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Direct and Inverse Proportions

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Direct and Inverse Proportions help us to understand how quantities are dependent on each other. Let’s say if you drive faster you will reach your destination in less time, similarly if a laborer works for more hours he will earn more. Here we see that speed and time are in opposite relation and hence are in inverse proportion while wage and working hours are in direct proportion. Direct and Inverse Proportion is a very important topic for class 8 to understand ratios and proportions.

Thus this article on Direct and Inverse Proportions is helpful in understanding the dependency of quantities on each other. In this article, we will explore the direct and inverse proportions in detail including its examples and solved problems as well. We will also consider many real-life scenarios to use direct and inverse proportions.

Direct and Inverse Proportions Definition

The proportion is said to be a direct proportion between two values when one is a multiple of the other while The value is said to be inversely proportional when one value increases, and the other decreases.

Direct Proportion

If x and y are any two quantities such that both of them increase or decrease together and x/y remains constant (say k), then we say that x and y are in Direct Proportion. This is written as x ∝ y and read as x is directly proportional to y.

Direct Proportion Formula

x ∝ y

(x/y) = k ⇒ x = ky 

Where k is constant of proportion.

 Similarly, if y1 and y2 are the values of y corresponding to the values of x1 and x2 of x respectively, then

\bold{\frac{x_1}{y_1} = \frac{x_2}{y_2} = k}

OR

\bold{x_1y_2 = x_2 y_1 = \text{Constant}}

Examples of Direct Proportion

Example of Direct Proportion

On the occasion of the School Anniversary, the Head Master of the school decided to take up a plantation of saplings. The number of students in each class is given below in the form of a table.

Class

VI

VII

VIII

IX

X

Number of Students

7

10

11

14

17

Each student has to plant two saplings. Find the number of saplings needed for the plantation for each class.

Class

VI

VII

VIII

IX

X

Number of Students

7

10

11

14

17

Number of Saplings required

14

20

22

28

34

What can you say regarding the number of saplings required? What kind of change do you observe in the number of students and number of saplings required? Either both increase or decrease.

 \bold{\frac{ \text{Number of saplings required}}{ \text{Number of Students}} = \frac{14}{7}= \frac{20}{10}= \frac{22}{11} \cdots = \frac{2}{1}}

Now, let’s help our friend Geethika in finding the total number of cups of water required to cook 6 cups of rice. As mentioned earlier, Geethika uses 4 cups of water to cook 2 cups of rice.

Number of cups of rice 

2

6

Number of cups of water

4

12

Here, it can be observed that the number of cups of water required increases with an increase in the number of cups of rice.

 \bold{\frac{ \text{ Number of cups of water}}{ \text{Number of cups of rice }} = \frac{4}{2}= \frac{12}{6}= \frac{2}{1}}

In this case, 2 is the constant of proportion. Here, the two quantities number of cups of water and the number of cups of rice are said to be in direct proportion to each other i.e.,

Number of cups of water ∝ Number of cups of rice

Solved Problems of Direct Proportion

Problem 1: A vertical pole of 10 m height casts a 20 m long shadow. Find the height of another pole that casts an 80m long shadow under similar conditions.

Solution: 

The length of the Shadow is directly proportional to the height of the pole.

Height of Pole

10

?

Length of Shadow

20

80

So, (x1 / y1) = (x2 / y2). Here, x1 = 10m   y1 = 20m   x2 = ?   and y2 = 80m.

Upon substituting the values,

(10 / 20) = (x2 / 80)

x2 = (10 x 80) / 20

x2 = 40m

Therefore, the height of another pole is x2 = 40m.

Problem 2: If the cost of 50m of cloth is Rs. 1500, then what will be the cost of 10m of that cloth?

Solution:

The cost of the cloth is directly proportional to the length of the cloth.

Length of cloth

50m

10m

Cost of cloth

₹1500

?

So, (x1/y1) = (x2 / y2). 

Here, x1 = 50m, y1 = Rs.1500, x2 = 10m, y2 = ?

Upon substituting the values,

(50/500) = (10/y2)

⇒ y2 = (10×1500)/50

⇒ y2 = 300

Therefore, the cost of 10m cloth is Rs.300.

