NCERT Textbook: Comparing Quantities | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 Page 1


? ??????????????????? ? ? ? ?
7.1  Recalling Ratios and Percentages
We know, ratio means comparing two quantities.
A basket has two types of fruits, say, 20 apples and 5 oranges.
Then, the ratio of the number of oranges to the number of apples = 5 : 20.
The comparison can be done by using fractions as, 
5
20
 = 
1
4
The number of oranges is 
1
4
th the number of apples. In terms of ratio, this is
1 : 4, read as, “1 is to 4”
Number of apples to number of oranges = 
20 4
51
=
  which means, the number of apples
is  4 times the number of oranges. This comparison can also be done using percentages.
There are 5 oranges out of 25 fruits.
So percentage of oranges is
 
5 4 20
20%
25 4 100
×= = OR
[Denominator made 100].
Since  contains only apples and oranges,
So, percentage of apples + percentage of oranges = 100
or percentage of apples + 20 = 100
or percentage of apples = 100 – 20 = 80
Thus the basket has 20% oranges and 80% apples.
Example 1: A picnic is being planned in a school for Class VII. Girls are 60% of the
total number of students and are 18 in number .
The picnic site is 55 km from the school and the transport company is charging at the rate
of ? 12 per km. The total cost of refreshments will be ? 4280.
??? ??? ???? ??? ???? ???
? ? ? ? ? ? ?
?
By unitary method:
Out of 25 fruits, number of oranges are 5.
So out of 100 fruits, number of oranges
= 
5
100
25
×
 = 20.
OR
Reprint 2024-25
Page 2


? ??????????????????? ? ? ? ?
7.1  Recalling Ratios and Percentages
We know, ratio means comparing two quantities.
A basket has two types of fruits, say, 20 apples and 5 oranges.
Then, the ratio of the number of oranges to the number of apples = 5 : 20.
The comparison can be done by using fractions as, 
5
20
 = 
1
4
The number of oranges is 
1
4
th the number of apples. In terms of ratio, this is
1 : 4, read as, “1 is to 4”
Number of apples to number of oranges = 
20 4
51
=
  which means, the number of apples
is  4 times the number of oranges. This comparison can also be done using percentages.
There are 5 oranges out of 25 fruits.
So percentage of oranges is
 
5 4 20
20%
25 4 100
×= = OR
[Denominator made 100].
Since  contains only apples and oranges,
So, percentage of apples + percentage of oranges = 100
or percentage of apples + 20 = 100
or percentage of apples = 100 – 20 = 80
Thus the basket has 20% oranges and 80% apples.
Example 1: A picnic is being planned in a school for Class VII. Girls are 60% of the
total number of students and are 18 in number .
The picnic site is 55 km from the school and the transport company is charging at the rate
of ? 12 per km. The total cost of refreshments will be ? 4280.
??? ??? ???? ??? ???? ???
? ? ? ? ? ? ?
?
By unitary method:
Out of 25 fruits, number of oranges are 5.
So out of 100 fruits, number of oranges
= 
5
100
25
×
 = 20.
OR
Reprint 2024-25
??? ? ? ? ????? ????
Can you tell.
1. The ratio of the number of girls to the number of boys in the class?
2. The cost per head if two teachers are also going with the class?
3. If their first stop is at a place 22 km from the school, what per cent of the total
distance of 55 km is this? What per cent of the distance is left to be covered?
Solution:
1. To find the ratio of girls to boys.
Ashima and John came up with the following answers.
They needed to know the number of boys and also the total number of students.
Ashima did this John used the unitary method
Let the total number of students There are 60 girls out of 100 students.
be x. 60% of x is girls. There is one girl out of 
100
60
 students.
Therefore, 60% of x = 18 So, 18 girls are out of  how many students?
60
100
x ×
 =  18 OR Number of students = 
100
18
60
×
or,  x = 
18 100
60
×
 = 30 = 30
Number of students = 30.
So, the number of boys = 30 – 18 = 12.
Hence, ratio of the number of girls to the number of boys is 18 : 12 or 
18
12
 = 
3
2
.
3
2
 is written as 3 : 2 and read as 3 is to 2.
2. To find the cost per person.
Transportation charge = Distance both ways × Rate
= ? (55 × 2) × 12
= ? 110 × 12 = ? 1320
Total expenses = Refreshment charge
+ Transportation charge
= ? 4280 + ? 1320
= ? 5600
Total number of persons =18 girls + 12 boys + 2 teachers
= 32 persons
Ashima and John then used unitary method to find the cost per head.
For 32 persons, amount spent would be ? 5600.
The amount spent for 1 person = ? 
5600
32
 = ? 175.
3. The distance of the place where first stop was made = 22 km.
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Page 3


