Page 1
12.1 Introduction
Chapter 12 Chapter 12 Chapter 12 Chapter 12 Chapter 12
Ratio and Ratio and
Ratio and Ratio and Ratio and
Proportion Proportion
Proportion Proportion Proportion
In our daily life, many a times we compare two
quantities of the same type. For example, Avnee and
Shari collected flowers for scrap notebook. Avnee
collected 30 flowers and Shari collected 45 flowers.
So, we may say that Shari collected 45 – 30 = 15
flowers more than Avnee.
Also, if height of Rahim is 150 cm and that of
Avnee is 140 cm then, we may say that the height of
Rahim is 150 cm – 140 cm = 10 cm more than Avnee.
This is one way of comparison by taking difference.
If we wish to compare the lengths of an ant and a
grasshopper, taking the difference does not express
the comparison. The grasshopper’s length, typically
4 cm to 5 cm is too long as compared to the ant’s
length which is a few mm. Comparison will be better
if we try to find that how many ants can be placed
one behind the other to match the length of
grasshopper. So, we can say that 20 to 30 ants have
the same length as a grasshopper.
Consider another example.
Cost of a car is ` 2,50,000 and that of a motorbike is ` 50,000. If we calculate
the difference between the costs, it is ` 2,00,000 and if we compare by division;
i.e.
2,50,000
50,000
=
5
1
Rationalised 2023-24
Page 2
12.1 Introduction
Chapter 12 Chapter 12 Chapter 12 Chapter 12 Chapter 12
Ratio and Ratio and
Ratio and Ratio and Ratio and
Proportion Proportion
Proportion Proportion Proportion
In our daily life, many a times we compare two
quantities of the same type. For example, Avnee and
Shari collected flowers for scrap notebook. Avnee
collected 30 flowers and Shari collected 45 flowers.
So, we may say that Shari collected 45 – 30 = 15
flowers more than Avnee.
Also, if height of Rahim is 150 cm and that of
Avnee is 140 cm then, we may say that the height of
Rahim is 150 cm – 140 cm = 10 cm more than Avnee.
This is one way of comparison by taking difference.
If we wish to compare the lengths of an ant and a
grasshopper, taking the difference does not express
the comparison. The grasshopper’s length, typically
4 cm to 5 cm is too long as compared to the ant’s
length which is a few mm. Comparison will be better
if we try to find that how many ants can be placed
one behind the other to match the length of
grasshopper. So, we can say that 20 to 30 ants have
the same length as a grasshopper.
Consider another example.
Cost of a car is ` 2,50,000 and that of a motorbike is ` 50,000. If we calculate
the difference between the costs, it is ` 2,00,000 and if we compare by division;
i.e.
2,50,000
50,000
=
5
1
Rationalised 2023-24
MATHEMATICS
178
We can say that the cost of the car is five times the cost of the motorbike.
Thus, in certain situations, comparison by division makes better sense than
comparison by taking the difference. The comparison by division is the Ratio.
In the next section, we shall learn more about ‘Ratios’.
12.2 Ratio
Consider the following:
Isha’s weight is 25 kg and her father’s weight is 75 kg. How many times
Father’s weight is of Isha’s weight? It is three times.
Cost of a pen is ` 10 and cost of a pencil is ` 2. How many times the cost of a pen
that of a pencil? Obviously it is five times.
In the above examples, we compared the two quantities in terms of
‘how many times’. This comparison is known as the Ratio. We denote
ratio using symbol ‘:’
Consider the earlier examples again. We can say,
The ratio of father’s weight to Isha’s weight =
75
25
=
3
1
= 3:1
The ratio of the cost of a pen to the cost of a pencil =
10
2
=
5
1
= 5:1
Let us look at this problem.
In a class, there are 20 boys and 40 girls. What is the ratio of
(a) Number of girls to the total number of students.
(b) Number of boys to the total number of students.
First we need to find the total number of students,
which is,
Number of girls + Number of boys = 20 + 40 = 60.
Then, the ratio of number of girls to the total
number of students is
40
60
=
2
3
= 2 : 3
Find the answer of part (b) in the similar manner.
Now consider the following example.
Length of a house lizard is 20 cm and the length of
a crocodile is 4 m.
