Direct and Inverse Proportions
- Variations: If the values of two quantities depend on each other in such a way that a change in one causes corresponding change in the other, then the two quantities are said to be in variation.
- Direct Variation or Direct Proportion:
Two quantities x and y are said to be in direct proportion if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant. That is if=k [k is a positive number, then x and y are said to vary directly. In such a case if y1, y2 are the values of y corresponding to the values x1, x of x respectively then
=
.
- If the number of articles purchased increases, the total cost also increases.
- More than money deposited in a bank more is the interest earned.
- Quantities increasing or decreasing together need not always be in direct proportion, same in the case of inverse proportion.
- When two quantities x and y are in direct proportion (or very directly), they are written as.
Symbol
stands for ‘is proportion to’.
- Inverse Proportion: Two quantities x and y are said to be in inverse proportion if an increase in x causes a proportional decrease in y (and vice-versa) in such a manner that the product of their corresponding values remains constant. That is, if xy = k, then x and y are said to vary inversely. In this case, if y1, y2 are the values of y corresponding to the values x1, x2 of x respectively then x1, Y1 = x2, y2 or
- When two quantities x and y are in inverse proportion (or vary inversely), they are written as x
. Example: If the number of workers increases, time taken to finish the job decreases. Or If the speed will increase the time required to cover a given distance will decrease.
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
When x and y are in direct proportions, we have:
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.