? ??????????????????? ? ? ? ?
7.1  Recalling Ratios and Percentages
We know, ratio means comparing two quantities.
A basket has two types of fruits, say, 20 apples and 5 oranges.
Then, the ratio of the number of oranges to the number of apples = 5 : 20.
The comparison can be done by using fractions as, 
5
20
 = 
1
4
The number of oranges is 
1
4
th the number of apples. In terms of ratio, this is
1 : 4, read as, “1 is to 4”
Number of apples to number of oranges = 
20 4
51
=
  which means, the number of apples
is  4 times the number of oranges. This comparison can also be done using percentages.
There are 5 oranges out of 25 fruits.
So percentage of oranges is
 
5 4 20
20%
25 4 100
×= = OR
[Denominator made 100].
Since  contains only apples and oranges,
So, percentage of apples + percentage of oranges = 100
or percentage of apples + 20 = 100
or percentage of apples = 100 – 20 = 80
Thus the basket has 20% oranges and 80% apples.
Example 1: A picnic is being planned in a school for Class VII. Girls are 60% of the
total number of students and are 18 in number .
The picnic site is 55 km from the school and the transport company is charging at the rate
of ? 12 per km. The total cost of refreshments will be ? 4280.
??? ??? ???? ??? ???? ???
? ? ? ? ? ? ?
?
By unitary method:
Out of 25 fruits, number of oranges are 5.
So out of 100 fruits, number of oranges
= 
5
100
25
×
 = 20.
OR
Reprint 2024-25
??? ? ? ? ????? ????
Can you tell.
1. The ratio of the number of girls to the number of boys in the class?
2. The cost per head if two teachers are also going with the class?
3. If their first stop is at a place 22 km from the school, what per cent of the total
distance of 55 km is this? What per cent of the distance is left to be covered?
Solution:
1. To find the ratio of girls to boys.
Ashima and John came up with the following answers.
They needed to know the number of boys and also the total number of students.
Ashima did this John used the unitary method
Let the total number of students There are 60 girls out of 100 students.
be x. 60% of x is girls. There is one girl out of 
100
60
 students.
Therefore, 60% of x = 18 So, 18 girls are out of  how many students?
60
100
x ×
 =  18 OR Number of students = 
100
18
60
×
or,  x = 
18 100
60
×
 = 30 = 30
Number of students = 30.
So, the number of boys = 30 – 18 = 12.
Hence, ratio of the number of girls to the number of boys is 18 : 12 or 
18
12
 = 
3
2
.
3
2
 is written as 3 : 2 and read as 3 is to 2.
2. To find the cost per person.
Transportation charge = Distance both ways × Rate
= ? (55 × 2) × 12
= ? 110 × 12 = ? 1320
Total expenses = Refreshment charge
+ Transportation charge
= ? 4280 + ? 1320
= ? 5600
Total number of persons =18 girls + 12 boys + 2 teachers
= 32 persons
Ashima and John then used unitary method to find the cost per head.
For 32 persons, amount spent would be ? 5600.
The amount spent for 1 person = ? 
5600
32
 = ? 175.
3. The distance of the place where first stop was made = 22 km.
Reprint 2024-25
? ??????????????????? ? ? ? ?
To find the percentage of distance:
Ashima used this method: John used the unitary method:
22 22 100
40%
55 55 100
=× =
Out of 55 km, 22 km are travelled.
OR Out of 1 km, 
22
55
km are travelled.
 Out of 100 km, 
22
55
 × 100 km are travelled.
That is 40% of the total distance is travelled.
TRY THESE
She is multiplying
100
the ratio by =1
100
and converting to
percentage.
  