Problem 3: Following are the vehicle parking charges near a Bus Station.

Number of Hours 

(x)

Parking Charges

(y)

up-to 4 hours

Rs.40

up-to 8 hours

Rs.80

up-to 12 hours

Rs.120

up-to 24 hours

Rs.240

Check if the parking charges and parking hours are in direct proportion.

Solution:

We can observe that the parking charges (y) increase with the increase in the number of hours (x). Let’s calculate the value of (x / y). If it is a  constant, then they are in direct proportion. Otherwise, they are not in direct proportion.

   x /y = 4/40 = 8/80 = 12/120  = 24/240 = 1/10

Here, (1/10) is constant and is called the constant of proportion. You can easily observe that all these ratios are equal. So they are in Direct Proportion.

Problem 4: If the cost of 35 rice bags of the same size is Rs. 28,000. What is the cost of 100 rice bags of the same kind?

Solution:

We know that if the number of rice bags purchased increases then the cost also increases. Therefore, the cost of rice bags varies directly with the number of rice bags purchased.

Number of rice bags (x)

35

100

Cost (y)

Rs. 28,000

?

So,  (x1/y1) = (x2/y2)  

Here x1 = 35, y1 = Rs. 28000, x2 = 100, and y2 = ?

Upon substituting the values,

(35/28000)  = (100/y2)

⇒ y2 = (100 ⨯ 28000)/35

⇒ y2 = 80,000

Therefore, the cost of 100 rice bags of the same size is y2  = Rs. 80,000

Inverse Proportion

Two quantities change in such a manner that, if one quantity increases, the other quantity decreases in the same proportion and vice versa, then it is called Inverse Proportion. 

Examples of Inverse Proportion

Examples of Inverse Proportion

A Parcel company has a certain number of parcels to deliver. If the company engages 36 persons, it takes 12 days. If there are only 18 people, it will take 24 days to finish the task. You see as the number of persons is the halved time taken is doubled if the company engages 72 people, will the time taken be half? Yes, it is. Let’s have a look at the table 

Number of Persons

36

18

9

72

108

Time Taken

12

24

48

6

4

How many persons shall a company engage if it wants to deliver the parcels within a day? This question can be answered using the inverse proportions as 

\bold{\text{ Number of days required } \propto \frac{1}{\text{Numbers of persons engaged}} }

Inverse Proportion Formula

In the above example, the number of persons engaged and the number of days are inversely proportional to each other. Symbolically, this is represented as 

If x and y are in inverse proportion, then x ∝ (1 / y)

x = k/y ⇒ xy = k

Where, k is the constant of proportionality.

For two cases of each variable, let’s consider  y1 and y2 are the values of y corresponding to the values of x1 and x2 of x respectively then

\bold{x_1y_1 = k  = x_2 y_2}

OR

\bold{\frac{x_1}{x_2} = \frac{y_1}{y_2}}

Difference between Direct and Inverse Proportions

The key difference between direct and Inverse Proportions is as follows:

Property

Direct Proportion

Inverse Proportion

Relationship When two variables change in 
the same direction
When two variables change 
in opposite directions
Formulay = kx (where k is a constant)y = k/x (where k is a constant)
GraphA straight line passes through 
the origin (0,0)
A hyperbola
ExampleThe more hours you work, the 
more money you earn
The more people sharing a pizza, 
the smaller the slice each person gets
Symbol∝ (proportional to) E.g. a ∝ b∝ (inversely proportional to) E.g. a ∝ 1/b
Equationy = kxxy = k

Note: In direct proportion, as one variable increases, the other variable increases proportionally. In inverse proportion, as one variable increases, the other variable decreases proportionally.

Related Resources,

Direct and Inverse Proportions for Class 8 NCERT Solutions

Solved Problems on Inverse Proportions

Problem 1: If 36 workers can build a wall in 12 days, how many days will 16 workers take to build the same wall? (assuming the number of working hours per day is constant)

Solution:

If the number of workers decreases, the time to take built the wall increases in the same proportion. Clearly, the number of workers varies inversely to the number of days.

