Recall that when some whole number of things are shared equally among some number of people, fractions tell us how much each share is.
Shabnam: Do you remember, if one roti is divided equally
between two children, how much
roti will each child get?
Mukta: Each child will get half a roti.
Shabnam: The fraction ‘one half’ is written as
12 . We also sometimes read this as
‘one upon two.’
Mukta: If one roti is equally shared among
4 children, how much roti will one child get?
Shabnam: Each child’s share is
14 roti.
Mukta: And which is more
12 roti or 14 roti?
Shabnam: When 2 children share 1 roti
equally, each child gets
12 roti. When
4 children share 1 roti equally, each
child gets
14 roti. Since, in the second
group more children share the
same one roti, each child gets a smaller share. So,
12 roti is
more than
14 roti.
12 > 14
7.1 Fractional Units and Equal Shares
Beni: Which fraction is greater —
15 or 19?
Arvin: 9 is bigger than 5. So I would guess that
19 is greater than 15. Am I right?Beni: No! That is a common mistake. Think of these fractions as shares.Arvin: If one roti is shared among 5 children, each one gets a share of 15 roti. If one roti is shared among 9 children, each one gets a share of 19 roti?Beni: Exactly! Now think again - which share is higher?
Arvin: If I share with more people, I will get less. So,
19 < 15.Beni: You got it!
Oh, so 1100 is bigger than 1200!
When one unit is divided into several equal parts, each part is called a fractional unit. These are all fractional units:
12 , 13 , 14 , 15 , 16 , …, 110 , …, 150 , …, 1100, etc.
We also sometimes refer to fractional units as ‘unit fractions.’
Figure it Out
Fill in the blanks with fractions.
1. Three guavas together weigh 1 kg. If they are roughly of the same size, each guava will roughly weigh ____ kg.
2. A wholesale merchant packed 1 kg of rice in four packets of equal weight. The weight of each packet is ___ kg.
Math Talk
3. Four friends ordered 3 glasses of sugarcane juice and shared it equally among themselves. Each one drank ____ glass of sugarcane juice.
4. The big fish weighs 12 kg. The small one weighs 14 kg.
Math Talk
Together they weigh ____ kg.
Knowledge from the past!
Fractions have been used and named in India since ancient times. In the Rig Veda, the fraction 34 is referred to as tri-pada. This has the same meaning as the words for 34 in many Indian languages today, e.g., ʻteen paavʼ in colloquial Hindi and ‘mukkaal’ in Tamil. Indeed, words for fractions used today in many Indian languages go back to ancient times.
Find out and discuss the words for fractions that are used in the different languages spoken in your home, city, or state. Ask your grandparents, parents, teachers, and classmates what words they use for different fractions, such as for one and a half, three quarters, one and a quarter, half, quarter, and two and a half, and write them here:
___________ ___________ ___________ ___________ ___________ ___________
5. Arrange these fraction words in order of size from the smallest to the biggest in the empty box below:
One and a half, three quarters, one and a quarter, half, quarter, two and a half.
Write your answer here.
7.2 Fractional Units as Parts of a Whole
The picture shows a whole chikki.
A whole chikkiA picture of the chikki broken into 2 pieces is shown below. How much of the original chikki is each piece?
14
We can see that the bigger piece has 3 pieces of
14 chikki in it. So, we
can measure the bigger piece using the fractional unit
14. We see that
the bigger piece is
34 chikki.
16
16
A whole chikki cut into 6 equal pieces.A whole chikki cut into 6 equal pieces in a different way.
By dividing the whole chikki into 6 equal parts in different
Math Talk
ways, we get
16 chikki pieces of different shapes. Are they of
the same size?
What is the fractional unit of chikki shown below?
We get this piece by breaking the chikki into 3 equal pieces. So this is
13 chikki.
13
A whole chikki
Figure it Out
The figures below show different fractional units of a whole chikki. How much of a whole chikki is each piece?
a.
b.c.d.
e.f.g.h.
7.3 Measuring Using Fractional Units
Take a strip of paper. We consider this paper strip to be one unit long.
1 Strip Paper
Fold the strip into two equal parts and then open up the strip again. Taking the strip to be one unit in length, what are the lengths of the two new parts of the strip created by the crease?
12
12
What will you get if you fold the previously-folded strip again into two equal parts? You will now get four equal parts.
14
2 times 14 = 24
3 times 14 = 34
4 times 14 = 44
Do it once more! Fill in the blank boxes.
2 times 18
4 times 18
6 times 18
8 times 18 88==1
Fractional quantities can be measured using fractional units.
Represents a full roti (whole)
Let us look at another example,
12
12 + 12 + 12
12 + 12 + 12 + 12
12 + 12 + 12 + 12 + 12
12 + 12
= 1 times half
= 3 times half
= 4 times half
= 5 times half
= 2 times half
We can describe how much the quantity is by collecting together the fractional units.
Figure it Out
1. Continue this table of 12 for 2 more steps.
2. Can you create a similar table for 14?
3. Make 13 using a paper strip. Can you use this to also make 16 ?
4. Draw a picture and write an addition statement as above to show:
a. 5 times 14 of a roti b. 9 times 14 of a roti
5. Match each fractional unit with the correct picture:
13
15
18
16
Reading Fractions
We usually read the fraction 34 as ‘three quarters’ or ‘three upon four’,
but reading it as ‘3 times 14’ helps us to understand the size of the
fraction because it clearly shows what the fractional unit is (14) and
how many such fractional units (3) there are.
