Direct variation is a special relationship between two variables. Suppose you have a recipe for a salad giving the measures of various ingredients required, which can serve four people. If you want to prepare the salad for a small party consisting of eight people, how would you alter the measurements of the ingredients for the preparation? Yes, you are correct! The number of people to be served 8 is twice the number servings for which the recipe is given. You will use twice the quantity given for each ingredient, as you have to prepare twice the amount of salad for which the recipe is given. If the measure of cooked mushrooms is given in the recipe as two cups you would use four cups of cooked mushrooms for your preparation. We increase the measurements proportionate to the servings required.
The relationship between the number of servings required 's' and the number of cups of cooked mushrooms used in the salad "c" as s = 2c.This relationship is known as direct variation as the variable s vary the same way as c. For any set of values for s and c, the ratio $\frac{s}{c}$ is a constant and equal to 2.

Direct variation is a special relationship between two variables if they vary in such a way, that the ratio of the two variables is always a constant.

The two variables so involved in the direct variation, increase or decrease by the same factor. The relationship between the variables is represented by a linear equation.
The relationship between the two variables in direct variation can be expressed as a linear equation of the form y = kx where k ≠ 0. This equation is also known as the direct variation equation. This equation is used in the test for direct variation. If any given equation involving two variables can be reduced or rewritten in this form, we conclude the two variables vary directly and otherwise not.
The constant k in the equation is called the constant of variation.For example if we have an equation y = 4x relating the variables x and y, then we say the variables x and y vary directly and the constant of variation is 4.

The linear equation representing a direct variation will not contain a constant term.
Often it is required to find the constant of variation k for the two variables x and y in direct variation. The direct variation equation y = kx is solved for k to get the formula for finding the constant of variation 'k'.
k = $\frac{y}{x}$

Example:
Two variables x and y vary directly. The value of y is 14 when x = 20. Find the constant of variation.

Using the formula, k = $\frac{y}{x}$ = $\frac{14}{20}$ = $\frac{7}{10}$ = 0.7
The graph of the equation y = kx representing a direct variation is straight line with slope = k. You may note that the y intercept of the equation is zero, hence the graph passes through the origin.

Graph Model

The graph of direct variation is done using 'k' the slope of the line. 
  1. Mark the origin (0, 0) as one of the points.
  2. Mark the second point on the line using the value of slope = k
  3. Join the two points to get the line.

The graphs representing the direct variations y = $\frac{2}{3}$ x and y = -4x are shown below:

Positive Slope
      
Negative Slope


Solved Examples

Question 1: Determine whether the given equations represent a direct variation. If yes find the constant of variation.
  1. 2x - 3y = 0         
  2. 4x - 3y = 12

Solution:
 
1. 2x -3y = 0
        -3y = -2x
          y = $\frac{2}{3}$x    
   The equation is solved for y is of the form y = kx. Hence the equation represents a direct variation and the constant of variation k = $\frac{2}{3}$.

2.  4x - 3y = 12
         - 3y = -4x + 12
            y = $\frac{4}{3}$ x - 4.
     The equation solved for y, contains a constant term -4 on the right side. Hence the equation does not represent a direct variation.
 

Question 2: For the data given in the table below does y vary directly with x ? If yes, find the equation of direct variation.

  X 
 Y 
  1
0.25
 - 8  -2
 9.6  2.4


Solution:
 
Let us find the ration $\frac{y}{x}$ for each set of values.

$\frac{Y}{X}$  = $\frac{0.25}{1}$ = 0.25

$\frac{Y}{X}$ = $\frac{-2}{-8}$ = 0.25

$\frac{Y}{X}$ = $\frac{2.4}{9.6}$ = 0.25

The ratio is same for all the three sets of values.  Hence the data values represent a direct variation with k = 0.25 or $\frac{1}{4}$ The equation of the direct variation is hence y = $\frac{1}{4}$ x or y = 0.25x
 


Solved Examples

Question 1: The waiters at a restaurant are tipped 15% of the Bill amount. Does it mean that the tip amount 't' varies directly with the bill amount 'b'? If so write a direct variation equation for the situation and find the constant of variation. Find also the tip paid if the bill amount is 24 dollars.
Solution:
 
If the tip paid is 15% of the bill amount t = 15% of b = 0.15 b.
    This equation represents a direct variation with k = 0.15
    When b = 24 dollars
     t = 0.15 x 24 = 3.60.
    Tip paid = 3.60 dollars.
 

Question 2: When you drive at a constant speed, the distance traveled 'd' varies directly with the time spent on driving 't'. Write an equation relating d and t. What is the constant of variation. If Mary drives at a constant speed of 55 Km per hour, find the distance traveled by her in 4 hrs.
Solution:
 
The equation representing the direct variation between distance and time  is d = st where 's' is the constant speed.
    The constant of variation is therefore the constant speed.
    When s = 55, the direct variation equation is written as d = 55t
    If time driven t = 4 hrs, distance traveled = 55 x 4 = 220 Km.