DIRECT VARIATION AND INVERSE VARIATION

About "Direct variation and inverse variation"

Direct variation and inverse variation is one of the important topics in school level math.

And there is no competitive exam without questions from direct and inverse variation. 

First let us come to know what is direct variation. 

Please look at the following situations.

Thus we can say, If an increase in one quantity produces a proportionate increase in another quantity, then the quantities are said to be in direct variation.  

or

If a decrease in one quantity produces a roportionate decrease in another quantity, then the quantities are said to be in direct variation.  

Change in both the quantities must be same. 

That is, 

Increase ---------------> Increase 

or

Decrease ---------------> Decrease 


Now, let us come to know what is inverse variation.

Please look at the following situations.

Thus we can say, If an increase in one quantity produces a proportionate decrease in another quantity, then the quantities are said to be in direct variation.  

or

If a decrease in one quantity produces proportionate increase in another quantity, then the quantities are said to be in direct variation.  

Change in the two quantities must be in different ways. 

That is, 

Increase ---------------> Decrease 

or

Decrease ---------------> Increase 

Direct variation and inverse variation - Shortcuts

To have better understanding on direct variation and inverse variation, let us look at some practice problems.

Problem 1 :

75 basketballs cost $1143.75. Find the cost of 26 basketballs 

Solution :

This is a situation of direct variation.

Because, less number of basket balls -----> cost will be less

Let "m" be the cost of 26 basket balls 

No. of basket balls 

75

26

Cost

1143.75

m

Since this is direct variation, we have to apply the shortcut "cross multiplication"

75 x m  =  26 x 1143.75

m  =  (26 x 1143.75) / 75

m  =  396.50

Hence, the cost of 26 basket balls is $ 396.50

Let us look at the next problem on "Direct variation and inverse variation"

Problem 2 :

7 men can complete a work in 52 days. In how many days will 13 men finish the same work? 

Solution :

This is a situation of inverse variation.

Because, more men -----> less days

Let "m" be the required no. of days. 

No. of men 

7

13

No. of days

52

m

Since this is inverse variation, we have to apply the shortcut "straight multiplication"

7 x 52  =  13 x m

(7 x 52) / 13  =  m

28  =  m

Hence, 13 men can complete the work in 28 days

Let us look at the next problem on "Direct variation and inverse variation"

Problem 3 :

If David sells 2 gallons of juice for $4, how much money will he get by selling 17 gallons of juice ?

Solution : 

This is a situation of direct variation.

Because, more gallons of juice -----> amount received will be more

Let "m" be the required amount of money

No. of gallons 

2

17

Value (in dollars) 

4

m

Since this is direct variation, we have to apply the shortcut "cross multiplication"

2 x m  =    17 x 4

m  =  (17 x 4) / 2

m  =  34

Hence, David will earn $34 by selling 17 gallons of juice

Let us look at the next problem on "Direct variation and inverse variation"

Problem 4 :

A book contains 120 pages and each page has 35 lines . How many pages will the book contain if every page has 24 lines per page?

Solution : 

This is a situation of inverse variation.

Because, less lines -----> more pages

Let "m" be the required number of pages

No. of pages 

120

m

No. of lines

35

24

Since this is inverse variation, we have to apply the shortcut "straight multiplication"

120 x 35  =  m x 24

(120 x 35) / 24  =  m

175  =  m

Hence, if every page has 24 lines per page, the book will contain 175 pages

Let us look at the next problem on "Direct variation and inverse variation"

Problem 5 :

The cost of a taxi is $40.50 for 15 miles. Find the cost for 20 miles. 

Solution : 

This is a situation of direct variation.

Because, more miles -----> more cost

Let "m" be the required cost

No. of miles 

15

20

Cost

40.50

m

Since this is direct variation, we have to apply the shortcut "cross multiplication"

15 x m  =  20 x 40.50

m  =  (20 x 40.50) / 15

m  =  54

Hence, the cost for 20 miles is $54

Let us look at the next problem on "Direct variation and inverse variation"

Problem 6 :

A truck covers a particular distance in 3 hours with the speed of 60 miles per hour. If the speed is increased by 30 miles per hour, find the time taken by the truck to cover the same distance

Solution : 

This is a situation of inverse variation.

Because, more speed -----> less time

If the given speed 60 mph is increased by 30 mph,

then the new speed = 90 mph 

Let "m" be the required time

No. of hours 

3

m

Speed

60

90

Since this is inverse variation, we have to apply the shortcut "straight multiplication"

3 x 60  =  m x 90

(3 x 60) / 90  =  m

2  =  m

Hence, if the speed is increased by 30 mph, time taken by the truck is 2 hours. 

Let us look at the next problem on "Direct variation and inverse variation"

Problem 7 : 

In a business, if A can earn $7500 in 2.5 years, At the same rate, find his earning for 4 years. 

Solution : 

This is a situation of direct variation.

