DIRECT PROPORTION AND INVERSE PROPORTION

Direct Proportion

If an increase in one quantity produces a proportionate increase in another quantity, then the two quantities are directly proportional to each other

or

If a decrease in one quantity produces a proportionate decrease in another quantity, then the two quantities are directly proportional to each other

Change in both the quantities must be same. 

That is, 

Increase ------> Increase 

or

Decrease ------> Decrease 

If 'y' is directly proportional to 'x', then 

y  =  kx

where 'k' is the constant of proportionality.

Inverse Proportion

If an increase in one quantity produces a proportionate  decrease in another quantity, then two quantities are inversely proportional to each other.   

or

If a decrease in one quantity produces proportionate increase in another quantity, then two quantities are inversely proportional to each other.  

Changes in the quantities must be opposite. 

That is, 

Increase ------> Decrease 

or

Decrease ------> Increase 

If 'y' is inversely proportional to 'x', then 

y  =  k/x

where 'k' is the constant of proportionality.

Solved Problems

Problem 1 :

y is directly proportional to x. Given that y = 144 and x = 12. Find the value of y when  x = 7.

Solution :

y directly proportional to x. 

Then, 

y  =  kx 

Substitute y = 144 and x = 12. 

144  =  12k 

12  =  k 

Then,

y  =  12x

Substitute x = 7. 

y  =  12(7)

y  =  84

Problem 2 :

y is inversely proportional to x. Given that y = 12 and x = 6. Find the value of y when  x = 8.

Solution :

y inversely proportional to x.

Then, 

y  =  k / x 

Substitute y = 12 and x = 6. 

12  =  k / 6

12 x 6  =  k 

72  =  k

Then,

y  =  72 / x

Substitute x = 8.

y  =  72 / 8

y  =  9

Problem 3 :

75 basketballs cost $1,143.75. Find the cost of 26 basketballs.

Solution :

This is a situation of direct proportion.

Because,

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

Let x be the no. of basket balls and y be the cost. 

Since this is direct proportion, we have 

y  =  kx

Substitute x = 75 and y = 1143.75.

1143.75  =  75k

15.25  =  k 

Then, 

y  =  15.25x

Substitute x = 26.

y  =  15.25(26)

y  =  396.50

So, the cost of 2626 basketballs is $396.50.

Problem 4 :

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 proportion.

Because,

more men -----> less days

Let x be the no. of men and y be the no. of days. 

Because this is inverse proportion, we have 

y  =  k / x

Substitute x = 7 and y = 52.

52  =  k / 7

364  =  k 

Then,

y  =  364 / x

Substitute x = 13.

y  =  364 / 13

y  =  28

So, 13 men will finish the same work in 28 days.

Problem 5 :

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 proportion.

Because,

more gallons of juice -----> more amount of money

Let x be the no. of gallons of juice and y be the cost.

Since this is direct proportion, we have 

y  =  kx

Substitute x = 2 and y = 4.

4  =  2k

2  =  k 

Therefore,  y  =  2x

Plug  x  =  17,

y  =  2(17)

y  =  34

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

Problem 6 :

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 proportion.

Because,

less lines -----> more pages

Let x be the no. of pages and y be the no. of lines.

Since this is inverse proportion, we have 

y  =  k / x

Substitute x = 120 and y = 35.

35  =  k / 120

4200  =  k 

Then, 

y  =  4200 / x

Substitute y = 24.

24  =  4200 / x

x  =  4200 / 24

x  =  175

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

Problem 7 :

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

Solution : 

This is a situation of direct proportion.

Because,

more miles -----> more cost

Let x be the no. of miles and y be the cost.

Since this is direct proportion, we have 

y  =  kx

Substitute x = 15 and y = 40.50.

40.50  =  15k

2.7  =  k 

Then,

y  =  2.7x

Substitute x = 20.

y  =  2.7(20)

y  =  54

So, the cost for 20 miles is $54.

Problem 8 :

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 proportion.

Because,

more speed -----> less time

Let x be the hours and y be the speed.

Since this is inverse proportion, we have 

y  =  k / x

Substitute x = 3 and y = 60.

60  =  k / 3

180  =  k

Then,  

y  =  180 / x

If the given speed 60 mph is increased by 30 mph, then the new speed is 90 mph.  

So, we have to find x when y = 90. 

Substitute y = 90,

90  =  180 / x

x  =  180 / 90

x  =  2

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

Problem 9 : 

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 proportion

Because, 

more time  -----> more earning  

Let x be the years and y be the earning. 

Since this is direct proportion, we have 

y  =  kx

Substitute x = 2.5 and y = 7500.

7500  = 2.5k

3000  =  k 

Then,

y  =  3000x

Substitute x = 4.

y  =  3000(4)

y  =  12000

So, the earning for 4 years is $12000.

Problem 10 : 

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 proportion.

Because,

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

Let x be the days and y be the hours.

Since this is inverse proportion, we have 

y  =  k / x

Substitute x = 6 and y = 8.

8  = k / 6

48  =  k 

Then, 

y  =  48 / x

Substitute x = 3.

y  =  48 / 3

y  =  16

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

Problem 11 :

In 36.5 weeks, Miguel raised $2372.50 for cancer research. How much money will he raise 20 weeks ? 

Solution :

This is a situation of direct proportion.

Because,

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

Let x be the weeks and y be the amount of money raised.

Since this is direct proportion, we have 

y  =  kx

Substitute x = 36.5 and y = 2372.50.

2372.50  = 36.5k

2372.50 / 36.5  =  k 

65  =  k

Then, 

y  =  65x

Substitute x = 20. 

y  =  65(20)

y  =  1300

So, the money raised in 20 weeks is $1300.

Problem 12 :

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 proportion.

Because,

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

Let x be the minutes and y be the days.

Since this is inverse proportion, we have 

y  =  k / x

Substitute x = 30 and y = 15.

15  = k / 30

450  =  k 

Then,

y  =  450 / x

1 hour 30 minutes  =  90 minutes 

So, we have to find y when x = 90. 

Substitute x = 90.

y  =  450 / 90

y  =  5

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

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