Recall what we call the top number and the bottom number of fractions.
In the fraction 56 , 5 is the numerator and 6 is the denominator.
Teacher’s Note
Give several opportunities to the children to explore the idea of fractional units with different shapes like circles, squares, rectangles, triangles, etc.
7.4 Marking Fraction Lengths on the Number Line
We have marked lengths equal to 1, 2, 3, … units on the number line. Now, let us try to mark lengths equal to fractions on the number line. What is the length of the blue line? Write the fraction that gives the length of the blue line in the box.
012
The distance between 0 and 1 is one unit long. It is divided into two equal parts. So, the length of each part is 12 unit. So, this blue line is 12 unit long.
Now, can you find the lengths of the various blue lines shown below? Fill in the boxes as well. 1. Here, the fractional unit is dividing a length of 1 unit into three
equal parts. Write the fraction that gives the length of the blue line in the box or in your notebook.
01213
2. Here, a unit is divided into 5 equal parts. Write the fraction that
gives the length of the blue lines in the respective boxes or in your notebook.
01215
35
3. Now, a unit is divided into 8 equal parts. Write the appropriate fractions in your notebook.
Figure it Out
1. On a number line, draw lines of lengths 110 , 310 , and 45 .
Math Talk
2. Write five more fractions of your choice and mark them on the
number line.
3. How many fractions lie between 0 and 1? Think, discuss with
your classmates, and write your answer.
4. What is the length of the blue line and black line shown below? The
distance between 0 and 1 is 1 unit long, and it is divided into two
equal parts. The length of each part is 12. So the blue line is 12 units
long. Write the fraction that gives the length of the black line in the
box.
12012
5. Write the fraction that gives the lengths of the black lines in the respective boxes.
01215
25
35
45
Teacher’s Note
Draw these lines on the board and ask the students to write the answers in their notebooks.
7.5 Mixed Fractions
Fractions greater than oneYou marked some fractions on the number line earlier. Did you notice that the lengths of all the blue lines were less than one and the lengths of all the black lines were more than 1?
Write down all the fractions you marked on the number line earlier. Now, let us classify these in two groups:
Lengths less than 1 unitLengths more than 1 unit
Did you notice something common between the fractions that are greater than 1?
In all the fractions that are less than 1 unit, the numerator is
smaller than the denominator, while in the fractions that are more
than 1 unit, the numerator is larger than the denominator.
We know that
32 , 52 and 72 are all greater than 1 unit. But can we
see how many whole units they contain?
32 = 12 + 12 + 12 = 1 + 12
52 = 12 + 12 + 12 + 12 + 12 = 2 + 12
I know that 13 + 13 + 13 = 33 = 1. If I add one more 13,
I will get more than 1 unit! So, 43 > 1.
Figure it Out
1. How many whole units are there in 72 ?
Math Talk
2. How many whole units are there in 43 and in 73 ?
Writing fractions greater than one as mixed numbers
We saw that: 32 = 1 + 12 .
We can write other fractions in a similar way. For example,
43 = 13 + 13 + 13 + 13 = 1 + 13 .
3 ×
13 = 1
Figure it Out
1. Figure out the number of whole units in each of the following fractions:
a.
83 b. 115 c. 94
We saw that
This number is thus also called ‘two and
83 = 2 + 23
two thirds’. We also write it as 2 23 .
Fraction Mixed number
2. Can all fractions greater than 1 be written as such mixed numbers?
A mixed number or mixed fraction contains a whole number (called the whole part) and a fraction that is less than 1 (called the fractional part).
3. Write the following fractions as mixed fractions (e.g., 92 = 4 12 ):
a.
92 b. 95 c. 2119 d. 479 e. 1211 f. 196
Can we write a mixed number (mixed fraction) as a regular fraction?
Yes! I figured out a way to write a mixed number as a regular fraction!
Jaya: When I have 3 +
34 , this means 1 + 1 + 1 + 34 . I know
1 =
14 + 14 + 14 + 14 .
So I get
(
14 + 14 + 14 + 14 ) + ( 14 + 14 + 14 + 14 ) + ( 14 + 14 + 14 + 14 ) + ( 14 + 14 + 14 ) = 154 .
Therefore, (4 ×
14 ) + (4 × 14 ) + (4 × 14 ) + (3 × 14 ) = 154 .
Figure it Out
Write the following mixed numbers as fractions: a. 3 14 b. 7 23 c. 9 49
Math Talk
d. 3 16 e. 2 311 f. 3 910
7.6 Equivalent Fractions
Using a fraction wall to find equal fractional lengths!
In the previous section, you used paper folding to represent various fractions using fractional units. Let us do some more activities with the same paper strips.
What do you
observe?
12
• Are the lengths 12 and 24 equal?
• Are the lengths 24 and 48 equal?
We can say that 12 = 24 = 48 .
24
48
These are ‘equivalent fractions’ that denote the same length, but they are expressed in terms of different fractional units.Now, check whether 13 and 26 are equivalent fractions or not, using paper strips.Make your own fraction wall using such strips as given in the picture below!
Answer the following questions after looking at the fraction wall:
1. Are the lengths
12 and 36 equal?
2. Are
23 and 46 equivalent
1 UNIT
12
22
fractions? Why?
