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so I am looking for a word that would best describe such a situation as; the popularity of A decreased because the popularity of B increased.

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    It's called inverse proportionality, but perhaps someone will come up with something snappier,
    – Andrew Leach
    Feb 8, 2021 at 21:52
  • No, you got it. Stick with A is inversely proportional to B, B is inversely proportional to A, A and B are in inverse proportion, and A and B are inverses for short. Feb 8, 2021 at 21:55
  • If you are looking for something less rigidly mathematical you could also say that "The popularity of B increased at the expense of that of A". This has the advantage of not requiring an exact equivalence, for instance there might be people who used to buy A all the time but now buy B some of the time and even buy C on occasions.
    – BoldBen
    Feb 9, 2021 at 5:35

2 Answers 2

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There are two ways: the second is a restricted sense of the first. It is convenient to illustrate the point with numerical examples.

First: they are certainly inversely related in the loose sense that one goes up as the other goes down.

For example, 48->24->6 (successive decreases of 50% and 75%) in one leads to 24->30->60 (successive increases of 25% and 100%) in the other is simply an inverse relationship.

Second: they may also be inversely proportional if the relative decrease in one is in constant proportion to the relative increase in the other.

For example, 20->10->5 (successive decreases always 50%) in one leads to 7->14->28 (successive increase always 100%) in the other is an inversely proportional relationship. Note that this equates to the reciprocity mentioned by Jim Mack in his answer.

This mathematical use of reciprocal follows is justified from the limited definition:

reciprocal = a number that, when multiplied by another number, results in 1

Cambridge

In this case the product of the two measures of popularity (however it is measured) would be constant (not necessarily 1). Because you give no feeling for what the measure is, and because it seems unlikely to be reliably numeric, this strict mathematical meaning is unlikely (although not impossible) to pertain to your question.

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In addition to the good comments suggesting inverse proportionality, consider that the two may be reciprocals in the mathematical sense given at TFD:

  1. Mathematics A number related to another in such a way that when multiplied together their product is 1. For example, the reciprocal of 7 is 1/7 ; the reciprocal of 2/3 is 3/2 .
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    This is something much more specific than what the question is about. The OP said the popularity of A decreased (by some unspecified amount) because the popularity of B increased (by some unspecified amount); the question does not say that the amount of the increase and the amount of the decrease stand in any particular relationship, nor that the total amounts after the change stand in any particular relationship.
    – jsw29
    Feb 8, 2021 at 22:24
  • @jsw29 I regret disagreeing but the use of "may" in the answer justifies what follows. The illustrative numbers are merely exemplary. The answer has a speculative generality that seems to apply quite well to the question.
    – Anton
    Feb 8, 2021 at 22:41
  • @Anton, the last sentence of your own answer seems to imply that you agree rather than disagree.
    – jsw29
    Feb 8, 2021 at 22:44
  • @jsw29 - Just adding another possibility to the mix. In addition to the strict mathematical usage, a reciprocal relationship colloquially implies that A and B vary in... inverse proportion.
    – Jim Mack
    Feb 8, 2021 at 23:32

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