Interpret equations involving a reciprocal

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Interpret equations involving a reciprocal

The statement X is inversely proportional to Y means X = k/Y, or X is directly proportional to 1/Y. The same relationship can be written XY = k, with the relevant nonzero-domain restriction retained.

To interpret a supplied equation, identify what stays constant and how one quantity responds to a change in the other. For X = 12/Y, doubling a positive Y halves X. The rule describes one fixed product, not any arbitrary downward trend.

Units and domain belong to the model. If X and Y are measured quantities, k has the units of their product. A division-based formula must not be evaluated at a zero denominator.

Worked example: Interpret X = 12/Y when Y changes from 2 to 4.

At Y = 2, X = 6.

At Y = 4, X = 3.

Compare the products XY.

Answer: X halves; the product remains 12.

Check your understanding: Is X = 12 − Y an inverse-proportion rule?

Answer: No.

Why: Its product XY is not constant.