13
23
33
3. How many pieces of
14
24
34
44
length
16 will make a
15
25
35
45
55
length of
12 ?
16
26
36
46
56
66
4. How many pieces of length
16 will make a length of 13 ?
We can extend this idea to make a fraction wall up to the fractional unit
110. (This fraction wall is given at the end of the book.)
1 UNIT
1213141516171819110
22
23
33
24
34
44
25
35
45
55
26
36
46
56
66
27
37
47
57
67
17
28
38
48
58
68
78
88
29
39
49
59
69
79
99
89
210
310
410
510
610
710
810
910
1010
Figure it Out
1. Are
36 , 48 , 510 equivalent fractions? Why?
2. Write two equivalent fractions for
26 .
3.
46 = = = = ............ (Write as many as you can)
Understanding equivalent fractions using equal shares
One roti was shared equally by four children. What fraction of the whole did each child get? The adjoining picture shows the division of a roti among four children.
Fraction of roti each child got is
14 .
The four shares must be
equal to each other!
You can also express this event through division facts, addition facts, and multiplication facts.
The division fact is 1 ÷ 4 =
14 .
The addition fact is 1 =
14 + 14 + 14 + 14 .
The multiplication fact is 1 = 4 ×
14 .
Figure it Out
1. Three rotis are shared equally by four children. Show the division in the picture and write a fraction for how much each child gets. Also, write the corresponding division facts, addition facts, and, multiplication facts.
Fraction of roti each child gets is ______.
Division fact:
Addition fact:
Multiplication fact:
Compare your picture and answers with your classmates!
2. Draw a picture to show how much each child gets when 2 rotis are shared equally by 4 children. Also, write the corresponding division facts, addition facts, and multiplication facts.
3. Anil was in a group where 2 cakes were divided equally among 5 children. How much cake would Anil get?
What if we put two such groups
Now, if there are 10 children
together? one group where
in my group, how many
2 cakes are divided equally
cakes will I need so that they
between 5 children, and another
get same amount of cake as
group again with 4 cakes and
Anil?
10 children.
Group 1
Group 2
So, 25 = 410!
So, the share of each child is the same in both these situations!
Let us examine the shares of each child in the following situations.• 1 roti is divided equally between 2 children.• 2 rotis are divided equally among 4 children. • 3 rotis are divided equally among 6 children.
Let us draw and share! Did you notice that in each situation the share of every child is the same? So, we can say that 12 = 24 = 36 .
2 rotis are divided
1 roti is divided
3 rotis are divided
equally among 4
equally between 2
equally among 6
12
12
24
24
24
24
36
36
36
36
36
36
Fractions where the shares are equal are called ‘equivalent fractions’.
So,
12 , 24 , and 36 are all equivalent fractions.
Find some more fractions equivalent to
12 . Write them in the
boxes here:
Equally divide the rotis in the situations shown below and write down the share of each child. Are the shares in each of these cases the same? Why?
6 rotis divided
2 rotis divided
4 rotis divided
equally among
equally among
equally among
9 children
3 children
6 children
23
23
23
23 is also called the simplest form of 46 . It is also the simplest form of 69 as well.
Do you notice anything about the relationship between the
numerator and denominator in each of these fractions?
Figure it Out
Find the missing numbers:
a. 5 glasses of juice shared equally among 4 friends is the same as ____ glasses of juice shared equally among 8 friends.
Math Talk
So, 54 = 8 .
b. 4 kg of potatoes divided equally in 3 bags is the same as 12 kgs of potatoes divided equally in ___ bags.
So, 43 = 12
c. 7 rotis divided among 5 children is the same as ____ rotis divided among _____ children.
So, 75 = .
In which group will each child get more chikki?
1 chikki divided between 2 children or 5 chikkis divided among 8 children.
Mukta: So, we must compare
12 and 58 . Which is more?
Shabnam: Well, we have seen that
12 = 48 ; and clearly 48 < 58 . So, the
children for whom 5 chikkis is divided equally among 8 will get more than those children for whom 1 chikki is divided equally among 2. The children of the second group will get more chikki each.
What about the following groups? In which group will each child get more?
1 chikki divided between 2 children or 4 chikkis divided among 7 children.
Shabnam: The children of which group will get more chikki this time?
Mukta: We must compare
17 and 47 .
Now
1 × 42 × 4 = 48 so, 12 = 48 .
Shabnam: But why did you multiply the numerator and denominator by 4 again?
Mukta: You will see!
When 4 chikkis are divided equally among 7 children, each one will get 47 chikki. When 4 chikkis are divided equally among 8 children, each one will get 48 chikki. So 47 > 48.
If the number of units that are shared is the same, but the number of children among whom the units are shared is more, then the share is less.
Therefore, 47 > 48 and 48 = 12 , so 47 > 12.
Now I understood why you multiplied the numerator and denominator by 4.
Suppose the number of children is kept the same, but the number of units that are being shared is increased? What can you say about each child’s share now? Why? Discuss how your reasoning explains
15 < 25 , 37 < 47 , and 12 < 58 .
Now, decide in which of the two groups will each child get a larger share:
1. Group 1 : 3 glasses of sugarcane juice divided equally among 4 children. Group 2: 7 glasses of sugarcane juice divided equally among 10 children.