Because, more time  -----> more earning  

No. of years 

2.5

4

Earning

7500

m

Since this is direct variation, we have to apply the shortcut "cross multiplication"

2.5 x m  =  4 x 7500

m  =  (4 x 7500) / 2.5

m  =  12000

Hence, the earning for 4 years is $12000

Let us look at the next problem on "Direct variation and inverse variation"

Problem 8 : 

David can complete a work in 6 days working 8 hours per day. If he works 3 hours per day, how many days will he take to complete the work ?

Solution : 

This is a situation of inverse variation.

Because, less hours per day-----> more days to complete the work

Let "m" be the required number of days.

No. of days

6

m

No. of hours 

8

3

Since this is inverse variation, we have to apply the shortcut "straight multiplication"

6 x 8  =  m x 3

(6 x 8) / 3  =  m

16  =  m

Hence, David can complete the work in 16 days working 3 hours per day.

Let us look at the next problem on "Direct variation and inverse variation"

Problem 9 :

In 36.5 weeks, Miguel raised $2,372.50 for cancer research. How much money will he raise 20 weeks ? 

Solution :

This is a situation of direct variation.

Because, less number of weeks ----> amount raised will be less

Let "m" be the required amount of money. 

Weeks

36.5

20

Amount of money

2372.50

m

Since this is direct variation, we have to apply the shortcut "cross multiplication"

36.5 x m  =  20 x 2372.50

m  =  (20 x 2372.50) / 36.5

m  =  1300

Hence, the money raised in 20 weeks is $1300

Let us look at the next problem on "Direct variation and inverse variation"

Problem 10 :

Alex  takes 15 days to reduce 30 kilograms of his weight by doing 30 minutes exercise per day. If he does exercise for 1 hour 30 minutes per day, how many days will he take to reduce the same weight ?

Solution :

This is a situation of inverse variation.

Because, more minutes per day----> less days to reduce the weight

1 hour 30 minutes per day -----> 90 minutes per day

Let "m" be the required number of days

No. of days

15

m

No. of minutes

30

90

Since this is inverse variation, we have to apply the shortcut "straight multiplication"

15 x 30  =  m x 90

(15 x 30) / 90  =  m

5  =  m

Hence, if Alex does exercise for 1 hour 30 minutes per day, it will take 5 days to reduce 30 kilograms of weight.

Let us look at the next problem on "Direct variation and inverse variation"

Problem 11 : 

Shanel gets 2/ 5 of a dollar for 1/7 hour of work.How much money does she get for 3 hours ? 

Solution : 

This is a situation of direct variation.

Because, more hours -----> more earning 

No. of hours

1/7

3

Dollars

2/5

m

Since this is direct variation, we have to apply the shortcut "cross multiplication"

(1/7) x m  =  3 x 2/5

m  =  7 x 6/5

m  =   42/5

m  =  8.4

Hence, Shanel gets $8.4 for 3 hours of work

Let us look at the next problem on "Direct variation and inverse variation"

Problem 12 : 

If 5 men can paint a house in 18 hours, how many men will be able to paint it in 10 hours ?

Solution : 

This is a situation of inverse variation.

Because, less hours  -----> more men 

Let "m" be the required number of men

No. of men

5

m

No. of hours

18

10

Since this is inverse variation, we have to apply the shortcut "straight multiplication"

5 x 18  =  m x 10

90 / 10  =  m

9  =  m

Hence, 9 men will be able to paint the house in 10 hours

Let us look at the next problem on "Direct variation and inverse variation"

Problem 13 : 

In a fort, 360 men have provisions for 21 days. If 60 more men join them, how long will the provision last ?

Solution : 

This is a situation of inversion  variation.

Because, more men -----> provision will last for less days

Given : 360 men -----> 21 days

If 60 more men join, total number of men  =  420

Let "m" be the required number of days.

No. of men

360

420

No. of days

21

m

Since this is inverse variation, we have to apply the shortcut "straight multiplication"

360 x 21  =  420 x m

(360 x 21) / 420  =  m

18  =  m

Hence, if 60 more men join, provision will last for 18 days

Let us look at the next problem on "Direct variation and inverse variation"

Problem 14 : 

John ordered 330 units of a product for $495. Then he reduced his order to 270 units. How much money does John have to pay for 270 units ?

Solution : 

This is a situation of direct variation.

Because, less units -----> less cost 

Let "m" be the required amount of money

No. of units

330

270

Money

495

m

Since this is direct variation, we have to apply the shortcut "cross multiplication"

330 x m  =  270 x 495

m  =  (270 x 495) / 330

m  =  405

Hence, John has to pay $405 for 270 units 

Let us look at the next problem on "Direct variation and inverse variation"

Problem 15 :

A man can type 9 pages of a book everyday and completes it in 50 days. How many days will he take to complete it, if he types 15 pages everyday ? 

Solution : 

This is a situation of inverse variation.

Because, more pages per day-----> less days to complete the book

Let "m" be the required number of days

No. of days

50

m

No. of pages

9

15

Since this is inverse variation, we have to apply the shortcut "straight multiplication"

50 x 9  =  m x 15

450 / 15  =  m

30  =  m

Hence, the man will complete the book in 30 days, if he types 15 pages per day. 

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