  
  
  
  
  
  
  
Both came out with the same answer that the distance from their school of the place where
they stopped at was 40% of the total distance they had to travel.
Therefore, the percent distance left to be travelled = 100% – 40% = 60%.
In a primary school, the parents were asked about the number of hours they spend per day
in helping their children to do homework. There were 90 parents who helped for 
1
2
 hour
to 
1
1
2
 hours. The distribution of parents according to the time for which,
they  said they helped is given in the adjoining figure ; 20% helped for
more than 
1
1
2
 hours per day;
30% helped for 
1
2
 hour to 
1
1
2
 hours; 50% did not help at all.
Using this, answer the following:
(i) How many parents were surveyed?
(ii) How many said that they did not help?
(iii) How many said that they helped for more than 
1
1
2
 hours?
EXERCISE 7.1
1. Find the ratio of the following.
(a) Speed of a cycle 15 km per hour to the speed of scooter 30 km per hour.
(b) 5 m to 10 km (c) 50 paise to ? 5
2. Convert the following ratios to percentages.
(a) 3 : 4 (b) 2 : 3
3. 72% of 25 students are interested in mathematics. How many are not interested
in mathematics?
4. A football team won 10 matches out of the total number of matches they played. If
their win percentage was 40, then how many matches did they play in all?
5. If Chameli had ? 600 left after spending 75% of her money, how much did she have
in the beginning?
Reprint 2024-25
Page 4


? ??????????????????? ? ? ? ?
7.1  Recalling Ratios and Percentages
We know, ratio means comparing two quantities.
A basket has two types of fruits, say, 20 apples and 5 oranges.
Then, the ratio of the number of oranges to the number of apples = 5 : 20.
The comparison can be done by using fractions as, 
5
20
 = 
1
4
The number of oranges is 
1
4
th the number of apples. In terms of ratio, this is
1 : 4, read as, “1 is to 4”
Number of apples to number of oranges = 
20 4
51
=
  which means, the number of apples
is  4 times the number of oranges. This comparison can also be done using percentages.
There are 5 oranges out of 25 fruits.
So percentage of oranges is
 
5 4 20
20%
25 4 100
×= = OR
[Denominator made 100].
Since  contains only apples and oranges,
So, percentage of apples + percentage of oranges = 100
or percentage of apples + 20 = 100
or percentage of apples = 100 – 20 = 80
Thus the basket has 20% oranges and 80% apples.
Example 1: A picnic is being planned in a school for Class VII. Girls are 60% of the
total number of students and are 18 in number .
The picnic site is 55 km from the school and the transport company is charging at the rate
of ? 12 per km. The total cost of refreshments will be ? 4280.
??? ??? ???? ??? ???? ???
? ? ? ? ? ? ?
?
By unitary method:
Out of 25 fruits, number of oranges are 5.
So out of 100 fruits, number of oranges
= 
5
100
25
×
 = 20.
OR
Reprint 2024-25
??? ? ? ? ????? ????
Can you tell.
1. The ratio of the number of girls to the number of boys in the class?
2. The cost per head if two teachers are also going with the class?
3. If their first stop is at a place 22 km from the school, what per cent of the total
distance of 55 km is this? What per cent of the distance is left to be covered?
Solution:
1. To find the ratio of girls to boys.
Ashima and John came up with the following answers.
They needed to know the number of boys and also the total number of students.
Ashima did this John used the unitary method
Let the total number of students There are 60 girls out of 100 students.
be x. 60% of x is girls. There is one girl out of 
100
60
 students.
Therefore, 60% of x = 18 So, 18 girls are out of  how many students?
60
100
x ×
 =  18 OR Number of students = 
100
18
60
×
or,  x = 
18 100
60
×
 = 30 = 30
Number of students = 30.
So, the number of boys = 30 – 18 = 12.
Hence, ratio of the number of girls to the number of boys is 18 : 12 or 
18
12
 = 
3
2
.
3
2
 is written as 3 : 2 and read as 3 is to 2.
2. To find the cost per person.
Transportation charge = Distance both ways × Rate
= ? (55 × 2) × 12
= ? 110 × 12 = ? 1320
Total expenses = Refreshment charge
+ Transportation charge
= ? 4280 + ? 1320
= ? 5600
Total number of persons =18 girls + 12 boys + 2 teachers
= 32 persons
Ashima and John then used unitary method to find the cost per head.
For 32 persons, amount spent would be ? 5600.
The amount spent for 1 person = ? 
5600
32
 = ? 175.
3. The distance of the place where first stop was made = 22 km.
Reprint 2024-25
? ??????????????????? ? ? ? ?
To find the percentage of distance:
Ashima used this method: John used the unitary method:
22 22 100
40%
55 55 100
=× =
Out of 55 km, 22 km are travelled.
OR Out of 1 km, 
22
55
km are travelled.
 Out of 100 km, 
22
55
 × 100 km are travelled.
That is 40% of the total distance is travelled.
TRY THESE
She is multiplying
100
the ratio by =1
100
and converting to
percentage.
  
