NCERT Textbook- Direct and Inverse Proportions | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 Page 1


? ????? ? ???? ? ???????? ?????????? ? ???
11.1  Introduction
Mohan prepares tea for himself and his sister. He uses 300 mL of
water, 2 spoons of sugar, 1 spoon of tea leaves and 50 m L of milk.
How much quantity of each item will he need, if he has to make tea
for five persons?
If two students take 20 minutes to arrange chairs for an assembly,
then how much time would five students take to do the same job?
We come across many such situations in our day-to-day life, where we
need to see variation in one quantity bringing in variation in the other
quantity.
For example:
(i) If the number of articles purchased increases, the total cost also increases.
(ii) More the money deposited in a bank, more is the interest earned.
(iii) As the speed of a vehicle increases, the time taken to cover the same distance
decreases.
(iv) For a given job, more the number of workers, less will be the time taken to complete
the work.
Observe that change in one quantity leads to change in the other quantity .
Write five more such situations where change in one quantity leads to change in
another quantity .
How do we find out the quantity of each item needed by Mohan? Or, the time five
students take to complete the job?
To answer such questions, we now study some concepts of variation.
11.2 Direct Proportion
If the cost of 1 kg of sugar is  ? 36, then what would be the cost of 3 kg sugar? It is ? 108.
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
???????????
? ? ? ? ? ? ?
??
Reprint 2024-25
Page 2


? ????? ? ???? ? ???????? ?????????? ? ???
11.1  Introduction
Mohan prepares tea for himself and his sister. He uses 300 mL of
water, 2 spoons of sugar, 1 spoon of tea leaves and 50 m L of milk.
How much quantity of each item will he need, if he has to make tea
for five persons?
If two students take 20 minutes to arrange chairs for an assembly,
then how much time would five students take to do the same job?
We come across many such situations in our day-to-day life, where we
need to see variation in one quantity bringing in variation in the other
quantity.
For example:
(i) If the number of articles purchased increases, the total cost also increases.
(ii) More the money deposited in a bank, more is the interest earned.
(iii) As the speed of a vehicle increases, the time taken to cover the same distance
decreases.
(iv) For a given job, more the number of workers, less will be the time taken to complete
the work.
Observe that change in one quantity leads to change in the other quantity .
Write five more such situations where change in one quantity leads to change in
another quantity .
How do we find out the quantity of each item needed by Mohan? Or, the time five
students take to complete the job?
To answer such questions, we now study some concepts of variation.
11.2 Direct Proportion
If the cost of 1 kg of sugar is  ? 36, then what would be the cost of 3 kg sugar? It is ? 108.
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
???????????
? ? ? ? ? ? ?
??
Reprint 2024-25
???? ? ? ? ????? ????
Similarly, we can find the cost of 5 kg or 8 kg of sugar . Study the following table.
Observe that as weight of sugar increases, cost also increases in such a manner that
their ratio remains constant.
Take one more example. Suppose a car uses 4 litres of petrol to travel a distance of
60 km. How far will it travel using 12 litres? The answer is 180 km. How did we calculate
it? Since petrol consumed in the second instance is 12 litres, i.e., three times of 4 litres, the
distance travelled will also be three times of 60 km. In other words, when the petrol
consumption becomes three-fold, the distance travelled is also three fold the previous
one. Let the consumption of petrol be x litres and the corresponding distance travelled be
y km . Now, complete the following table:
Petrol in litres (x) 4 8 12 15 20 25
Distance in km (y) 60 ... 180 ... ... ...
We find that as the value of x increases, value of y also increases in such a way that the
ratio 
x
y
 does not change; it remains constant (say k). In this case, it is 
1
15
 (check it!).
We say that x and y are in direct proportion, if  =
x
k
y
 or x = ky.
