Question

How many ways are there to

adjust two quantities so that they are in a
given proportional relationship? Explain your
reasoning.

Answer 1

[AI-generated answer — Bot GPT 3.5]

There are multiple ways to adjust two quantities so that they are in a given proportional relationship.

To understand why, let's consider two quantities x and y that have a proportional relationship. This means that their ratio, x:y, remains constant.

Now, to adjust these quantities while maintaining their proportional relationship, we can use multiplication or division.

Multiplication: We can multiply both x and y by the same non-zero constant to adjust their values while keeping their ratio constant. For example, if the initial ratio is x:y, we can multiply both x and y by any non-zero constant c, yielding cx:cy. This adjustment maintains the proportional relationship between x and y.

Division: Alternatively, we can divide both x and y by the same non-zero constant to adjust their values while maintaining their ratio. For example, if the initial ratio is x:y (where x > y), we can divide both x and y by the same non-zero constant c, resulting in (x/c):(y/c). Again, this adjustment preserves the proportional relationship.

Therefore, there are infinitely many ways to adjust two quantities so that they are in a given proportional relationship because we can use different non-zero constants for multiplication or division.