Need for Concepts of Variation
To find out the quantity of each item needed by Mohan or, the time taken by five students to complete the job, we need study some concepts of variation.
We will study the following types of variation:
- Direct variation
- Inverse variation
Direct proportion
Two quantities x and y are said to be in direct proportion if whenever the value of x increases (or decreases), then the value of y increases (or decreases) in such a way that the ratio xy remams constant.
When x and y are in direct proportions, we have:
x1y1=x2y2=x3y3
Inverse proportion
Two quantities x and y are said to be in inverse proportion, if whenever the value of x increases (or decreases), then the value of y decreases (or increases) in such a way that xy remains constant.
When x and y are in inverse proportion, then
x1 × y1 = x2 × y2 = x3 × y3, and so on.
Two quantities may be linked in two ways:
- Both increase or decrease together proportionally.
- If one increases, the other decreases proportionally and vice-versa.
The first way is named as direct variation whereas the second way is named as an inverse variation.
Direct Proportion
If two quantities are related in such a way that an increase in one leads to a corresponding proportional increase in the other, then such a variation is called direct variation.
Thus, two numbers x and y are said to vary in direct proportion if xy = k or x = ky where k is a constant.
In this case, if y1, y2 are the values ofy corresponding to the values x1, x2 of x respectively, then
x1y1=x2y2
Inverse Proportion
If two quantities are related in such a way that in increase in one quantity leads to a corresponding proportional decrease in the other and vice-versa, then such a variation is called inverse proportion.
Thus, two quantities x and y are said to vary in inverse proportion if xy = k where k is a constant of proportionality.
In this case, if y1, y2 are the values of y corresponding to the values x1, x2 of x respectively, then
x1y1 = x2y2 or x1x2=y2y1