In this example, 
4 12
60 180
=
, where 4 and 12 are the quantities of petrol consumed in
litres (x) and 60 and 180 are the distances (y) in km. So when x and y are in direct
proportion, we can write 
12
12
xx
yy
=
. [y
1
, y
2
 are values of y corresponding to the values x
1
,
x
2
 of x respectively]
The consumption of petrol and the distance travelled by a car is a case of direct
proportion. Similarly, the total amount spent and the number of articles purchased is also
an example of direct proportion.
Reprint 2024-25
Page 3


? ????? ? ???? ? ???????? ?????????? ? ???
11.1  Introduction
Mohan prepares tea for himself and his sister. He uses 300 mL of
water, 2 spoons of sugar, 1 spoon of tea leaves and 50 m L of milk.
How much quantity of each item will he need, if he has to make tea
for five persons?
If two students take 20 minutes to arrange chairs for an assembly,
then how much time would five students take to do the same job?
We come across many such situations in our day-to-day life, where we
need to see variation in one quantity bringing in variation in the other
quantity.
For example:
(i) If the number of articles purchased increases, the total cost also increases.
(ii) More the money deposited in a bank, more is the interest earned.
(iii) As the speed of a vehicle increases, the time taken to cover the same distance
decreases.
(iv) For a given job, more the number of workers, less will be the time taken to complete
the work.
Observe that change in one quantity leads to change in the other quantity .
Write five more such situations where change in one quantity leads to change in
another quantity .
How do we find out the quantity of each item needed by Mohan? Or, the time five
students take to complete the job?
To answer such questions, we now study some concepts of variation.
11.2 Direct Proportion
If the cost of 1 kg of sugar is  ? 36, then what would be the cost of 3 kg sugar? It is ? 108.
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
???????????
? ? ? ? ? ? ?
??
Reprint 2024-25
???? ? ? ? ????? ????
Similarly, we can find the cost of 5 kg or 8 kg of sugar . Study the following table.
Observe that as weight of sugar increases, cost also increases in such a manner that
their ratio remains constant.
Take one more example. Suppose a car uses 4 litres of petrol to travel a distance of
60 km. How far will it travel using 12 litres? The answer is 180 km. How did we calculate
it? Since petrol consumed in the second instance is 12 litres, i.e., three times of 4 litres, the
distance travelled will also be three times of 60 km. In other words, when the petrol
consumption becomes three-fold, the distance travelled is also three fold the previous
one. Let the consumption of petrol be x litres and the corresponding distance travelled be
y km . Now, complete the following table:
Petrol in litres (x) 4 8 12 15 20 25
Distance in km (y) 60 ... 180 ... ... ...
We find that as the value of x increases, value of y also increases in such a way that the
ratio 
x
y
 does not change; it remains constant (say k). In this case, it is 
1
15
 (check it!).
We say that x and y are in direct proportion, if  =
x
k
y
 or x = ky.
In this example, 
4 12
60 180
=
, where 4 and 12 are the quantities of petrol consumed in
litres (x) and 60 and 180 are the distances (y) in km. So when x and y are in direct
proportion, we can write 
12
12
xx
yy
=
. [y
1
, y
2
 are values of y corresponding to the values x
1
,
x
2
 of x respectively]
The consumption of petrol and the distance travelled by a car is a case of direct
proportion. Similarly, the total amount spent and the number of articles purchased is also
an example of direct proportion.
Reprint 2024-25
? ????? ? ???? ? ???????? ?????????? ? ???
DO THIS
Think of a few more examples for direct proportion. Check whether Mohan [in the initial example] will
take 750 mL of water, 5 spoons of sugar, 
1
2
2
 spoons of tea leaves and 125 mL of milk to prepare tea for
five persons! Let us try to understand further the concept of direct proportion through the following activities.
(i) • Take a clock and fix its minute hand at 12.
