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
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.
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.
How to solve an inverse variation problem with a change of variables?
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.
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.
How to solve direct variation and indirect/inverse variation word problems?
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 or directly?
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
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