“I am 5
times bigger
than you”, says
the lizard. As
we can see this
1. In a class, there are
20 boys and 40
girls. What is the
ratio of the number
of boys to the
number of girls?
2. Ravi walks 6 km
in an hour while
Roshan walks
4 km in an hour.
What is the ratio
of the distance
covered by Ravi
to the distance
covered by Roshan?
I am bigger
You are smaller
Rationalised 2023-24
Page 3
12.1 Introduction
Chapter 12 Chapter 12 Chapter 12 Chapter 12 Chapter 12
Ratio and Ratio and
Ratio and Ratio and Ratio and
Proportion Proportion
Proportion Proportion Proportion
In our daily life, many a times we compare two
quantities of the same type. For example, Avnee and
Shari collected flowers for scrap notebook. Avnee
collected 30 flowers and Shari collected 45 flowers.
So, we may say that Shari collected 45 – 30 = 15
flowers more than Avnee.
Also, if height of Rahim is 150 cm and that of
Avnee is 140 cm then, we may say that the height of
Rahim is 150 cm – 140 cm = 10 cm more than Avnee.
This is one way of comparison by taking difference.
If we wish to compare the lengths of an ant and a
grasshopper, taking the difference does not express
the comparison. The grasshopper’s length, typically
4 cm to 5 cm is too long as compared to the ant’s
length which is a few mm. Comparison will be better
if we try to find that how many ants can be placed
one behind the other to match the length of
grasshopper. So, we can say that 20 to 30 ants have
the same length as a grasshopper.
Consider another example.
Cost of a car is ` 2,50,000 and that of a motorbike is ` 50,000. If we calculate
the difference between the costs, it is ` 2,00,000 and if we compare by division;
i.e.
2,50,000
50,000
=
5
1
Rationalised 2023-24
MATHEMATICS
178
We can say that the cost of the car is five times the cost of the motorbike.
Thus, in certain situations, comparison by division makes better sense than
comparison by taking the difference. The comparison by division is the Ratio.
In the next section, we shall learn more about ‘Ratios’.
12.2 Ratio
Consider the following:
Isha’s weight is 25 kg and her father’s weight is 75 kg. How many times
Father’s weight is of Isha’s weight? It is three times.
Cost of a pen is ` 10 and cost of a pencil is ` 2. How many times the cost of a pen
that of a pencil? Obviously it is five times.
In the above examples, we compared the two quantities in terms of
‘how many times’. This comparison is known as the Ratio. We denote
ratio using symbol ‘:’
Consider the earlier examples again. We can say,
The ratio of father’s weight to Isha’s weight =
75
25
=
3
1
= 3:1
The ratio of the cost of a pen to the cost of a pencil =
10
2
=
5
1
= 5:1
Let us look at this problem.
In a class, there are 20 boys and 40 girls. What is the ratio of
(a) Number of girls to the total number of students.
(b) Number of boys to the total number of students.
First we need to find the total number of students,
which is,
Number of girls + Number of boys = 20 + 40 = 60.
Then, the ratio of number of girls to the total
number of students is
40
60
=
2
3
= 2 : 3
Find the answer of part (b) in the similar manner.
Now consider the following example.
Length of a house lizard is 20 cm and the length of
a crocodile is 4 m.
“I am 5
times bigger
than you”, says
the lizard. As
we can see this
1. In a class, there are
20 boys and 40
girls. What is the
ratio of the number
of boys to the
number of girls?
2. Ravi walks 6 km
in an hour while
Roshan walks
4 km in an hour.
What is the ratio
of the distance
covered by Ravi
to the distance
covered by Roshan?
I am bigger
You are smaller
Rationalised 2023-24
RATIO AND PROPORTION
179
is really absurd. A lizard’s length cannot be 5 times of the length of a crocodile.
So, what is wrong? Observe that the length of the lizard is in centimetres and
length of the crocodile is in metres. So, we have to convert their lengths into
the same unit.
Length of the crocodile = 4 m = 4 × 100 = 400 cm.
Therefore, ratio of the length of the crocodile to the length of the lizard
=
400
20
= =
20
1
20 1 :
.
Two quantities can be compared only if they are in the same unit.
Now what is the ratio of the length of the lizard to the length of the crocodile?
It is
20
400
= =
1
20
1 20 : .
Observe that the two ratios 1 : 20 and 20 : 1 are different from each other.