• Record the angle turned through by the minute hand from its original position
and the time that has passed, in the following table:
Time Passed (T) (T
1
) (T
2
) (T
3
) (T
4
)
(in minutes) 15 30 45 60
Angle turned (A) (A
1
) (A
2
) (A
3
) (A
4
)
(in degree) 90 ... ... ...
T
A
... ... ... ...
What do you observe about T and A? Do they increase together?
Is  
T
A
 same every time?
Is the angle turned through by the minute hand directly proportional
to the time that has passed? Yes!
From the above table, you can also see
T
1
 : T
2
 = A
1
 : A
2
 , because
T
1
 : T
2
 = 15 : 30 = 1:2
A
1
 : A
2
 = 90 : 180 = 1:2
Check if T
2
 : T
3
 = A
2
 : A
3
  and  T
3
 : T
4
  = A
3
 : A
4
You can repeat this activity by choosing your own time interval.
(ii) Ask your friend to fill the following table and find the ratio of his age to the
corresponding age of his mother .
Age Present Age
five years ago age after five years
Friend’s age (F)
Mother’s age (M)
F
M
What do you observe?
Do F and M increase (or decrease) together? Is 
F
M
 same every time? No!
You can repeat this activity with other friends and write down your observations.
Reprint 2024-25
Page 4


? ????? ? ???? ? ???????? ?????????? ? ???
11.1  Introduction
Mohan prepares tea for himself and his sister. He uses 300 mL of
water, 2 spoons of sugar, 1 spoon of tea leaves and 50 m L of milk.
How much quantity of each item will he need, if he has to make tea
for five persons?
If two students take 20 minutes to arrange chairs for an assembly,
then how much time would five students take to do the same job?
We come across many such situations in our day-to-day life, where we
need to see variation in one quantity bringing in variation in the other
quantity.
For example:
(i) If the number of articles purchased increases, the total cost also increases.
(ii) More the money deposited in a bank, more is the interest earned.
(iii) As the speed of a vehicle increases, the time taken to cover the same distance
decreases.
(iv) For a given job, more the number of workers, less will be the time taken to complete
the work.
Observe that change in one quantity leads to change in the other quantity .
Write five more such situations where change in one quantity leads to change in
another quantity .
How do we find out the quantity of each item needed by Mohan? Or, the time five
students take to complete the job?
To answer such questions, we now study some concepts of variation.
11.2 Direct Proportion
If the cost of 1 kg of sugar is  ? 36, then what would be the cost of 3 kg sugar? It is ? 108.
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
???????????
? ? ? ? ? ? ?
??
Reprint 2024-25
???? ? ? ? ????? ????
Similarly, we can find the cost of 5 kg or 8 kg of sugar . Study the following table.
Observe that as weight of sugar increases, cost also increases in such a manner that
their ratio remains constant.
Take one more example. Suppose a car uses 4 litres of petrol to travel a distance of
60 km. How far will it travel using 12 litres? The answer is 180 km. How did we calculate
it? Since petrol consumed in the second instance is 12 litres, i.e., three times of 4 litres, the
distance travelled will also be three times of 60 km. In other words, when the petrol
consumption becomes three-fold, the distance travelled is also three fold the previous
one. Let the consumption of petrol be x litres and the corresponding distance travelled be
y km . Now, complete the following table:
Petrol in litres (x) 4 8 12 15 20 25
Distance in km (y) 60 ... 180 ... ... ...
We find that as the value of x increases, value of y also increases in such a way that the
ratio 
x
y
 does not change; it remains constant (say k). In this case, it is 
1
15
 (check it!).
We say that x and y are in direct proportion, if  =
x
k
y
 or x = ky.
In this example, 
4 12
60 180
=
, where 4 and 12 are the quantities of petrol consumed in
litres (x) and 60 and 180 are the distances (y) in km. So when x and y are in direct
proportion, we can write 
12
12
xx
yy
=
. [y
1
, y
2
 are values of y corresponding to the values x
1
,
x
2
 of x respectively]
The consumption of petrol and the distance travelled by a car is a case of direct
proportion. Similarly, the total amount spent and the number of articles purchased is also
an example of direct proportion.
