Inverse Variation
Contents
Summary
When we say that a variable varies inversely as another variable, or is inversely proportionate to another variable, we mean that when a variable takes an -fold increase, then the other variable decreases by times. This is also one of the representative types of correlation, along with the direct variation.
Suppose that Carlos is practicing his 100-meter sprints. If he runs at a speed of 5 meters per second, it will take him seconds to complete the sprint. If he runs at 10 meters per second, then he will finish the sprint in seconds. Observe that if he runs two times faster, then it takes him half the time to complete the sprint.
This kind of correlation can be represented as a hyperbolic function whose equation is The two variables we are considering are and while is the constant of variation. If we let be the time (in seconds) it takes Carlos to finish the sprint, and be the speed at which Carlos runs, then we can set the equation as where the constant of variation is 100 meters.
Note that the equation can be rewritten as which implies that the product of and is always equal to
In summary, an inverse variation has the following characteristics:
- It can be described by the hyperbolic equation
- If is increased by times, then experiences an -fold decrease.
- If is decreased by times, then experiences an -fold increase.
- The product of the two variables is always equal to
Example Problems
If varies inversely as and when then what is the equation that describes this inverse variation?
If varies inversely as then there is the following relationship between and where is the non-zero constant of variation. For this problem, we have Therefore, the equation is
If varies inversely as and the constant of variation is what is when
From the equation of the inverse variation we have
Suppose that is inversely proportional to and that when What is when
From the equation of the inverse variation we have
Then the equation of the inverse variation is which implies
There is a job that men can do in days. How many days will it take if men do the same job?
As the man power increases, the number of days needed to complete the same job decreases, implying this is an inverse variation.
Let be the number of men workers and let be the number of days needed to complete the work. Then where is the constant of inverse variation. Thus, we set the following equation to obtain Therefore, our answer is days.