Math Talk
2. Group 1 : 4 glasses of sugarcane juice divided equally among 7 children. Group 2: 5 glasses of sugarcane juice divided equally among 7 children.
Which groups were easier to compare? Why?
When the number of
children is same, it is easier
Shabnam: To compare the first two groups,
to compare, isn’t it?
we have to find fractions equivalent to the fractions
34 and 710.
Mukta: How about
68 = 34 and 2130 = 710?
Shabnam: There is a condition. The fractional unit used for the two
fractions have to be the same! Like
26 and 36 both use the
same fractional unit
16 (i.e., the denominators are the same).
But
68 and 2130 do not use the same fractional units (they have
different denominators).
Mukta: Okay, so let us start making equivalent fractions then:
34 = 68 = 912 = 1216 = 1520 … But when do I stop?
Shabnam: Got it! How about we go on till 4 × 10 = 40.
Mukta: You mean the product of the two denominators? Sounds good!
We have
34 and 710. The product of the two denominators
(4 and 10) is 40.
34 = 68 = 912 = 1216 = 1520 = 1824 = … = 2736 = 3040 .
Go till we reach the
denominator 40.
710 = 1420 = 2130 = 2840.
1520 and
1420 also had
Yes! We just needed to get the
But notice that
same fractional units for each
the same denominator!
fraction.
Shabnam: So, fractions equivalent to
34 and 710 with the same fractional
unit (same denominators) are
3040 and 2840, or 1520 and 1420.
Since clearly
3040 > 2840, we conclude that 34 > 710.
Find equivalent fractions for the given pairs of fractions such that the fractional units are the same.
a. 72 and 35 b. 83 and 56 c. 34 and 35 d. 67 and 85
e. 94 and 52 f. 110 and 29 g. 83 and 114 h. 136 and 19
Expressing a fraction in lowest terms (or in its simplest form)
In any fraction, if its numerator and denominator have no common factor except 1, then the fraction is said to be in lowest terms or in its simplest form. In other words, a fraction is said to be in lowest terms if its numerator and denominator are as small as possible.
Any fraction can be expressed in lowest terms by finding an equivalent fraction whose numerator and denominator are as small as possible. Let’s see how to express fractions in lowest terms.
Example: Is the fraction
1620 in lowest terms? No, 4 is a common factor
of 16 and 20. Let us reduce
1620 to lowest terms.
We know that both 16 (numerator) and 20 (denominator) are divisible by 4.
So,
16 ÷ 420 ÷ 4 = 45.
Now, there is no common factor between 4 and 5. Hence,
1620
expressed in lowest terms is
45. So, 45 is called the simplest form of 1620,
since 4 and 5 have no common factor other than 1.
Any fraction can be converted to
lowest terms by dividing both the
numerator and denominator by the
highest common factor between
them.
Expressing a fraction in lowest terms can also be done in steps. Suppose we want to express
3660 in lowest terms. First, we notice that both the numerator and denominator are even. So, we divide both by 2, and see that 3660 = 1830.
Both the numerator and denominator are even again, so we can divide them each by 2 again; we get
1830 = 915.
We now notice that 9 and 15 are both multiples of 3, so we divide both by 3 to get
915 = 35 .
Now, 3 and 5 have no common factor other than 1, so,
3660 in lowest
terms is
35.
Alternatively, we could have noticed that in
3660 , both the numerator
and denominator are multiples of 12 : we see that 36 = 3 × 12 and
60 = 5 × 12. Therefore, we could have concluded that
3660 = 35 straight away.
Either method works and will give the same answer! But sometimes it can be easier to go in steps.
Figure it Out
Express the following fractions in lowest terms:
a.
1751 b. 64144 e. 126147 d. 525112
7.7 Comparing Fractions
Which is greater,
45 or 79? It can be difficult to compare two such fractions directly. However, we know how to find fractions equivalent to two fractions with the same denominator. Let us see how we can use it:
45 = 4×95×9 = 3645
79 = 7×59×5 = 3545.
45 is a common multiple
of 5 and 9, so we can
use 45 as a common
denominator.
Clearly,
3645 > 3545
So,
45 > 79 !
Let us try this for another pair:
79 and 1721.
63 is a common multiple of 9 and 21. We can then write:
79 = 7×79×7 = 4963 , 1721 = 17×321×3 = 5163.
Clearly,
4963 < 5163 . So, 79 < 1721!
Let’s Summarise!
Steps to compare the sizes of two or more given fractions:
Step 1: Change the given fractions to equivalent fractions so that they all are expressed with the same denominator or same fractional unit.
Step 2: Now, compare the equivalent fractions by simply comparing the numerators, i.e., the number of fractional units each has.
Figure it Out
1. Compare the following fractions and justify your answers:
a. 83 , 52 b. 49 , 37 c. 710 , 914
d. 125 , 85 e. 94 , 52
2. Write the following fractions in ascending order.
a. 710 , 1115 , 25 b. 1924 , 56 , 712
3. Write the following fractions in descending order.
a. 2516, 78, 134 , 1732 b. 34 , 125 , 712, 54
7.8 Addition and Subtraction of Fractions
Meena’s father made some chikki. Meena ate
12
of it and her younger brother ate
14 of it. How
much of the total chikki did Meena and her brother eat together?
We can arrive at the answer by visualising it. Let us take a piece of chikki and divide it into two halves first like this.
Meena ate
12 of it as
shown in the picture.