Reprint 2024-25
? ????? ? ???? ? ???????? ?????????? ? ???
DO THIS
Think of a few more examples for direct proportion. Check whether Mohan [in the initial example] will
take 750 mL of water, 5 spoons of sugar, 
1
2
2
 spoons of tea leaves and 125 mL of milk to prepare tea for
five persons! Let us try to understand further the concept of direct proportion through the following activities.
(i) • Take a clock and fix its minute hand at 12.
• Record the angle turned through by the minute hand from its original position
and the time that has passed, in the following table:
Time Passed (T) (T
1
) (T
2
) (T
3
) (T
4
)
(in minutes) 15 30 45 60
Angle turned (A) (A
1
) (A
2
) (A
3
) (A
4
)
(in degree) 90 ... ... ...
T
A
... ... ... ...
What do you observe about T and A? Do they increase together?
Is  
T
A
 same every time?
Is the angle turned through by the minute hand directly proportional
to the time that has passed? Yes!
From the above table, you can also see
T
1
 : T
2
 = A
1
 : A
2
 , because
T
1
 : T
2
 = 15 : 30 = 1:2
A
1
 : A
2
 = 90 : 180 = 1:2
Check if T
2
 : T
3
 = A
2
 : A
3
  and  T
3
 : T
4
  = A
3
 : A
4
You can repeat this activity by choosing your own time interval.
(ii) Ask your friend to fill the following table and find the ratio of his age to the
corresponding age of his mother .
Age Present Age
five years ago age after five years
Friend’s age (F)
Mother’s age (M)
F
M
What do you observe?
Do F and M increase (or decrease) together? Is 
F
M
 same every time? No!
You can repeat this activity with other friends and write down your observations.
Reprint 2024-25
???? ? ? ? ????? ????
TRY THESE
Thus, variables increasing (or decreasing) together need not always be in direct
proportion. For example:
(i) physical changes in human beings occur with time but not necessarily in a predeter-
mined ratio.
(ii) changes in weight and height among individuals are not in any known proportion and
(iii) there is no direct relationship or ratio between the height of a tree and the number
of leaves growing on its branches. Think of some more similar examples.
1. Observe the following tables and find if x and y are directly proportional.
   (i) x 20 17 14 11 8 52
y 40 34 28 22 16 10 4
  (ii) x 6 10 14 18 22 26 30
y 4 8 12 16 20 24 28
 (iii) x 5 8 12 15 18 20
y 15 24 36 60 72 100
2. Principal = ? 1000, Rate = 8% per annum. Fill in the following table and find
which type of interest (simple or compound) changes in direct proportion with
time period.
Time period 1 year 2 years 3 years
Simple Interest (in ?)
Compound Interest (in ?)
P P 1
100
+
?
?
?
?
?
?
-
r
t
P
100
rt ××
If we fix time period and the rate of interest, simple interest changes proportionally
with principal. Would there be a similar relationship for compound interest? Why?
Let us consider some solved examples where we would use the concept of
direct proportion.
Example 1: The cost of 5 metres of a particular quality of cloth is ? 210. Tabulate the
cost of 2, 4, 10 and 13 metres of cloth of the same type.
Solution: Suppose the length of cloth is x metres and its cost, in ?, is y.
x 2 4 5 10 13
y y
2
y
3
210 y
4
y
5
THINK, DISCUSS AND WRITE
Reprint 2024-25
Page 5


? ????? ? ???? ? ???????? ?????????? ? ???
11.1  Introduction
Mohan prepares tea for himself and his sister. He uses 300 mL of
water, 2 spoons of sugar, 1 spoon of tea leaves and 50 m L of milk.
How much quantity of each item will he need, if he has to make tea
for five persons?
