There are many situations in daily living, science, engineering and industry where the value of one quantity depends upon that of another.  Those who encounter these situations are constantly in search of relationships between quantities which can be expressed as formulas or equations.  Relationships between changing variables can be ascertained through inspection or by using a model for the data which illustrates how the change in one variable affects a second variable.

Direct Variation

Making a table is one way to visualize a relationship between two variables.

From the table we can see that the variable y is always 4 times the variable x.  A formula for this relationship is y = 4 x or y/x = 4 or x/y = 1/4. The variable y is said to vary directly with x when the ratio of y to x is constant.

Definition:

Direct variation can be expressed as a linear equation.  The graph of a direct variation is shown below.

Direct Variation

The table, equation and graph show that y increases when x increases and that y increases four times as fast as x.

Examples of Direct Variation

1. The circumference of a circle varies directly as the diameter, C = kD
2. The distance an object will fall varies directly as the square of the time, d = kt²
3. The volume of a gas in a container at a constant pressure varies directly as the absolute temp, V = kT

Example 1

Write the equation for the following conditions: y varies directly with x and when x = 12, y = 36. 

Solution: The equation in most general form is y = kx.  Replace x with 12 and y with 36 to find the value of k: 36= 12k.  Solving for k, we get k = 3 and the equation becomes y = 3x.

Example 2

Suppose that y varies directly with x. When x = 10, y = 25. Find y when x = 6.

Solution: Use the first conditions to find the constant of variation in y = kx. We substitute known values to get 25 = k (10). So k = 25/10 = 5/2 = 2.5. Now replace k and the value given for x in the equation to obtain y = 2.5 (6) = 15.

Indirect (Inverse) Variation

Examine the table below.

The table shows pairs of variables x and y which are related by the equation xy = 28.  

Note that y decreases as x increases and the product is constant.  Since xy = 28, x = 28/y, or y = 28/x. In this case y varies indirectly or inversely with x or x varies indirectly with y.

Definition

Symbolically, y varies indirectly with x when xy = k or y = k/x.   Graphically an indirect variation is a curve, not a straight line as in the direct variation.  Refer to Figure 2.

Fig. 2  Indirect Variation

Examples of Indirect Variation

1. The volume of a gas in a container with a constant temperature varies indirectly with the pressure, V P = k
2. When a constant force is applied to a massive object the acceleration experienced by the object varies indirectly with the mass, m a = k
3. To travel a certain distance at a constant speed, the speed varies indirectly with the time it takes to make the trip, s t = k
4. The severity of an itch varies indirectly with the ability to reach it, S R = k

Example 3

Find the constant of variation if y varies indirectly with x and y = 30 when x = 5. 

Solution: The most general equation is xy = k.  Substitute x and y to get k:  5(30) = k, and k = 150.  The equation for the given conditions is  xy = 150.

Example 4

Suppose that h varies indirectly with g.  When g = 3, h = 12.  Find h when g = 15.

Solution: First, we find the constant of variation. The equation is hg = k, so (12)(3) = k = 36.  Now substitue for g and k to get h(15) = 36. The solution is h = 36/15 = 2.4.