  
  
  
  
  
  
  
Both came out with the same answer that the distance from their school of the place where
they stopped at was 40% of the total distance they had to travel.
Therefore, the percent distance left to be travelled = 100% – 40% = 60%.
In a primary school, the parents were asked about the number of hours they spend per day
in helping their children to do homework. There were 90 parents who helped for 
1
2
 hour
to 
1
1
2
 hours. The distribution of parents according to the time for which,
they  said they helped is given in the adjoining figure ; 20% helped for
more than 
1
1
2
 hours per day;
30% helped for 
1
2
 hour to 
1
1
2
 hours; 50% did not help at all.
Using this, answer the following:
(i) How many parents were surveyed?
(ii) How many said that they did not help?
(iii) How many said that they helped for more than 
1
1
2
 hours?
EXERCISE 7.1
1. Find the ratio of the following.
(a) Speed of a cycle 15 km per hour to the speed of scooter 30 km per hour.
(b) 5 m to 10 km (c) 50 paise to ? 5
2. Convert the following ratios to percentages.
(a) 3 : 4 (b) 2 : 3
3. 72% of 25 students are interested in mathematics. How many are not interested
in mathematics?
4. A football team won 10 matches out of the total number of matches they played. If
their win percentage was 40, then how many matches did they play in all?
5. If Chameli had ? 600 left after spending 75% of her money, how much did she have
in the beginning?
Reprint 2024-25
??? ? ? ? ????? ????
6. If 60% people in a city like cricket, 30% like football and the remaining like other
games, then what per cent of the people like other games? If the total number of
people is 50 lakh, find the exact number who like each type of game.
7.2  Finding Discounts
Discount is a reduction given on the Marked Price
(MP) of the article.
This is generally given to attract customers to buy
goods or to promote sales of the goods. You can find
the discount by subtracting its sale price from its
marked price.
So, Discount = Marked price – Sale price
Example 2: An item marked at ? 840 is sold for ? 714. What is the discount and
discount %?
Solution: Discount = Marked Price – Sale Price
= ? 840 – ? 714
= ? 126
Since discount is on marked price, we will have to use marked price as the base.
On marked price of ? 840, the discount is ? 126.
On MP of ? 100, how much will the discount be?
Discount =
126
100%
840
×
 = 15%
You can also find discount when discount % is given.
Example 3: The list price of a frock is ? 220.
A discount of 20% is announced on sales. What is the amount
of discount on it and its sale price.
Solution: Marked price is same as the list price.
20% discount means that on ? 100 (MP), the discount is ? 20.
By unitary method, on ?1 the discount will be ? 
20
100
.
On ? 220,  discount = ? 
20
220
100
× = ? 44
The sale price = ( ? 220 – ? 44) or  ? 176
Rehana found the sale price like this —
A discount of 20% means for a MP of ? 100, discount is ? 20. Hence the sale price is
? 80. Using unitary method, when MP is ? 100, sale price is ? 80;
Reprint 2024-25
Page 5


? ??????????????????? ? ? ? ?
7.1  Recalling Ratios and Percentages
We know, ratio means comparing two quantities.
A basket has two types of fruits, say, 20 apples and 5 oranges.
Then, the ratio of the number of oranges to the number of apples = 5 : 20.
The comparison can be done by using fractions as, 
5
20
 = 
1
4
The number of oranges is 
1
4
th the number of apples. In terms of ratio, this is
1 : 4, read as, “1 is to 4”
Number of apples to number of oranges = 
20 4
51
=
  which means, the number of apples
is  4 times the number of oranges. This comparison can also be done using percentages.
There are 5 oranges out of 25 fruits.
So percentage of oranges is
 
5 4 20
20%
25 4 100
×= = OR
[Denominator made 100].
Since  contains only apples and oranges,
So, percentage of apples + percentage of oranges = 100
or percentage of apples + 20 = 100
or percentage of apples = 100 – 20 = 80
Thus the basket has 20% oranges and 80% apples.
Example 1: A picnic is being planned in a school for Class VII. Girls are 60% of the
total number of students and are 18 in number .
The picnic site is 55 km from the school and the transport company is charging at the rate
of ? 12 per km. The total cost of refreshments will be ? 4280.
??? ??? ???? ??? ???? ???
? ? ? ? ? ? ?
?
By unitary method:
Out of 25 fruits, number of oranges are 5.
So out of 100 fruits, number of oranges
= 
5
100
25
×
 = 20.
OR
Reprint 2024-25
??? ? ? ? ????? ????
Can you tell.
1. The ratio of the number of girls to the number of boys in the class?
2. The cost per head if two teachers are also going with the class?
3. If their first stop is at a place 22 km from the school, what per cent of the total
distance of 55 km is this? What per cent of the distance is left to be covered?
Solution:
1. To find the ratio of girls to boys.
Ashima and John came up with the following answers.
They needed to know the number of boys and also the total number of students.
Ashima did this John used the unitary method
Let the total number of students There are 60 girls out of 100 students.
be x. 60% of x is girls. There is one girl out of 
100
60
 students.
Therefore, 60% of x = 18 So, 18 girls are out of  how many students?
60
100
x ×
 =  18 OR Number of students = 
100
18
60
×
or,  x = 
18 100
60
×
 = 30 = 30
Number of students = 30.
So, the number of boys = 30 – 18 = 12.
Hence, ratio of the number of girls to the number of boys is 18 : 12 or 
18
12
 = 
3
2
.
3
2
 is written as 3 : 2 and read as 3 is to 2.
2. To find the cost per person.
Transportation charge = Distance both ways × Rate
= ? (55 × 2) × 12
= ? 110 × 12 = ? 1320
Total expenses = Refreshment charge
+ Transportation charge
= ? 4280 + ? 1320
= ? 5600
Total number of persons =18 girls + 12 boys + 2 teachers
= 32 persons
Ashima and John then used unitary method to find the cost per head.
For 32 persons, amount spent would be ? 5600.
The amount spent for 1 person = ? 
5600
32
 = ? 175.
3. The distance of the place where first stop was made = 22 km.
Reprint 2024-25
? ??????????????????? ? ? ? ?
To find the percentage of distance:
Ashima used this method: John used the unitary method:
22 22 100
40%
55 55 100
=× =
Out of 55 km, 22 km are travelled.
OR Out of 1 km, 
22
55
km are travelled.
 Out of 100 km, 
22
55
 × 100 km are travelled.
That is 40% of the total distance is travelled.
TRY THESE
She is multiplying
100
the ratio by =1
100
and converting to
percentage.
  
  
  
  
  
  
  
  
Both came out with the same answer that the distance from their school of the place where
they stopped at was 40% of the total distance they had to travel.
Therefore, the percent distance left to be travelled = 100% – 40% = 60%.
In a primary school, the parents were asked about the number of hours they spend per day
in helping their children to do homework. There were 90 parents who helped for 
1
2
 hour
to 
1
1
2
 hours. The distribution of parents according to the time for which,
they  said they helped is given in the adjoining figure ; 20% helped for
more than 
1
1
2
 hours per day;
30% helped for 
1
2
 hour to 
1
1
2
 hours; 50% did not help at all.
Using this, answer the following:
(i) How many parents were surveyed?
(ii) How many said that they did not help?
(iii) How many said that they helped for more than 
1
1
2
 hours?
EXERCISE 7.1
1. Find the ratio of the following.
(a) Speed of a cycle 15 km per hour to the speed of scooter 30 km per hour.
(b) 5 m to 10 km (c) 50 paise to ? 5
2. Convert the following ratios to percentages.
(a) 3 : 4 (b) 2 : 3
3. 72% of 25 students are interested in mathematics. How many are not interested
in mathematics?
4. A football team won 10 matches out of the total number of matches they played. If
their win percentage was 40, then how many matches did they play in all?
5. If Chameli had ? 600 left after spending 75% of her money, how much did she have
in the beginning?
Reprint 2024-25
??? ? ? ? ????? ????
6. If 60% people in a city like cricket, 30% like football and the remaining like other
games, then what per cent of the people like other games? If the total number of
people is 50 lakh, find the exact number who like each type of game.
7.2  Finding Discounts
Discount is a reduction given on the Marked Price
(MP) of the article.
This is generally given to attract customers to buy
goods or to promote sales of the goods. You can find
the discount by subtracting its sale price from its
marked price.
So, Discount = Marked price – Sale price
Example 2: An item marked at ? 840 is sold for ? 714. What is the discount and
discount %?
Solution: Discount = Marked Price – Sale Price
= ? 840 – ? 714
= ? 126
Since discount is on marked price, we will have to use marked price as the base.
On marked price of ? 840, the discount is ? 126.
On MP of ? 100, how much will the discount be?
Discount =
126
100%
840
×
 = 15%
You can also find discount when discount % is given.
Example 3: The list price of a frock is ? 220.
A discount of 20% is announced on sales. What is the amount
of discount on it and its sale price.
Solution: Marked price is same as the list price.
20% discount means that on ? 100 (MP), the discount is ? 20.
By unitary method, on ?1 the discount will be ? 
20
100
.
On ? 220,  discount = ? 
20
220
100
× = ? 44
The sale price = ( ? 220 – ? 44) or  ? 176
Rehana found the sale price like this —
A discount of 20% means for a MP of ? 100, discount is ? 20. Hence the sale price is
? 80. Using unitary method, when MP is ? 100, sale price is ? 80;
Reprint 2024-25
? ??????????????????? ? ? ? ?
TRY THESE
When MP is ? 1, sale price is ? 
80
100
.
Hence when MP is ? 220, sale price = ? 
80
220
100
× = ? 176.
1. A shop gives 20% discount. What would the sale price of each of these be?
(a) A dress marked at ? 120 (b) A pair of shoes marked at ? 750
(c) A bag marked at ? 250
2. A table marked at ? 15,000 is available for ? 14,400. Find the discount  given and
the discount per cent.
3. An almirah is sold at ? 5,225 after allowing a discount of 5%. Find its marked price.
7.2.1 Estimation in percentages
Your bill in a shop is ? 577.80 and the shopkeeper gives a discount of 15%. How would
you estimate the amount to be paid?
(i) Round off the bill to the nearest tens of ? 577.80, i.e., to ? 580.
(ii) Find 10% of this, i.e., ? 
10
580 58
100
×= ? .
(iii) Take half of this, i.e., 
1
58 29
2
×= ? .
(iv) Add the amounts in (ii) and (iii) to get ? 87.
You could therefore reduce your bill amount by ? 87 or by about ? 85, which will be
? 495 approximately.
1. Try estimating 20% of the same bill amount. 2. Try finding 15% of ? 375.
7.3  Sales Tax/Value Added Tax/Goods and
Services Tax
The teacher showed the class a bill in which the following heads were written.
Bill No. Date
Menu
S.No. Item Quantity Rate Amount
Bill amount
+ ST (5%)
Total
Even though the
discount was not
found, I could find
the sale price
directly.
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