If two students take 20 minutes to arrange chairs for an assembly,
then how much time would five students take to do the same job?
We come across many such situations in our day-to-day life, where we
need to see variation in one quantity bringing in variation in the other
quantity.
For example:
(i) If the number of articles purchased increases, the total cost also increases.
(ii) More the money deposited in a bank, more is the interest earned.
(iii) As the speed of a vehicle increases, the time taken to cover the same distance
decreases.
(iv) For a given job, more the number of workers, less will be the time taken to complete
the work.
Observe that change in one quantity leads to change in the other quantity .
Write five more such situations where change in one quantity leads to change in
another quantity .
How do we find out the quantity of each item needed by Mohan? Or, the time five
students take to complete the job?
To answer such questions, we now study some concepts of variation.
11.2 Direct Proportion
If the cost of 1 kg of sugar is  ? 36, then what would be the cost of 3 kg sugar? It is ? 108.
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
???????????
? ? ? ? ? ? ?
??
Reprint 2024-25
???? ? ? ? ????? ????
Similarly, we can find the cost of 5 kg or 8 kg of sugar . Study the following table.
Observe that as weight of sugar increases, cost also increases in such a manner that
their ratio remains constant.
Take one more example. Suppose a car uses 4 litres of petrol to travel a distance of
60 km. How far will it travel using 12 litres? The answer is 180 km. How did we calculate
it? Since petrol consumed in the second instance is 12 litres, i.e., three times of 4 litres, the
distance travelled will also be three times of 60 km. In other words, when the petrol
consumption becomes three-fold, the distance travelled is also three fold the previous
one. Let the consumption of petrol be x litres and the corresponding distance travelled be
y km . Now, complete the following table:
Petrol in litres (x) 4 8 12 15 20 25
Distance in km (y) 60 ... 180 ... ... ...
We find that as the value of x increases, value of y also increases in such a way that the
ratio 
x
y
 does not change; it remains constant (say k). In this case, it is 
1
15
 (check it!).
We say that x and y are in direct proportion, if  =
x
k
y
 or x = ky.
In this example, 
4 12
60 180
=
, where 4 and 12 are the quantities of petrol consumed in
litres (x) and 60 and 180 are the distances (y) in km. So when x and y are in direct
proportion, we can write 
12
12
xx
yy
=
. [y
1
, y
2
 are values of y corresponding to the values x
1
,
x
2
 of x respectively]
The consumption of petrol and the distance travelled by a car is a case of direct
proportion. Similarly, the total amount spent and the number of articles purchased is also
an example of direct proportion.
Reprint 2024-25
? ????? ? ???? ? ???????? ?????????? ? ???
DO THIS
Think of a few more examples for direct proportion. Check whether Mohan [in the initial example] will
take 750 mL of water, 5 spoons of sugar, 
1
2
2
 spoons of tea leaves and 125 mL of milk to prepare tea for
five persons! Let us try to understand further the concept of direct proportion through the following activities.
(i) • Take a clock and fix its minute hand at 12.
• Record the angle turned through by the minute hand from its original position
and the time that has passed, in the following table:
Time Passed (T) (T
1
) (T
2
) (T
3
) (T
4
)
(in minutes) 15 30 45 60
Angle turned (A) (A
1
) (A
2
) (A
3
) (A
4
)
(in degree) 90 ... ... ...
T
A
... ... ... ...
What do you observe about T and A? Do they increase together?
Is  
T
A
 same every time?
Is the angle turned through by the minute hand directly proportional
to the time that has passed? Yes!
From the above table, you can also see
T
1
 : T
2
 = A
1
 : A
2
 , because
T
1
 : T
2
 = 15 : 30 = 1:2
A
1
 : A
2
 = 90 : 180 = 1:2
Check if T
2
 : T
3
 = A
2
 : A
3
  and  T
3
 : T
4
  = A
3
 : A
4
You can repeat this activity by choosing your own time interval.
