underlying principle governs every case.

Input Quantity Scale Change

Defining a Scale Change to the Input

A scale change to the input means multiplying a known input value by some factor, producing a new input value that is larger, smaller, or otherwise adjusted relative to the original, without yet considering what happens to the corresponding output.

Why the Input Can Be Scaled Freely

Because a direct variation relationship's rule applies to any valid input within its domain, scaling a known input by a chosen factor simply produces another valid input for which a corresponding output can be determined.

Setting Up the Scaling Question

Once an input has been scaled by a chosen factor, the central question this topic addresses is how the corresponding output must change to remain consistent with the same underlying proportional relationship.

Matching Output Scale Change

The Key Scaling Principle

For any direct variation relationship, scaling the input by a given factor scales the output by that exact same factor, meaning if the input becomes three times as large, the output also becomes three times as large.

Why the Output Must Match the Input's Scaling

This matching behavior follows directly from the rule $$y=kx$$, since multiplying $$x$$ by a factor and substituting into the rule produces an output multiplied by that same factor, as the constant $$k$$ remains unchanged throughout.

Using This Principle Without the Full Rule

Because this matching behavior holds for any direct variation relationship, it can be applied directly to a known pair to find a new pair without first calculating the constant of proportionality, offering a faster route to a new pair than the full direct variation rule construction process.

Pair Doubling

Doubling as a Specific Scaling Case

Doubling refers to scaling by a factor of two, and applying the matching output scale change principle means that doubling a known input produces a new pair whose output is also exactly double the original output.

An Example of Pair Doubling

Given the known pair $$\left(3,12\right)$$ from a direct variation relationship, doubling the input to $$6$$ produces the new pair $$\left(6,24\right)$$, since the output must also double.

Verifying a Doubled Pair

A doubled pair can be verified by checking that its output-to-input ratio still matches the original constant of proportionality, confirming that the doubling was applied correctly to both values.

Pair Halving

Halving as a Specific Scaling Case

Halving refers to scaling by a factor of one-half, and applying the same matching principle means that halving a known input produces a new pair whose output is also exactly half the original output.

An Example of Pair Halving

Given the known pair $$\left(8,20\right)$$, halving the input to $$4$$ produces the new pair $$\left(4,10\right)$$, since the output must also be halved.

Verifying a Halved Pair

As with doubling, a halved pair can be verified by confirming that its output-to-input ratio still equals the original constant of proportionality, providing the same kind of check used for any scaled pair.

Fractional Pair Scaling

Scaling by a Fraction Other Than One-Half

Beyond halving, an input can be scaled by any fraction, such as one-third or three-fourths, with the output required to scale by that same fraction regardless of its specific value.

An Example of Fractional Scaling

Given the known pair $$\left(9,18\right)$$, scaling the input by $$\frac{1}{3}$$ produces a new input of $$3$$ and, applying the same fraction to the output, a new output of $$6$$, giving the pair $$\left(3,6\right)$$.

Handling Scale Factors Greater Than One That Are Fractions

A fractional scale factor greater than one, such as $$\frac{3}{2}$$, still follows the same matching principle, increasing both the input and output by that same fractional amount rather than decreasing them, since the size of the fraction relative to one determines whether the scaling increases or decreases the values.

Signed Pair Scaling

Scaling by a Negative Factor

An input can also be scaled by a negative factor, and the matching output scale change principle continues to hold: the output is multiplied by that same negative factor, which reverses the sign of both the input and the output relative to the original pair.

An Example of Signed Scaling

Given the known pair $$\left(5,15\right)$$, scaling by $$-1$$ produces the new pair $$\left(-5,-15\right)$$, with both coordinates negated together.

Why Signed Scaling Still Preserves the Ratio

Because both the input and output are multiplied by the same negative factor, their ratio remains unchanged, since dividing two negative numbers, or a negative by a negative, produces the same positive constant of proportionality as the original pair.

Direct Variation Ratio Preservation

Why Scaling Never Changes the Ratio

Regardless of the specific scale factor used, whether a whole number, a fraction, or a negative value, scaling a known pair by any factor never changes the underlying output-to-input ratio, since both coordinates are multiplied by the identical factor and that factor cancels out of the ratio calculation.

Confirming Ratio Preservation Algebraically

Starting from a known ratio $$k=\frac{y}{x}$$, scaling both coordinates by a factor $$c$$ produces the new ratio $$\frac{cy}{cx}=\frac{y}{x}=k$$, confirming that the constant remains exactly the same after scaling.

Using Ratio Preservation as a General Check

Because ratio preservation holds for every valid scale factor, checking that a newly scaled pair still produces the original constant of proportionality serves as a reliable, general-purpose verification for any proportional pair scaling calculation, regardless of which specific type of scale factor was applied.