Meena ate
Let us now divide the remaining half into two further halves as shown. Each of these pieces is
14 of the whole chikki.
Meena’s brother ate
14
of the whole chikki, as is shown in the picture.
Meena ate
Brother ate
The total chikki eaten is
12 (by Meena) and 14 (by her
brother)
The total chikki eaten =
12 + 14
Total chikki eaten
=
14 + 14 + 14
= 3 ×
14 = 34.
How much of the total chikki is remaining?
Adding fractions with the same fractional unit or denominator
Example: Find the sum of
25 and 15.
Let us represent both using the rectangular strips. In both fractions, the fractional unit is the same
15 , so, each strip will be divided into 5 equal parts.
So
25 will be represented as —
And
15 will be represented as —
Adding the two given fractions is the same as finding out the total number of shaded parts, each of which represent the same fractional unit 15.
In this case, the total number of shaded parts is 3. Since, each shaded part represents the fractional unit
15 , we see that the 3 shaded
parts together represent the fraction
35.
Therefore,
25 + 15 = 35?
Example: Find the sum of
47 and 67.
Let us represent both again using the rectangular strip model. Here in both fractions, the fractional unit is the same, i.e., 17 , so each strip will
be divided into 7 equal parts.
Then
47 will be represented as —
and
67 will be represented as —
In this case, the total number of
While adding fractions
shaded parts is 10, and each shaded part
with the same fractional
unit, just add the number of
represents the fractional unit
17, so, the
fractional units from each
10 shaded parts together represent the
fraction.
fraction
107 as seen here.
Therefore,
47 + 67 = 107
= 1 +
37
= 1
37.
Try adding
47 + 67 using a number line. Do you get the same answer?
Adding fractions with different fractional units or denominators
Example: Find the sum of
14 and 13.
To add fractions with different fractional units, first convert the fractions into equivalent fractions with the same denominator or
fractional unit. In this case, the common denominator can be made
3 × 4 = 12, i.e., we can find equivalent fractions with fractional unit
112.
Let us write the equivalent fraction for each given fraction.
14 = 1 × 34 × 3 = 312 , 13 = 1 × 43 × 4 = 412 .
Now,
312 and 412 have the same fractional unit, i.e., 112 .
Therefore,
14 + 13 = 312 + 412 = 712 .
This method of addition, which works for adding any number of fractions, was first explicitly described in general by Brahmagupta in the year 628 CE! We will describe the history of the development of fractions in more detail later in the chapter. For now, we simply summarise the steps in Brahmagupta’s method for addition of fractions.
Brahmagupta’s method for adding fractions
1. Find equivalent fractions so that the fractional unit is common for all fractions. This can be done by finding a common multiple of the denominators (e.g., the product of the denominators, or the smallest common multiple of the denominators).
2. Add these equivalent fractions with the same fractional units. This can be done by adding the numerators and keeping the same denominator.
3. Express the result in lowest terms if needed.
Let us carry out another example of Brahmagupta’s method.
Example: Find the sum of
23 and 15.
The denominators of the given fractions are 3 and 5. The lowest common multiple of 3 and 5 is 15. Then we see that
23 = 2 × 53 × 5 = 1015 , 15 = 1 × 35 × 3 = 315 .
Therefore,
23 + 15 = 1015 + 315 = 1315 .
Example: Find the sum of
16 and 13.
The smallest common multiple of 6 and 3 is 6.
16 will remain 16 .
13 = 1 × 23 × 2 = 26
Therefore,
16 + 13 = 16 + 26 = 36.
The fraction
36 can now be re-expressed in lowest terms, if
desired. This can be done by dividing both the numerator and denominator by 3 (the biggest common factor of 3 and 6):
36 = 3 ÷ 36 ÷ 3 = 12.
Therefore,
16 + 13 = 12.
Figure it Out
1. Add the following fractions using Brahmagupta’s method:
a. 27 + 57 + 67 b. 34 + 13 c. 23 + 56 d. 23 + 27 e. 34 + 13 + 15
f. 23 + 45 g. 45 + 23 h. 35 + 58 i. 92 + 54 j. 83 + 27
k. 34 + 13 + 15 l. 23 + 45 + 37 m. 92 + 54 + 76
2. Rahim mixes 23 litres of yellow paint with 34 litres of blue paint to
make green paint. What is the volume of green paint he has made?
3. Geeta bought 25 meter of lace and Shamim bought 34 meter of the
same lace to put a complete border on a table cloth whose perimeter is 1 meter long. Find the total length of the lace they both have bought. Will the lace be sufficient to cover the whole border?
Subtraction of fractions with the same fractional unit or denominator
Brahmagupta’s method also applies when subtracting fractions! Let us start with the problem of subtracting
47 from 67, i.e., what is
67 – 47?
To solve this problem, we can again use the rectangular strips. In both fractions, the fractional unit is the same, i.e., 17. Let us first represent the bigger fraction using a rectangular strip model as shown:
67
Each shaded part represents
17. Now, we need to subtract 47. To do
this let us remove 4 of the shaded parts:
Fractional parts to
be removed.
We can do this here directly
because both fractions have
the same fractional units.
So, we are left with 2 shaded parts, i.e.,
67 – 47 = 27.
Try doing this same exercise using the number line.
Figure it Out
1. 58 – 38 2. 79 – 59 3. 1027 – 127
Subtraction of fractions with different fractional units or denominators
Example: What is
34 – 23?