(ii) Ask your friend to fill the following table and find the ratio of his age to the
corresponding age of his mother .
Age Present Age
five years ago age after five years
Friend’s age (F)
Mother’s age (M)
F
M
What do you observe?
Do F and M increase (or decrease) together? Is 
F
M
 same every time? No!
You can repeat this activity with other friends and write down your observations.
Reprint 2024-25
???? ? ? ? ????? ????
TRY THESE
Thus, variables increasing (or decreasing) together need not always be in direct
proportion. For example:
(i) physical changes in human beings occur with time but not necessarily in a predeter-
mined ratio.
(ii) changes in weight and height among individuals are not in any known proportion and
(iii) there is no direct relationship or ratio between the height of a tree and the number
of leaves growing on its branches. Think of some more similar examples.
1. Observe the following tables and find if x and y are directly proportional.
   (i) x 20 17 14 11 8 52
y 40 34 28 22 16 10 4
  (ii) x 6 10 14 18 22 26 30
y 4 8 12 16 20 24 28
 (iii) x 5 8 12 15 18 20
y 15 24 36 60 72 100
2. Principal = ? 1000, Rate = 8% per annum. Fill in the following table and find
which type of interest (simple or compound) changes in direct proportion with
time period.
Time period 1 year 2 years 3 years
Simple Interest (in ?)
Compound Interest (in ?)
P P 1
100
+
?
?
?
?
?
?
-
r
t
P
100
rt ××
If we fix time period and the rate of interest, simple interest changes proportionally
with principal. Would there be a similar relationship for compound interest? Why?
Let us consider some solved examples where we would use the concept of
direct proportion.
Example 1: The cost of 5 metres of a particular quality of cloth is ? 210. Tabulate the
cost of 2, 4, 10 and 13 metres of cloth of the same type.
Solution: Suppose the length of cloth is x metres and its cost, in ?, is y.
x 2 4 5 10 13
y y
2
y
3
210 y
4
y
5
THINK, DISCUSS AND WRITE
Reprint 2024-25
? ????? ? ???? ? ???????? ?????????? ? ???
As the length of cloth increases, cost of the cloth also increases in the same ratio. It is
a case of direct proportion.
We make use of the relation of type 
12
12
xx
yy
=
(i) Here x
1
 = 5,  y
1
 = 210 and x
2
 = 2
Therefore,  
12
12
xx
yy
= gives 
2
52
210 y
=
 or 5y
2 
= 2 × 210 or 
2
2 210
5
y
×
=
  = 84
(ii) If x
3
 = 4, then 
3
54
210 y
=
 or 5y
3 
= 4 × 210 or 
3
4 210
5
y
×
=
 = 168
[Can we use 
3 2
23
x x
yy
=
 here? Try!]
(iii) If x
4
 = 10, then 
4
5 10
210 y
=
 or 
4
10 210
5
y
×
=
= 420
(iv) If x
5
 = 13, then 
5
5 13
210 y
= or 
5
13 210
5
y
×
=
 = 546
Note that here we can also use or or in the place 
2
84
4
168
10
420
o of
5
210
?
?
?
?
?
?
Example 2: An electric pole, 14 metres high, casts a shadow of 10 metres. Find the
height of a tree that casts a shadow of 15 metres under similar conditions.
Solution: Let the height of the tree be x metres. We form a table as shown below:
height of the object (in metres) 14 x
length of the shadow (in metres) 10 15
Note that more the height of an object, the more would be the length of its shadow.
Hence, this is a case of direct proportion. That is, 
1
1
x
y
= 
2
2
x
y
We have
14
10
 = 
15
x
   (Why?)
or
14
15
10
×
 = x
or
14 3
2
×
 = x
So 21 = x
Thus, height of the tree is 21 metres.
Alternately, we can write 
12
12
xx
yy
=  as 
11
22
xy
xy
=
Reprint 2024-25