Direct and Inverse Proportions | Study

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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. (Scroll down till end of the page)

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Direct Variation or Direct Proportion:

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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, y₂are 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 vary 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, y₂are the values of y

corresponding to the values x₁, x₂of x respectively then

^x1^{, Y}1 = ^x2^{, y}2 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 decreases.

1. Distance and Time

Time (hours)	Distance (km)	Formula
0	0	$d = 60 \times t$ $\Rightarrow d = 60 \times t$
1	60
2	120
3	180

2. Cost of Groceries

Number of Apples	Total Cost ($)	Formula
0	0	$C = 2n ⇒$
1	2
2	4
5	10

3. Cooking Ingredients

Number of Servings	Amount of Flour (cups)	Formula
0	0	$F = 0.5 s$
4	2
6	3
8	4

4. Speed and Fuel Consumption

Distance (km)	Fuel Consumed (liters)	Formula
0	0	$F = \frac{d}{15}$
15	1
30	2
45	3

5. Salary and Hours Worked

Hours Worked	Total Earnings ($)	Formula
0	0	$E = 15 h$
1	15
5	75
10	150

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