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    Word Problems: Inverse Variation

    Inverse variation type word problems encompass one variable, which is inverse proportional to another variable:

    y = k/x,

    where k is referred to as the constant of proportionality.

    Direct variation word problems often contain verbiage such as:
    • "varies inversely as"
    • " varied inversely as"
    • "inversely proporation to"
    Similar to
    direct variation word problems, those dealing with inverse variation may be solved using the following steps:

    1) Recognizing the word problem consists of inverse variation as exemplified by presence of verbiage listed above;

    2) Writing an equation containing known values for both variables resulting in an expression containing only the unknown representing the constant of proportionality;

    3) Solving the equation for the constant of proportionality;

    4) Using the calculated constant of proportionality to determine the value of one of the variables given the other.


    Bob's dentist determined the number of cavities developed in his patient's mouth each year is inversely proportional to the total number of minutes spent brushing during each session.  If Bob developed four cavities during the year he spent only 30 seconds brushing his teeth each time, how many annual cavities will Bob develop if he increases his brushing time to two minutes per session?


    Step 1:

    The problem may be recognized as relating to inverse variation due to the presence of the verbiage "is inversely proportional to";

    Step 2:

    = Number of Cavities Developed Each Year
    x = Number of Minutes Spent Brushing Per Session
    k = Constant of Proporationality

    y = k/x

    Knowing four cavities developed during year when brushing sessions were 30 seconds ( = 0.5 minute): 

    4 = k/(0.5)

    Step 3:

    k/(0.5) = 4

    k = 4(0.5)

    k = 2

    Step 4:

    When x = 2 minutes, what is the number of cavities, y, using k = 20?

    y = (2)/(2)

    y = 1 cavity