In these lessons, we will learn about inverse variation and how to solve applications that involve inverse variation.
Direct Variation Word Problems
, More Algebra Word Problems
What is Inverse Variation?
There are many situations in our daily lives that involve inverse variation
For example, the number of days required to build a bridge is varies inversely to the number of workers. As the number of workers increases, the number of days required to build would decrease.
In general, when two variables x
are such that
is a non-zero constant, we say that y varies inversely
In notation, inverse variation is written as
Suppose that y varies inversely as x and that y = 8 when x = 3.
a) Form an equation connecting x and y.
b) Calculate the value of y when x = 10.
i.e. xy = k where k is a non-zero constant
a) Substitute x = 3 and y = 8 into the equation to obtain k
3 × 8 = k ⇒ k = 24
The equation is xy = 24
b) When x = 10, 10 × y = 24 ⇒ y =
Suppose that y varies inversely as x 2 and that y = 10 when x = .
a) Find the equation connecting x and y .
b) Find the value of y when x = 3.
i.e. yx2 = k
a) Substitute x = and y = 10 into the equation to obtain k
The equation is yx2 =
b) When x = 3,
How to define inverse variation and how to solve inverse variation problems?
1. y varies inversely as x. y = 4 when x = 2. Determine the inverse variation equation. Then determine y when x = 16.
2. The time, t, required to empty a tank varies inversely as the rate, r, of pumping. If a pump can empty a tank in 2.5 hours at a rate of 400 gallons per minute, how long will it take to empty a tank at 500 gallons per minute?
3. The force, F, needed to break a board varies inversely with the length, L, of the board. If it takes 24 pounds of pressure to break a board 2 feet long, how many pounds of pressure would it take to break a board that is 5 feet long?
4. y varies inversely as the square root of x. y = 6 when x = 16. Determine the inverse variation equation. Then determine y when x = 4.
This video provides an example of how to solve a basic inverse variation problem.
Example: y varies inversely as x. y = 3 when x = 10. Determine the inverse variation equation. Then determine y when x = 6.
Example: Given m varies inversely as t, and m = 9 when t = 6, find the variation constant and the inverse variation equation. Then determine m when t = 27.
This video provides an example of how to solve a inverse variation problem when k is a fraction
Example: y varies inversely as x. y = 1/2 when x = 2/3. Find the variation constant and the inverse variation equation. Then determine y when x = 2/15.
Some examples of Inverse Variation Applications
The following video gives some practical examples of direct variation and indirect/inverse variation.
Example: On a string instrument, the length of a string varies inversely as the frequency of its vibrations. An 11-inch string has a frequency of 400 cycles per second. Find the constant of proportionality and the frequency of a 10-inch string.
How to tell if two variables vary inversely?
Recognizing Direct and Inverse Variation
This video looks at inverse variation: identifying inverse variations from ordered pairs, writing inverse variation equations, graphing inverse variations, and finding missing values.
Example: Let x1
= 4, y1
= 12 and x2
= 3. Let y vary inversely as x. Find y2
You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.
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