The ratio 1 : 20 is the ratio of the length of the lizard to the length of the
crocodile whereas, 20 : 1 is the ratio of the length of the crocodile to the
length of the lizard.
Now consider another example.
Length of a pencil is 18 cm and its
diameter is 8 mm. What is the ratio of
the diameter of the pencil to that of its
length? Since the length and the
diameter of the pencil are given in
different units, we first need to convert
them into same unit.
Thus, length of the pencil = 18 cm
= 18 × 10 mm = 180 mm.
The ratio of the diameter of the
pencil to that of the length of the pencil
=
8
180
=
2
45
=2:45 .
Think of some
more situations
where you compare
two quantities of same type in different units.
We use the concept of ratio in many situations of our
daily life without realising that we do so.
Compare the drawings A and B. B looks more natural
than A. Why?
A B
1. Saurabh takes 15 minutes to reach
school from his house and Sachin
takes one hour to reach school
from his house. Find the ratio of
the time taken by Saurabh to the
time taken by Sachin.
2. Cost of a toffee is 50 paise and
cost of a chocolate is ` 10. Find the
ratio of the cost of a toffee to the
cost of a chocolate.
3. In a school, there were 73
holidays in one year. What is the
ratio of the number of holidays
to the number of days in one year?
Rationalised 2023-24
Page 4
12.1 Introduction
Chapter 12 Chapter 12 Chapter 12 Chapter 12 Chapter 12
Ratio and Ratio and
Ratio and Ratio and Ratio and
Proportion Proportion
Proportion Proportion Proportion
In our daily life, many a times we compare two
quantities of the same type. For example, Avnee and
Shari collected flowers for scrap notebook. Avnee
collected 30 flowers and Shari collected 45 flowers.
So, we may say that Shari collected 45 – 30 = 15
flowers more than Avnee.
Also, if height of Rahim is 150 cm and that of
Avnee is 140 cm then, we may say that the height of
Rahim is 150 cm – 140 cm = 10 cm more than Avnee.
This is one way of comparison by taking difference.
If we wish to compare the lengths of an ant and a
grasshopper, taking the difference does not express
the comparison. The grasshopper’s length, typically
4 cm to 5 cm is too long as compared to the ant’s
length which is a few mm. Comparison will be better
if we try to find that how many ants can be placed
one behind the other to match the length of
grasshopper. So, we can say that 20 to 30 ants have
the same length as a grasshopper.
Consider another example.
Cost of a car is ` 2,50,000 and that of a motorbike is ` 50,000. If we calculate
the difference between the costs, it is ` 2,00,000 and if we compare by division;
i.e.
2,50,000
50,000
=
5
1
Rationalised 2023-24
MATHEMATICS
178
We can say that the cost of the car is five times the cost of the motorbike.
Thus, in certain situations, comparison by division makes better sense than
comparison by taking the difference. The comparison by division is the Ratio.
In the next section, we shall learn more about ‘Ratios’.
12.2 Ratio
Consider the following:
Isha’s weight is 25 kg and her father’s weight is 75 kg. How many times
Father’s weight is of Isha’s weight? It is three times.
Cost of a pen is ` 10 and cost of a pencil is ` 2. How many times the cost of a pen
that of a pencil? Obviously it is five times.
In the above examples, we compared the two quantities in terms of
‘how many times’. This comparison is known as the Ratio. We denote
ratio using symbol ‘:’
Consider the earlier examples again. We can say,
The ratio of father’s weight to Isha’s weight =
75
25
=
3
1
= 3:1
The ratio of the cost of a pen to the cost of a pencil =
10
2
=
5
1
= 5:1
Let us look at this problem.
In a class, there are 20 boys and 40 girls. What is the ratio of
(a) Number of girls to the total number of students.
(b) Number of boys to the total number of students.
First we need to find the total number of students,
which is,
Number of girls + Number of boys = 20 + 40 = 60.
Then, the ratio of number of girls to the total
number of students is
40
60
=
2
3
= 2 : 3
Find the answer of part (b) in the similar manner.
Now consider the following example.
Length of a house lizard is 20 cm and the length of
a crocodile is 4 m.
“I am 5
times bigger
than you”, says
the lizard. As
we can see this
1. In a class, there are
20 boys and 40
girls. What is the
ratio of the number
of boys to the
number of girls?
2. Ravi walks 6 km
in an hour while
Roshan walks
4 km in an hour.
What is the ratio
of the distance
covered by Ravi
to the distance
covered by Roshan?
I am bigger
You are smaller
Rationalised 2023-24
RATIO AND PROPORTION
179
is really absurd. A lizard’s length cannot be 5 times of the length of a crocodile.
So, what is wrong? Observe that the length of the lizard is in centimetres and
length of the crocodile is in metres. So, we have to convert their lengths into
the same unit.
Length of the crocodile = 4 m = 4 × 100 = 400 cm.
Therefore, ratio of the length of the crocodile to the length of the lizard
=
400
20
= =
20
1
20 1 :
.
Two quantities can be compared only if they are in the same unit.
Now what is the ratio of the length of the lizard to the length of the crocodile?
It is
20
400
= =
1
20
1 20 : .
Observe that the two ratios 1 : 20 and 20 : 1 are different from each other.
The ratio 1 : 20 is the ratio of the length of the lizard to the length of the
crocodile whereas, 20 : 1 is the ratio of the length of the crocodile to the
length of the lizard.
Now consider another example.
Length of a pencil is 18 cm and its
diameter is 8 mm. What is the ratio of
the diameter of the pencil to that of its
length? Since the length and the
diameter of the pencil are given in
different units, we first need to convert
them into same unit.
Thus, length of the pencil = 18 cm
= 18 × 10 mm = 180 mm.
The ratio of the diameter of the
pencil to that of the length of the pencil
=
8
180
=
2
45
=2:45 .
Think of some
more situations
where you compare
two quantities of same type in different units.
We use the concept of ratio in many situations of our
daily life without realising that we do so.
Compare the drawings A and B. B looks more natural
than A. Why?
A B
1. Saurabh takes 15 minutes to reach
school from his house and Sachin
takes one hour to reach school
from his house. Find the ratio of
the time taken by Saurabh to the
time taken by Sachin.
2. Cost of a toffee is 50 paise and
cost of a chocolate is ` 10. Find the
ratio of the cost of a toffee to the
cost of a chocolate.
3. In a school, there were 73
holidays in one year. What is the
ratio of the number of holidays
to the number of days in one year?
Rationalised 2023-24
MATHEMATICS
180
The legs in the picture A are too long in comparison to the other body parts.
This is because we normally expect a certain ratio of the length of legs to the
length of whole body.
Compare the two pictures of a pencil. Is the
first one looking like a full pencil? No.
Why not? The reason is that the thickness and
the length of the pencil are not in the correct ratio.
Same ratio in different situations :
Consider the following :
l
Length of a room is 30 m and its breadth is 20 m. So, the ratio of length of
the room to the breadth of the room =
30
20
=
3
2
=3:2
l l l l l
There are 24 girls and 16 boys going for a picnic. Ratio of the number of
girls to the number of boys =
24
16
=
3
2
=3:2
The ratio in both the examples is 3 : 2.
l l l l l
Note the ratios 30 : 20 and 24 : 16 in lowest form are same as 3 : 2. These
are equivalent ratios.
l l l l l
Can you think of some more examples having the ratio 3 : 2?
It is fun to write situations that give rise to a certain ratio. For example,
write situations that give the ratio 2 : 3.
l
Ratio of the breadth of a table to the length of the table is 2 : 3.
l
Sheena has 2 marbles and her friend Shabnam has 3 marbles.
Then, the ratio of marbles that Sheena and Shabnam have is 2 : 3.
Can you write some more situations for this ratio? Give any ratio to your
friends and ask them to frame situations.
Ravi and Rani started a business and invested
money in the ratio 2 : 3. After one year the total
profit was ` 4,00,000.
Ravi said “we would divide it equally”, Rani
said “I should get more as I have invested more”.
It was then decided that profit will be divided
in the ratio of their investment.
Here, the two terms of the ratio 2 : 3 are 2
and 3.
Sum of these terms = 2 + 3 = 5
What does this mean?
This means if the profit is ` 5 then Ravi should get ` 2 and Rani should get
` 3. Or, we can say that Ravi gets 2 parts and Rani gets 3 parts out of the 5 parts.
Rationalised 2023-24
Page 5
12.1 Introduction
Chapter 12 Chapter 12 Chapter 12 Chapter 12 Chapter 12
Ratio and Ratio and
Ratio and Ratio and Ratio and
Proportion Proportion
Proportion Proportion Proportion
In our daily life, many a times we compare two
quantities of the same type. For example, Avnee and
Shari collected flowers for scrap notebook. Avnee
collected 30 flowers and Shari collected 45 flowers.
So, we may say that Shari collected 45 – 30 = 15
flowers more than Avnee.
Also, if height of Rahim is 150 cm and that of
Avnee is 140 cm then, we may say that the height of
Rahim is 150 cm – 140 cm = 10 cm more than Avnee.
This is one way of comparison by taking difference.
If we wish to compare the lengths of an ant and a
grasshopper, taking the difference does not express
the comparison. The grasshopper’s length, typically
4 cm to 5 cm is too long as compared to the ant’s
length which is a few mm. Comparison will be better
if we try to find that how many ants can be placed
one behind the other to match the length of
grasshopper. So, we can say that 20 to 30 ants have
the same length as a grasshopper.
Consider another example.
Cost of a car is ` 2,50,000 and that of a motorbike is ` 50,000. If we calculate
the difference between the costs, it is ` 2,00,000 and if we compare by division;
i.e.
2,50,000
50,000
=
5
1
Rationalised 2023-24
MATHEMATICS
178
We can say that the cost of the car is five times the cost of the motorbike.
Thus, in certain situations, comparison by division makes better sense than
comparison by taking the difference. The comparison by division is the Ratio.
In the next section, we shall learn more about ‘Ratios’.
12.2 Ratio
Consider the following:
Isha’s weight is 25 kg and her father’s weight is 75 kg. How many times
Father’s weight is of Isha’s weight? It is three times.
Cost of a pen is ` 10 and cost of a pencil is ` 2. How many times the cost of a pen
that of a pencil? Obviously it is five times.
In the above examples, we compared the two quantities in terms of
‘how many times’. This comparison is known as the Ratio. We denote
ratio using symbol ‘:’
Consider the earlier examples again. We can say,
The ratio of father’s weight to Isha’s weight =
75
25
=
3
1
= 3:1
The ratio of the cost of a pen to the cost of a pencil =
10
2
=
5
1
= 5:1
Let us look at this problem.
In a class, there are 20 boys and 40 girls. What is the ratio of
(a) Number of girls to the total number of students.
(b) Number of boys to the total number of students.
First we need to find the total number of students,
which is,
Number of girls + Number of boys = 20 + 40 = 60.
Then, the ratio of number of girls to the total
number of students is
40
60
=
2
3
= 2 : 3
Find the answer of part (b) in the similar manner.
Now consider the following example.
Length of a house lizard is 20 cm and the length of
a crocodile is 4 m.
“I am 5
times bigger
than you”, says
the lizard. As
we can see this
1. In a class, there are
20 boys and 40
girls. What is the
ratio of the number
of boys to the
number of girls?
2. Ravi walks 6 km
in an hour while
Roshan walks
4 km in an hour.
What is the ratio
of the distance
covered by Ravi
to the distance
covered by Roshan?
I am bigger
You are smaller
Rationalised 2023-24
RATIO AND PROPORTION
179
is really absurd. A lizard’s length cannot be 5 times of the length of a crocodile.
So, what is wrong? Observe that the length of the lizard is in centimetres and
length of the crocodile is in metres. So, we have to convert their lengths into
the same unit.
Length of the crocodile = 4 m = 4 × 100 = 400 cm.
Therefore, ratio of the length of the crocodile to the length of the lizard
=
400
20
= =
20
1
20 1 :
.
Two quantities can be compared only if they are in the same unit.
Now what is the ratio of the length of the lizard to the length of the crocodile?
It is
20
400
= =
1
20
1 20 : .
Observe that the two ratios 1 : 20 and 20 : 1 are different from each other.
The ratio 1 : 20 is the ratio of the length of the lizard to the length of the
crocodile whereas, 20 : 1 is the ratio of the length of the crocodile to the
length of the lizard.
Now consider another example.
Length of a pencil is 18 cm and its
diameter is 8 mm. What is the ratio of
the diameter of the pencil to that of its
length? Since the length and the
diameter of the pencil are given in
different units, we first need to convert
them into same unit.
Thus, length of the pencil = 18 cm
= 18 × 10 mm = 180 mm.
The ratio of the diameter of the
pencil to that of the length of the pencil
=
8
180
=
2
45
=2:45 .
Think of some
more situations
where you compare
two quantities of same type in different units.
We use the concept of ratio in many situations of our
daily life without realising that we do so.
Compare the drawings A and B. B looks more natural
than A. Why?
A B
1. Saurabh takes 15 minutes to reach
school from his house and Sachin
takes one hour to reach school
from his house. Find the ratio of
the time taken by Saurabh to the
time taken by Sachin.
2. Cost of a toffee is 50 paise and
cost of a chocolate is ` 10. Find the
ratio of the cost of a toffee to the
cost of a chocolate.
3. In a school, there were 73
holidays in one year. What is the
ratio of the number of holidays
to the number of days in one year?
Rationalised 2023-24
MATHEMATICS
180
The legs in the picture A are too long in comparison to the other body parts.
This is because we normally expect a certain ratio of the length of legs to the
length of whole body.
Compare the two pictures of a pencil. Is the
first one looking like a full pencil? No.
Why not? The reason is that the thickness and
the length of the pencil are not in the correct ratio.
Same ratio in different situations :
Consider the following :
l
Length of a room is 30 m and its breadth is 20 m. So, the ratio of length of
the room to the breadth of the room =
30
20
=
3
2
=3:2
l l l l l
There are 24 girls and 16 boys going for a picnic. Ratio of the number of
girls to the number of boys =
24
16
=
3
2
=3:2
The ratio in both the examples is 3 : 2.
l l l l l
Note the ratios 30 : 20 and 24 : 16 in lowest form are same as 3 : 2. These
are equivalent ratios.
l l l l l
Can you think of some more examples having the ratio 3 : 2?
It is fun to write situations that give rise to a certain ratio. For example,
write situations that give the ratio 2 : 3.
l
Ratio of the breadth of a table to the length of the table is 2 : 3.
l
Sheena has 2 marbles and her friend Shabnam has 3 marbles.
Then, the ratio of marbles that Sheena and Shabnam have is 2 : 3.
Can you write some more situations for this ratio? Give any ratio to your
friends and ask them to frame situations.
Ravi and Rani started a business and invested
money in the ratio 2 : 3. After one year the total
profit was ` 4,00,000.
Ravi said “we would divide it equally”, Rani
said “I should get more as I have invested more”.
It was then decided that profit will be divided
in the ratio of their investment.
Here, the two terms of the ratio 2 : 3 are 2
and 3.
Sum of these terms = 2 + 3 = 5
What does this mean?
This means if the profit is ` 5 then Ravi should get ` 2 and Rani should get
` 3. Or, we can say that Ravi gets 2 parts and Rani gets 3 parts out of the 5 parts.
Rationalised 2023-24
RATIO AND PROPORTION
181
i.e., Ravi should get
2
5
of the total profit and Rani should get
3
5
of the total
profit.
If the total profit were ` 500
Ravi would get `
2
5
× 500 = ` 200
and Rani would get
3
5
× 500 = ` 300
Now, if the profit were ` 4,00,000 could you find the share of each?
Ravi’s share = `
2
5
×
× 4,00,000 = ` 1,60,000
And Rani’s share = `
3
5
× 4,00,000 = ` 2,40,000
Can you think of some more examples where you have to divide a number
of things in some ratio? Frame three such examples and ask your friends to
solve them.
Let us look at the kind of problems we have solved so far.
Example 1 : Length and breadth of a rectangular field are 50 m and 15 m
respectively. Find the ratio of the length to the breadth of the field.
Solution : Length of the rectangular field = 50 m
Breadth of the rectangular field = 15 m
The ratio of the length to the breadth is 50 : 15
The ratio can be written as = 10 : 3
Thus, the required ratio is 10 : 3.
1. Find the ratio of number of notebooks to the number of
books in your bag.
2. Find the ratio of number of desks and chairs in your
classroom.
3. Find the number of students above twelve years of age in your class.
Then, find the ratio of number of students with age above twelve years
and the remaining students.
4. Find the ratio of number of doors and the number of windows in your
classroom.
5. Draw any rectangle and find the ratio of its length to its breadth.
Rationalised 2023-24
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