Question

Population of the country and
are of land per person (indirect variation and which are inverse variation ​

Answer 1

Step-by-step explanation:

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Direct and Indirect Variation

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PROBLEMS ON DIRECT AND INVERSE VARIATION - EXAMPLE

Direct Variation:

If 3 balls weigh 24 kg. What is the weight of 5 balls?

The number of balls and weight are in direct variation.

Let the weight of 5 balls be x.

So,

24

3

=

x

5

x=

3

5×24

x=40 kg

Inverse variation:

If 3 people complete a task in 24 days. Find the number of days taken by 8 people to complete the same task.

Number of people and time are in inverse variation.

3×24=8×d

d=9

8 people can complete the same task in 9 days.

INVERSE VARIATION - DEFINITION

x 1 2 3 4

y 12 6 4 3

Observe the given table, we have two quantities denoted by x and y.

As we move from left to right the value of x increases from 1 to 4. However, the value of y decreases from 12 to 3.

Also, the variation in y is proportional to the variation in x

Such a relation, where value of one variable increases,as the value of other decreases,is known as inverse variation.

Inverse variation between y and x is denoted by,

yα

x

1

and read as y varies inversely as x

Example:

some workers complete a task in in a week. If we increase the number of workers the time to complete the task decreases.

Thus there exist an inverse variation between number of workers and time taken to complete a task.

DIRECT VARIATION - DEFINITION

x 1 2 3 4

y 5 10 15 20

OBSERVE THE GIVEN TABLE.

WE HAVE TWO QUANTITIES DENOTED BY VARIABLE X AND Y.

THE VALUE OF X INCREASES FROM 1 TO 4. AS WE MOVE TOWARDS RIGHT.

THE VALUE OF Y ALSO INCREASES FROM 5 TO 20 AS WE MOVE TOWARDS RIGHT

SUCH A RELATION IN WHICH THE VALUE OF ONE VARIABLE INCREASES WITH THE INCREASE IN VALUE OF OTHER VARIABLE,IS KNOWN AS DIRECT VARIATION.

WE REPRESENT DIRECT VARIATION BETWEEN VARIABLE X AND Y MATHEMATICALLY AS, YΑX.

IF YΑX,

THEN ,Y=KX

WHERE K IS KNOWN AS CONSTANT OF PROPORTIONALITY (OR VARIATION)

EXAMPLE:

A RECIPE FOR 6 CUPCAKES NEEDS 1 CUP OF FLOUR.THE NUMBER OF CUPCAKES YOU CAN MAKE VARIES DIRECTLY WITH THE AMOUNT OF FLOUR ,WHICH MEANS FOR MAKING MORE CUP CAKES WE REQUIRE MORE FLOUR.HOW MANY CUPCAKES CAN YOU MAKE WITH 4 CUPS OF FLOUR?

SOLUTION:LETX = AMOUNT OF FLOUR AND Y = NUMBER OF CUPCAKES

SINCE YΑX,

Y=KX . . . (1)

FOR 6 CUP CAKES WE REQUIRE 1 CUP OF FLOUR.

∴ FOR X=6, Y=1

PUT THE ABOVE VALUES IN EQ (1)

6=K×1

6=K

∴K=6

∴Y=6X . . . (2)

NOW WE HAVE TO FIND THE NUMBER CUP CAKES THAT CAN BE MADE IN FOUR CUPS OF FLOUR.

WHEN X=4 Y=?

USING EQ (2)

Y=6X=6×4

∴Y=24

SO YOU CAN MAKE 24 CUPCAKES IN 4 CUPS OF FLOUR.

Answer 2

Answer:

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Step-by-step explanation:

i) We know that more is the number of workers to do a job, less is the time taken to finish the job.

So, it is a case of inverse proportion.

(ii) Considering uniform speed, as the distance to be travelled increases, the time taken to cover it will also increase.

So, this is a case of direct variation.

(iii) Clearly, more is the area of cultivated land, more is the crop harvested.

So, it is a case of direct proportion.

(iv) We know that the more will be the speed of the vehicle, the distance will be covered in lesser time.

So, it is a case of inverse proportion.

(v) Clearly, more is the population, less is the area of land per person in a country.

So, it is a case of inverse proportion.