As we already know the procedure for subtraction of fractions with the same fractional units, let us convert each of the given fractions into equivalent fractions with the same fractional units.
Yes! By doing this we can easily
subtract the two fractions.
34 = (3×3)(4×3) = 912
Think! Why did we choose to
multiply both the numerator and
denominator by 3?
and similarly,
Again! Why did we choose to multiply
23 = (2×4)(3×4) = 812 .
both the numerator and denominator
here by 4?
Therefore,
34 – 23 = 912 – 812 = 112 .
Brahmagupta’s method for subtracting two fractions —
1. Convert the given fractions into equivalent fractions with the same fractional unit, i.e., the same denominator.
2. Carry out the subtraction of fractions having the same fractional units. This can be done by subtracting the numerators and keeping the same denominator.
3. Simplify the result into lowest terms if needed.
Figure it Out
1. Carry out the following subtractions using Brahmagupta’s method:
a. 815 – 315 b. 25 – 415 c. 56 – 49 d. 23 – 12
2. Subtract as indicated:
a. 134 from 103 b. 185 from 233 c. 297 from 457
3. Solve the following problems:
a. Jaya’s school is 710 km from her home. She takes an auto for
12 km from her home daily, and then walks the remaining
distance to reach her school. How much does she walk daily
to reach the school?
b. Jeevika takes 103 minutes to take a complete round of the
park and her friend Namit takes 134 minutes to do the same.
Who takes less time and by how much?
7.9 A Pinch of History
Do you know what a fraction was called in ancient India? It was called bhinna in Sanskrit, which means ‘broken’. It was also called bhaga or ansha meaning ‘part’ or ‘piece’.
The way we write fractions today, globally, originated in India. In ancient Indian mathematical texts, such as the Bakshali manuscript (from around the year 300 CE), when they wanted to write 12, they wrote it as 12 which is indeed very similar to the way we write it
today! This method of writing and working with fractions continued to be used in India for the next several centuries, including by Aryabhata (499 CE), Brahmagupta (628 CE), Sridharacharya (c. 750 CE), and Mahaviracharya (c. 850 CE), among others. The line segment between the numerator and denominator in ‘12’ and in other
fractions was later introduced by the Moroccan mathematician Al-Hassar (in the 12th century). Over the next few centuries the notation then spread to Europe and around the world.Fractions had also been used in other cultures such as the ancient Egyptian and Babylonian civilisations, but they primarily used only fractional units, that is, fractions with a 1 in the numerator. More general fractions were expressed as sums of fractional units, now called ‘Egyptian fractions’. Writing numbers as the sum of fractional units, e.g., 1924 = 12 + 16 + 18, can be quite an art and leads to beautiful
puzzles. We will consider one such puzzle below. General fractions (where the numerator is not necessarily 1) were first introduced in India, along with their rules of arithmetic operations like addition, subtraction, multiplication, and even division of fractions. The ancient Indian treatises called the ‘Sulba-sutras’ shows that even during Vedic times, Indians had discovered the rules for operations with fractions. General rules and procedures for working with and computing with fractions were first codified formally and in a modern form by Brahmagupta. Brahmagupta’s methods for working with and computing with fractions are still what we use today. For example, Brahmagupta described how to add and subtract fractions as follows:“By the multiplication of the numerator and the denominator of each of the fractions by the other denominators, the fractions are reduced to a common denominator. Then, in case of addition, the numerators (obtained after the above reduction) are added. In case of subtraction, their difference is taken.’’ (Brahmagupta, Brahmasphuṭasiddhānta, Verse 12.2, 628 CE)The Indian concepts and methods involving fractions were transmitted to Europe via the Arabs over the next few centuries and they came into general use in Europe in around the 17th century and then spread worldwide.
Puzzle! It is easy to add up fractional units to obtain the sum 1, if one uses the same fractional unit, for example,
12 + 12 = 1, 13 + 13 + 13 = 1, 14 + 14 + 14 + 14 = 1, etc.
However, can you think of a way to add fractional units that are all different to get 1?
It is not possible to add two different fractional units to get 1. The reason is that
12 is the largest fractional unit, and 12 + 12 = 1.
To get different fractional units, we would have to replace at
least one of the
12’s with some smaller fractional unit - but then
the sum would be less than 1! Therefore, it is not possible for
two different fractional units to add up to 1.
We can try to look instead for a way to write 1 as the sum of three different fractional units.
1. Can you find three different fractional units that add
TryThis
up to 1?
It turns out there is only one solution to this problem (up to changing the order of the 3 fractions)! Can you find it? Try to find it before reading further.
Here is a systematic way to find the solution. We know that
13 + 13 + 13 = 1. To get the fractional units to be different, we will have
to increase at least one of the
13’s, and decrease at least one of the
other
13’s to compensate for that increase. The only way to increase
13 to another fractional unit is to replace it by 12. So 12 must be one of
the fractional units.
Now
12 + 14 + 14 = 1. To get the fractional units to be different, we
will have to increase one of the
14’s and decrease the other 14 to
compensate for that increase. Now the only way to increase
14 to
another fractional unit, that is different from
12, is to replace it by 13.
So two of the fractions must be
12 and 13! What must be third fraction
then, so that the three fractions add up to 1?
This explains why there is only one solution to the above problem.
