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Inverse proportion

If two variables are inversely proportional to each other, it means that if one of the variables doubles, for example, the other variable will be halved.

In other words, the product of the two variables remains constant.

Importantly, if is inversely proportional to , then is also inversely proportional to . is equal to a constant divided by , and is equal to a constant divided by .

Expressing inverse proportion

Section titled “Expressing inverse proportion”
  • If is inversely proportional to , we can write this as:
  • Because is inversely proportional to , it must be equal to a constant divided by :
    • where is the constant of proportionality.
  • Similarly, because is inversely proportional to , it must be equal to a constant divided by :
    • where is the constant of proportionality in this case.

Proof

Section titled “Proof”

We can show that the inverse proportion can be expressed as using the rule stated above that is always constant in inverse proportion.

  • If is constant, we can call this constant :
  • Rearrange to make the subject:
  • This shows we can express inverse proportion as .

is inversely proportional to . Write as an equation in terms of and the constant of proportionality .

Section titled “ is inversely proportional to . Write as an equation in terms of and the constant of proportionality .”
  • Answer:

is inversely proportional to . Write as an equation in terms of and the constant of proportionality .

Section titled “ is inversely proportional to . Write as an equation in terms of and the constant of proportionality .”
  • Answer:

Finding the constant of proportionality

Section titled “Finding the constant of proportionality”

In a question, you will likely be given a pair of values for the two variables that are inversely proportional to each other. You can use these values to find the constant of proportionality - just by substituting them into the equation , if is inversely proportional to .

. When , . Find the constant of proportionality.

Section titled “. When , . Find the constant of proportionality.”
  • Substitute and :
  • Rearranging to find :
  • Answer:

. When , . Find the constant of proportionality.

Section titled “. When , . Find the constant of proportionality.”
  • Substitute and :
  • Rearranging to find :
  • Answer:

Using inverse proportion to find unknown values

Section titled “Using inverse proportion to find unknown values”

Once you have found the constant of proportionality, you can use it to find unknown values of either variable.

. The constant of proportionality is . Find when .

Section titled “. The constant of proportionality is . Find when .”
  • Substitute and :
  • Calculating :
  • Answer:

. When , . Find when .

Section titled “. When , . Find when .”
  • First, find the constant of proportionality:
    • Substitute and :
    • Rearranging to find :
  • Now, use to find when :
    • Substitute and :
    • Calculating :
  • Answer: