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

 Page 1


MATHEMATICS 108
Comparing
Quantities Chapter  7
7.1  PERCENTAGE – ANOTHER WAY OF COMPARING
QUANTITIES
Anita’s Report Rita’s Report
Total 320/400 Total 300/360
Percentage: 80 Percentage: 83.3
Anita said that she has done better as she got 320 marks whereas Rita got only 300. Do
you agree with her? Who do you think has done better?
Mansi told them that they cannot decide who has done better by just comparing the
total marks obtained because the maximum marks out of which they got the marks are not
the same.
She said why don’t you see the Percentages given in your report cards?
Anita’s Percentage was 80 and Rita’s was 83.3. So, this shows Rita has done better.
Do you agree?
Percentages are numerators of fractions with denominator 100 and have been
used in comparing results. Let us try to understand in detail about it.
7.1.1  Meaning of Percentage
Per cent is derived from Latin word ‘per centum’ meaning ‘per hundred’.
Per cent is represented by the symbol % and means hundredths too. That is 1% means
1 out of hundred or one hundredth. It can be written as:  1% = 
1
100
 = 0.01
T o understand this, let us consider the following example.
2024-25
Page 2


MATHEMATICS 108
Comparing
Quantities Chapter  7
7.1  PERCENTAGE – ANOTHER WAY OF COMPARING
QUANTITIES
Anita’s Report Rita’s Report
Total 320/400 Total 300/360
Percentage: 80 Percentage: 83.3
Anita said that she has done better as she got 320 marks whereas Rita got only 300. Do
you agree with her? Who do you think has done better?
Mansi told them that they cannot decide who has done better by just comparing the
total marks obtained because the maximum marks out of which they got the marks are not
the same.
She said why don’t you see the Percentages given in your report cards?
Anita’s Percentage was 80 and Rita’s was 83.3. So, this shows Rita has done better.
Do you agree?
Percentages are numerators of fractions with denominator 100 and have been
used in comparing results. Let us try to understand in detail about it.
7.1.1  Meaning of Percentage
Per cent is derived from Latin word ‘per centum’ meaning ‘per hundred’.
Per cent is represented by the symbol % and means hundredths too. That is 1% means
1 out of hundred or one hundredth. It can be written as:  1% = 
1
100
 = 0.01
T o understand this, let us consider the following example.
2024-25
COMP ARING QUANTITIES 109
Rina made a table top of 100 different coloured tiles. She counted yellow, green, red
and blue tiles separately and filled the table below. Can you help her complete the table?
 Colour Number Rate per Fraction Written as Read as
of Tiles Hundred
 Y ellow 14 14
14
100
14% 14 per cent
 Green 26 26
26
100
26% 26 per cent
 Red 35 35  ---- ---- ----
 Blue 25 --------  ---- ---- ----
Total 100
1. Find the Percentage of children of different heights for the following data.
Height Number of Children In Fraction In Percentage
110 cm 22
120 cm 25
128 cm 32
130 cm 21
T otal 100
2. A shop has the following number of shoe pairs of different
sizes.
Size 2 : 20 Size 3 : 30 Size 4 : 28
Size 5 : 14 Size 6 : 8
Write this information in tabular form as done earlier and
find the Percentage of each shoe size available in the shop.
Percentages when total is not hundred
In all these examples, the total number of items add up to 100. For example, Rina had 100
tiles in all, there were 100 children and 100 shoe pairs. How do we calculate Percentage
of an item if the total number of items do not add up to 100? In such cases, we need to
convert the fraction to an equivalent fraction with denominator 100. Consider the following
example. Y ou have a necklace with twenty beads in two colours.
TRY THESE
2024-25
Page 3


MATHEMATICS 108
Comparing
Quantities Chapter  7
7.1  PERCENTAGE – ANOTHER WAY OF COMPARING
QUANTITIES
Anita’s Report Rita’s Report
Total 320/400 Total 300/360
Percentage: 80 Percentage: 83.3
Anita said that she has done better as she got 320 marks whereas Rita got only 300. Do
you agree with her? Who do you think has done better?
Mansi told them that they cannot decide who has done better by just comparing the
total marks obtained because the maximum marks out of which they got the marks are not
the same.
She said why don’t you see the Percentages given in your report cards?
Anita’s Percentage was 80 and Rita’s was 83.3. So, this shows Rita has done better.
Do you agree?
Percentages are numerators of fractions with denominator 100 and have been
used in comparing results. Let us try to understand in detail about it.
7.1.1  Meaning of Percentage
Per cent is derived from Latin word ‘per centum’ meaning ‘per hundred’.
Per cent is represented by the symbol % and means hundredths too. That is 1% means
1 out of hundred or one hundredth. It can be written as:  1% = 
1
100
 = 0.01
T o understand this, let us consider the following example.
2024-25
COMP ARING QUANTITIES 109
Rina made a table top of 100 different coloured tiles. She counted yellow, green, red
and blue tiles separately and filled the table below. Can you help her complete the table?
 Colour Number Rate per Fraction Written as Read as
of Tiles Hundred
 Y ellow 14 14
14
100
14% 14 per cent
 Green 26 26
26
100
26% 26 per cent
 Red 35 35  ---- ---- ----
 Blue 25 --------  ---- ---- ----
Total 100
1. Find the Percentage of children of different heights for the following data.
Height Number of Children In Fraction In Percentage
110 cm 22
120 cm 25
128 cm 32
130 cm 21
T otal 100
2. A shop has the following number of shoe pairs of different
sizes.
Size 2 : 20 Size 3 : 30 Size 4 : 28
Size 5 : 14 Size 6 : 8
Write this information in tabular form as done earlier and
find the Percentage of each shoe size available in the shop.
Percentages when total is not hundred
In all these examples, the total number of items add up to 100. For example, Rina had 100
tiles in all, there were 100 children and 100 shoe pairs. How do we calculate Percentage
of an item if the total number of items do not add up to 100? In such cases, we need to
convert the fraction to an equivalent fraction with denominator 100. Consider the following
example. Y ou have a necklace with twenty beads in two colours.
TRY THESE
2024-25
MATHEMATICS 110
Colour Number Fraction Denominator  Hundred In Percentage
of Beads
Red 8
8
20
8
20
100
100
40
100
× =
40%
Blue 12
12
20
12
20
100
100
60
100
× =
60%
T otal  20
We see that these three methods can be used to find the Percentage when the total
does not add to give 100. In the method shown in the table, we multiply the fraction by
100
100
. This does not change the value of the fraction. Subsequently , only 100 remains in the
denominator .
Anwar has used the unitary method. Asha has multiplied by 
5
5
 to get 100 in the
denominator. Y ou can use whichever method you find suitable. May be, you can make
your own method too.
The method used by Anwar can work for all ratios. Can the method used by Asha also
work for all ratios? Anwar says Asha’s method can be used only if you can find a natural
number which on multiplication with the denominator gives 100. Since denominator was 20,
she could multiply it by 5 to get 100. If the denominator was 6, she would not have been
able to use this method. Do you agree?
1. A collection of 10 chips with different colours is given .
Colour Number Fraction Denominator Hundred In Percentage
Green
Blue
Red
T otal
Fill the table and find the percentage of chips of each colour.
Asha does it like this
8
20
8 5
20 5
=
×
×
= 
=
40
100
 =  40%
Anwar found the Percentage of red beads like this
Out of 20 beads, the number of red beads is 8.
Hence, out of 100, the number of red beads is
8
100 40
20
× = (out of hundred) = 40%
TRY THESE
G G G G
R R
R
B B B
2024-25
Page 4


MATHEMATICS 108
Comparing
Quantities Chapter  7
7.1  PERCENTAGE – ANOTHER WAY OF COMPARING
QUANTITIES
Anita’s Report Rita’s Report
Total 320/400 Total 300/360
Percentage: 80 Percentage: 83.3
Anita said that she has done better as she got 320 marks whereas Rita got only 300. Do
you agree with her? Who do you think has done better?
Mansi told them that they cannot decide who has done better by just comparing the
total marks obtained because the maximum marks out of which they got the marks are not
the same.
She said why don’t you see the Percentages given in your report cards?
Anita’s Percentage was 80 and Rita’s was 83.3. So, this shows Rita has done better.
Do you agree?
Percentages are numerators of fractions with denominator 100 and have been
used in comparing results. Let us try to understand in detail about it.
7.1.1  Meaning of Percentage
Per cent is derived from Latin word ‘per centum’ meaning ‘per hundred’.
Per cent is represented by the symbol % and means hundredths too. That is 1% means
1 out of hundred or one hundredth. It can be written as:  1% = 
1
100
 = 0.01
T o understand this, let us consider the following example.
2024-25
COMP ARING QUANTITIES 109
Rina made a table top of 100 different coloured tiles. She counted yellow, green, red
and blue tiles separately and filled the table below. Can you help her complete the table?
 Colour Number Rate per Fraction Written as Read as
of Tiles Hundred
 Y ellow 14 14
14
100
14% 14 per cent
 Green 26 26
26
100
26% 26 per cent
 Red 35 35  ---- ---- ----
 Blue 25 --------  ---- ---- ----
Total 100
1. Find the Percentage of children of different heights for the following data.
Height Number of Children In Fraction In Percentage
110 cm 22
120 cm 25
128 cm 32
130 cm 21
T otal 100
2. A shop has the following number of shoe pairs of different
sizes.
Size 2 : 20 Size 3 : 30 Size 4 : 28
Size 5 : 14 Size 6 : 8
Write this information in tabular form as done earlier and
find the Percentage of each shoe size available in the shop.
Percentages when total is not hundred
In all these examples, the total number of items add up to 100. For example, Rina had 100
tiles in all, there were 100 children and 100 shoe pairs. How do we calculate Percentage
of an item if the total number of items do not add up to 100? In such cases, we need to
convert the fraction to an equivalent fraction with denominator 100. Consider the following
example. Y ou have a necklace with twenty beads in two colours.
TRY THESE
2024-25
MATHEMATICS 110
Colour Number Fraction Denominator  Hundred In Percentage
of Beads
Red 8
8
20
8
20
100
100
40
100
× =
40%
Blue 12
12
20
12
20
100
100
60
100
× =
60%
T otal  20
We see that these three methods can be used to find the Percentage when the total
does not add to give 100. In the method shown in the table, we multiply the fraction by
100
100
. This does not change the value of the fraction. Subsequently , only 100 remains in the
denominator .
Anwar has used the unitary method. Asha has multiplied by 
5
5
 to get 100 in the
denominator. Y ou can use whichever method you find suitable. May be, you can make
your own method too.
The method used by Anwar can work for all ratios. Can the method used by Asha also
work for all ratios? Anwar says Asha’s method can be used only if you can find a natural
number which on multiplication with the denominator gives 100. Since denominator was 20,
she could multiply it by 5 to get 100. If the denominator was 6, she would not have been
able to use this method. Do you agree?
1. A collection of 10 chips with different colours is given .
Colour Number Fraction Denominator Hundred In Percentage
Green
Blue
Red
T otal
Fill the table and find the percentage of chips of each colour.
Asha does it like this
8
20
8 5
20 5
=
×
×
= 
=
40
100
 =  40%
Anwar found the Percentage of red beads like this
Out of 20 beads, the number of red beads is 8.
Hence, out of 100, the number of red beads is
8
100 40
20
× = (out of hundred) = 40%
TRY THESE
G G G G
R R
R
B B B
2024-25
COMP ARING QUANTITIES 111
 2. Mala has a collection of bangles. She has 20 gold bangles and 10 silver bangles.
What is the percentage of bangles of each type? Can you put it in the tabular form
as done in the above example?
THINK, DISCUSS AND WRITE
1. Look at the examples below and in each of them, discuss which is better for
comparison.
In the atmosphere, 1 g of air contains:
.78 g Nitrogen 78% Nitrogen
.21 g Oxygen or 21% Oxygen
.01 g Other gas 1% Other gas
2. A shirt has:
3
5
 Cotton 60% Cotton
2
5
 Polyster 40% Polyster
7.1.2  Converting Fractional Numbers to Percentage
Fractional numbers can have different denominator. To compare fractional numbers, we
need a common denominator and we have seen that it is more convenient to compare if
our denominator is 100. That is, we are converting the fractions to Percentages. Let us try
converting different fractional numbers to Percentages.
EXAMPLE1 Write 
1
3
 as per cent.
SOLUTION We have, 
1 1 100 1
100%
3 3 100 3
= × = ×
= 
100 1
% 33 %
3 3
=
EXAMPLE 2 Out of 25 children in a class, 15 are girls. What is the percentage of girls?
SOLUTION Out of 25 children, there are 15 girls.
Therefore, percentage of girls = 
15
25
×100 = 60. There are 60% girls in the class.
EXAMPLE 3 Convert 
5
4
 to per cent.
SOLUTION We have, 
5
4
5
4
100 125 = × = % %
or
2024-25
Page 5


MATHEMATICS 108
Comparing
Quantities Chapter  7
7.1  PERCENTAGE – ANOTHER WAY OF COMPARING
QUANTITIES
Anita’s Report Rita’s Report
Total 320/400 Total 300/360
Percentage: 80 Percentage: 83.3
Anita said that she has done better as she got 320 marks whereas Rita got only 300. Do
you agree with her? Who do you think has done better?
Mansi told them that they cannot decide who has done better by just comparing the
total marks obtained because the maximum marks out of which they got the marks are not
the same.
She said why don’t you see the Percentages given in your report cards?
Anita’s Percentage was 80 and Rita’s was 83.3. So, this shows Rita has done better.
Do you agree?
Percentages are numerators of fractions with denominator 100 and have been
used in comparing results. Let us try to understand in detail about it.
7.1.1  Meaning of Percentage
Per cent is derived from Latin word ‘per centum’ meaning ‘per hundred’.
Per cent is represented by the symbol % and means hundredths too. That is 1% means
1 out of hundred or one hundredth. It can be written as:  1% = 
1
100
 = 0.01
T o understand this, let us consider the following example.
2024-25
COMP ARING QUANTITIES 109
Rina made a table top of 100 different coloured tiles. She counted yellow, green, red
and blue tiles separately and filled the table below. Can you help her complete the table?
 Colour Number Rate per Fraction Written as Read as
of Tiles Hundred
 Y ellow 14 14
14
100
14% 14 per cent
 Green 26 26
26
100
26% 26 per cent
 Red 35 35  ---- ---- ----
 Blue 25 --------  ---- ---- ----
Total 100
1. Find the Percentage of children of different heights for the following data.
Height Number of Children In Fraction In Percentage
110 cm 22
120 cm 25
128 cm 32
130 cm 21
T otal 100
2. A shop has the following number of shoe pairs of different
sizes.
Size 2 : 20 Size 3 : 30 Size 4 : 28
Size 5 : 14 Size 6 : 8
Write this information in tabular form as done earlier and
find the Percentage of each shoe size available in the shop.
Percentages when total is not hundred
In all these examples, the total number of items add up to 100. For example, Rina had 100
tiles in all, there were 100 children and 100 shoe pairs. How do we calculate Percentage
of an item if the total number of items do not add up to 100? In such cases, we need to
convert the fraction to an equivalent fraction with denominator 100. Consider the following
example. Y ou have a necklace with twenty beads in two colours.
TRY THESE
2024-25
MATHEMATICS 110
Colour Number Fraction Denominator  Hundred In Percentage
of Beads
Red 8
8
20
8
20
100
100
40
100
× =
40%
Blue 12
12
20
12
20
100
100
60
100
× =
60%
T otal  20
We see that these three methods can be used to find the Percentage when the total
does not add to give 100. In the method shown in the table, we multiply the fraction by
100
100
. This does not change the value of the fraction. Subsequently , only 100 remains in the
denominator .
Anwar has used the unitary method. Asha has multiplied by 
5
5
 to get 100 in the
denominator. Y ou can use whichever method you find suitable. May be, you can make
your own method too.
The method used by Anwar can work for all ratios. Can the method used by Asha also
work for all ratios? Anwar says Asha’s method can be used only if you can find a natural
number which on multiplication with the denominator gives 100. Since denominator was 20,
she could multiply it by 5 to get 100. If the denominator was 6, she would not have been
able to use this method. Do you agree?
1. A collection of 10 chips with different colours is given .
Colour Number Fraction Denominator Hundred In Percentage
Green
Blue
Red
T otal
Fill the table and find the percentage of chips of each colour.
Asha does it like this
8
20
8 5
20 5
=
×
×
= 
=
40
100
 =  40%
Anwar found the Percentage of red beads like this
Out of 20 beads, the number of red beads is 8.
Hence, out of 100, the number of red beads is
8
100 40
20
× = (out of hundred) = 40%
TRY THESE
G G G G
R R
R
B B B
2024-25
COMP ARING QUANTITIES 111
 2. Mala has a collection of bangles. She has 20 gold bangles and 10 silver bangles.
What is the percentage of bangles of each type? Can you put it in the tabular form
as done in the above example?
THINK, DISCUSS AND WRITE
1. Look at the examples below and in each of them, discuss which is better for
comparison.
In the atmosphere, 1 g of air contains:
.78 g Nitrogen 78% Nitrogen
.21 g Oxygen or 21% Oxygen
.01 g Other gas 1% Other gas
2. A shirt has:
3
5
 Cotton 60% Cotton
2
5
 Polyster 40% Polyster
7.1.2  Converting Fractional Numbers to Percentage
Fractional numbers can have different denominator. To compare fractional numbers, we
need a common denominator and we have seen that it is more convenient to compare if
our denominator is 100. That is, we are converting the fractions to Percentages. Let us try
converting different fractional numbers to Percentages.
EXAMPLE1 Write 
1
3
 as per cent.
SOLUTION We have, 
1 1 100 1
100%
3 3 100 3
= × = ×
= 
100 1
% 33 %
3 3
=
EXAMPLE 2 Out of 25 children in a class, 15 are girls. What is the percentage of girls?
SOLUTION Out of 25 children, there are 15 girls.
Therefore, percentage of girls = 
15
25
×100 = 60. There are 60% girls in the class.
EXAMPLE 3 Convert 
5
4
 to per cent.
SOLUTION We have, 
5
4
5
4
100 125 = × = % %
or
2024-25
MATHEMATICS 112
From these examples, we find that the percentages related to proper fractions are less
than 100 whereas percentages related to improper fractions are more than 100.
THINK, DISCUSS AND WRITE
(i) Can you eat 50% of a cake? Can you eat 100% of a cake?
Can you eat 150% of a cake?
(ii) Can a price of an item go up by 50%? Can a price of an item go up by 100%?
Can a price of an item go up by 150%?
7.1.3  Converting Decimals to Percentage
W e have seen how fractions can be converted to per cents. Let us now find how decimals
can be converted to per cents.
EXAMPLE 4  Convert the given decimals to per cents:
(a) 0.75 (b) 0.09 (c) 0.2
SOLUTION
(a) 0.75 = 0.75 × 100 % (b) 0.09 = 
9
100
 = 9 %
= 
75
100
 × 100 % = 75%
(c) 0.2 = 
2
10
 × 100% = 20 %
 1. Convert the following to per cents:
(a)
12
16
(b) 3.5 (c)
49
50
(d)
2
2
(e) 0.05
 2. (i) Out of 32 students, 8 are absent. What per cent of the students are absent?
(ii) There are 25 radios, 16 of them are out of order. What per cent of radios are
out of order?
(iii) A shop has 500 items, out of which 5 are defective. What per cent are defective?
(iv) There are 120 voters, 90 of them voted yes. What per cent voted yes?
7.1.4  Converting Percentages to Fractions or Decimals
We have so far converted fractions and decimals to percentages. We can also do the
reverse. That is, given per cents, we can convert them to decimals or fractions. Look at the
table, observe and complete it:
TRY THESE
2024-25