12 + 13 + 16 = 1
What if we look for four different fractional units that add up to 1?
2. Can you find four different fractional units that add
up to 1?
TryThis
It turns out that this problem has six solutions! Can you find at least one of them? Can you find them all? You can try using similar reasoning as in the cases of two and three fractional units — or find your own method! Once you find one solution, try to divide a circle into parts like in the figure above to visualise it!
Fraction as equal share: When a whole number of units is divided into equal parts and shared equally, a fraction results.
Fractional Units: When one whole basic unit is divided into equal parts, then each part is called a fractional unit.
Reading Fractions: In a fraction such as 56, 5 is called the numerator
and 6 is called the denominator.
Mixed fractions contain a whole number part and a fractional part.
Number line: Fractions can be shown on a number line. Every fraction has a point associated with it on the number line.
Equivalent Fractions: When two or more fractions represent the same share or number, they are called equivalent fractions.
Lowest terms: A fraction whose numerator and denominator have no common factor other than 1 is said to be in lowest terms or in its simplest form.
Brahmagupta’s method for adding fractions: When adding fractions, convert them into equivalent fractions with the same fractional unit (i.e., the same denominator), and then add the number of fractional units in each fraction to obtain the sum. This is accomplished by adding the numerators while keeping the same denominator.
Brahmagupta’s method for subtracting fractions: When subtracting fractions, convert them into equivalent fractions with the same fractional unit (i.e., the same denominator), and then subtract the number of fractional units. This is accomplished by subtracting the numerators while keeping the same denominator.
CHAPTER 7 — SOLUTIONS
Section 7.1
Page no. 152
Figure it out
Q1. Three guavas together weigh 1 kg. If they are roughly of the same size, each guava
will roughly weigh ____kg.
Ans.
Q2. A wholesale merchant packed 1 kg of rice in four packets of equal weight. The weight
of each packet is_______ kg.
Ans.
Q3. Four friends ordered 3 glasses of sugarcane juice and shared it equally among
themselves. Each one drank ____ glass of sugarcane juice
Ans.
Q4. The big fish weighs
𝟐 kg. The small one weighs
𝟒 kg. Together they weigh ____ kg.
Ans.
Q5. Arrange these fraction words in order of size from the smallest to the biggest in the
empty box below:
One and a half, three quarters, one and a quarter, half, quarter, two and a half.
4,
2,
4, 1
4, 1
2, 2
Ans.
Section 7.2
Page No. 154
Q. By dividing the whole chikki into 6 equal parts in different ways, we get 1/6 chikki
pieces of different shapes. Are they of the same size?
Ans. Yes, they are of the same size.
Page No. 155
Figure it out
Q. The figures below show different fractional units of a whole chikki. How much of a
whole chikki is each piece?
Ans. a.
12 b.
4 c.
8 d.
6 e.
8 f.
6 g.
24 h.
Section 7.3
Figure it out Page No. 158
Q1. continue this table of
𝟐 for 2 more steps.
2 +
2 +
2 +
2 +
2 +
2 = 6 times
Ans.
2 +
2 +
2 +
2 +
2 +
2 +
2 = 7 times
𝟒 ?
Q2. Can you create a similar table for
Ans.
4 =
4 +
4 =
4 +
4 +
4 =
4 +
4 +
4 +
4=
4 times quarter
1-time quarter
2 times quarter
3 times quarter
Try Further!
Q4. Draw a picture and write an addition statement as above to show:
𝟒 of a roti b. 9 times
a. 5 times
𝟒 of a roti
Ans. a.
4 +
4 +
4 +
4 +
4 =
4 = 5 times quarter
b.
4 +
4 +
4 +
4 +
4 +
4 +
4 +
4 +
4 =
4 = 9 times quarter = 2 +
Q5. Match each fractional unit with the correct picture:
Ans.
Section 7.4
Page no. 159
Q1. Here, the fractional unit is dividing a length of 1 unit into three equal parts. Write
the fraction that gives the length of the blue line in the box or in your notebook.
Ans.
Q2. Here, a unit is divided into 5 equal parts. Write the fraction that gives the length of
the blue lines in the respective boxed or in your notebook.
5,
Ans.
Q3. Now, a unit is divided into 8 equal parts. Write the appropriate fractions in your
notebook.
8,
8,
8 ,…
Ans.
Page no. 160
Figure it out
𝟏𝟎,
𝟏𝟎, and
𝟓.
Q1. On a number line, draw lines of lengths
Ans.
45 1
Q3. How many fractions lie between 0 and 1? Think, discuss with your classmates, and
write your answer.
Ans. Uncountable number of fractions
Q4. What is the length of the blue line and black line shown below? The distance between
0 and 1 is 1 unit long, and it is divided into two equal parts. The length of each part
is
𝟐. So the blue line is
𝟐 units long. Write the fraction that gives the length of the
black line in the box.
Ans.
Q5. Write the fraction that gives the lengths of the black lines in the respective boxes.
5,
5,
5,
Ans.
Section 7.5
Page No. 162
Figure it out
Q1. How many whole units are there in
𝟐?
Ans. There are 3 whole units in
Q2. How many whole units are there in
𝟑 and in
𝟑?
Ans. There is 1 whole unit in
3 and 2 whole units in
3.
Figure it out
Q1. Figure out the number of whole units in each of the following fractions:
a.
𝟑 b.
𝟓 c.