So here, x1 y1 = x2 y2 

Where x1 = 36 workers, x2 = 16 workers, and y1 = 12 days and y2 = (?) days

No. of Workers

No. of days

36

12

16

y2

Since the number of workers are decreasing

36 ÷ x = 16  

⇒ x = 36 / 16

So the number of days will increase in the same proportion i.e,

⇒ (36 / 16) × 12 = 27 days

Substitute, (36/16) = (y2/12)

⇒ y2 = (12 × 36)/16  = 27 days.

Therefore 16 workers will build the same wall in 27 days.

Problem 2: A car takes 4 hours to reach the destination by traveling at a speed of 60 km/h. How long will it take if the car travels at a speed of 80 Km/h?  

Solution:

Method 1: As speed increases, time is taken decreases in the same proportion. So the time is taken and varies inversely to the speed of the vehicle, for the same distance.

Speed

Time

60

4

80

x

(60 / 80) = (x / 4)

60 x 4  =  (80  x  x)

x = (60 x 4) / 80 =  3hrs.

Method 2:

Speed

Time

60

4 ÷ x

80

y

(60)(x) = 80 and 4 ÷ x = y

⇒ x = 80 / 60

⇒ 4 ÷ (80 / 60) = y

⇒ y = (4 x 60) / 80 = 3hrs.

Therefore, the time taken to cover the distance at a speed of 80 Km/h is 3hrs.

Problem 3: 6 pumps are required to fill a tank in 1 hour 40 minutes. How long will it take if only 10 pumps of the same type are used?

Solution:

Let the desired time to fill the tank be x minutes. Thus, we have the following table.

Number of pumps

6

10

Time (in minutes)

100

x

The lesser the number of pumps more will be the time required to fill the tank. 

So, this is a case of inverse proportion. 

Hence, (100)(6) = (x)(10)  

[As in direct proportion x1 y1 = x2 y2]

⇒ (100 x 6) / 10 = x

⇒ x = 60 minutes

Thus, time taken to fill the tank by 10 pumps is 60 minutes or 1 hour.

Problem 4: A school has 7 periods a day each of 45 minutes duration. How long would each period become, if the school has 5 periods a day? (assuming the number of school hours to be the same)

Solution:

Let the desired duration of each period be x minutes. Thus, we have the following table.

Number of periods

5

Time for each period (in minutes)

45

x

The lesser the number of periods a day, the more will be the duration of each period.

so, this is a case of inverse proportion.

Hence, (7)(45) = (x)(5)

[As in direct proportion x1 y1 = x2 y2]

⇒ (7 x 45) / 5 = x

⇒ x = 63 minutes

Thus, time duration of each period if the school has 5 periods a day is 63 minutes or 1 hour 3 minutes.

Read More,

FAQs on Direct and Inverse Proportions

1. What is Direct Proportion?

Direct proportion is a relationship between two variables where they change in the same direction, meaning that if one variable increases, the other variable also increases, and vice versa. 

2. What is Inverse Proportion?

Inverse proportion is a relationship between two variables where they change in opposite directions, meaning that if one variable increases, the other variable decreases, and vice versa.

3. How can we determine if two variables are Directly Proportional?

We can determine if two variables are directly proportional by plotting them on a graph and checking if the graph is a straight line passing through the origin (0,0). Also, we can also check if the ratio of the two variables is constant.

4. How can we determine if two variables are Inversely Proportional?

We can determine if two variables are inversely proportional by plotting them on a graph and checking if the graph is a hyperbola. Also we can also check if the product of the two variables is constant.

5. What are some Examples of Direct Proportion?

Some examples of direct proportion are:

  • The more hours you work, the more money you earn.
  • The more ingredients you use, the more cookies you can bake.
  • The faster you run, the more distance you cover.

6. What are some Examples of Inverse Proportion?

Some examples of inverse proportion are:

  • The more people sharing a pizza, the smaller the slice each person gets.
  • The more you dilute a solution, the weaker it becomes.
  • The the narrower the strap of your bag, the more pressure you feel on your shoulder.


Last Updated : 01 Sep, 2023
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