Ans. a. 2
b. 2 c. 2
Q2. Can all fractions greater than 1 be written as such mixed numbers?
Ans. No. For example:
4 = 2 cannot be written as a mixed number.
𝟐 = 𝟒
Q3. Write the following fractions as mixed fractions (e.g.,
𝟐):
a.
𝟐 b.
𝟓 c.
𝟏𝟗 d.
𝟗 e.
𝟏𝟏 f.
Ans. a.
2 = 4
b.
5 = 1
c.
19 = 1
d.
9 = 5
e.
11 = 1
f.
6 = 3
Page No. 163
Figure it out
Q1. Write the following mixed numbers as fractions:
a. 3
𝟒 b. 7
𝟑 c. 9
𝟗 d. 3
𝟔 e. 2
𝟏𝟏 f. 3
Ans. a. 3
= 3 +
= 1 + 1 + 1 +
= (
4 +
4 +
4 +
4) + (
4 +
4 +
4 +
4) + (
4 +
4 +
4 +
4) +
= (4 ×
4) + (4 ×
4) + (4 ×
4) +
b.
c.
d.
e.
f.
Section 7.6
Page no. 164
Q1. Are the lengths
𝟐 and
𝟔 equal?
Ans. Yes.
Q2. Are
𝟑 and
𝟔 equivalent fractions? Why?
Ans. Yes, since they are of equal length which can be seen in fractional wall also.
Q3. How many pieces of length
𝟔 will make a length of
𝟐?
Ans. 3 pieces.
Q4. How many pieces of length
𝟔 will make a length of
𝟑?
Ans. 2 pieces.
Page no. 165
Figure it out
𝟔,
Q1. Are
𝟖, and
𝟏𝟎 are equivalent fractions? Why?
Ans. Yes. In the fraction wall their lengths can be seen to be equal.
Q2. Write two equivalent fractions for
𝟔.
Ans. Two equivalent fractions for
6 are
3 and
9. (Try for other equivalent fractions also)
Q3 .
𝟔 = = = = = ……. (Write as many as you can)
6 =
3 =
9 =
12 =
15 = …….
Ans.
Page no. 166
Figure it out
Q1. Three rotis are shared equally by four children. Show the division in the picture and
write a fraction for how much each child gets. Also, write the corresponding division facts, addition facts, and, multiplication facts.
Fraction of roti each child gets is ______.
Division fact:
Addition fact:
Multiplication fact:
Compare your picture and answers with your classmates!
Ans. Each child gets
4 roti.
Division fact: 3 4 =
4 +
4 +
4 +
Addition fact: 3 =
Multiplication fact: 3 = 4 ×
Q2. Draw a picture to show how much each child gets when 2 rotis are shared equally by
4 children. Also, write the corresponding division facts, addition facts, and multiplication facts.
2 Ans. 2 Rotis equally shared by 4 children. Each child
gets
roti.
Division fact :2 4 =
4 =
2 +
2 +
2 +
Addition fact: 2 =
Multiplication fact: 2 = 4 ×
Q3. Anil was in a group where 2 cakes were divided equally among 5 children. How much
cake would Anil get?
Ans. Anil would get
5 cake.
Page no. 168
Section 7.6
Figure it out
Q1. Find the missing numbers:
a. 5 glasses of juice shared equally among 4 friends is the same as ____ glasses of
juice shared equally among 8 friends.
𝟒 = 𝟖
So,
Ans. 10
b. 4 kg of potatoes divided equally in 3 bags is the same as 12 kgs of potatoes divided
equally in __ bags.
𝟑 =
So,
Ans. 9
c. 7 rotis divided among 5 children is the same as____rotis divided among _____
children.
𝟓 =
So,
Ans. One of the choices is 14, 10 (Try for other options also.)
Page no. 170
Section 7.6
Q. Suppose the number of children is kept the same, but the number of units that are
being shared is increased? What can you say about each child’s share now? Why? Discuss how your reasoning explains
𝟓 <
𝟓,
𝟕 <
𝟕 and
𝟐 <
𝟖 .
Ans. Each child will have larger share now.
If the number of units to be shared increase with same number of children then each child will have larger share.
Q. Now, decide in which of the two groups will each child get a larger share:
1. Group 1: 3 glasses of sugarcane juice divided equally among 4 children.
Group 2: 7 glasses of sugarcane juice divided equally among 10 children.
2. Group 1: 4 glasses of sugarcane juice divided equally among 7 children.
Group 2: 5 glasses of sugarcane juice divided equally among 7 children.
Which groups were easier to compare? Why?
Ans. Each child’s share in group 1 =
Each child’s share in group 2 =
7 >
And
The groups in second part were easier to compare because the number of children is same.
Page no. 172
Section 7.6
Q. Find equivalent fractions for the given pairs of fractions such that the fractional units
are the same.
a.
𝟐 and
𝟓 b.
𝟑 and
𝟔 c.
𝟒 and
𝟓 d.
𝟕 and
e.
𝟒 and
𝟐 f.
𝟏𝟎 and
𝟗 g.
𝟑 and
𝟒 h.
𝟔 and
Ans.
a.
10 and
b.
6 and
20 and
𝑐.
35 and
𝑑.
4 and
𝑒.
90 and
𝑓.
12 and
𝑔.
18 and
ℎ.
Page no. 173
Section 7.6
Figure it out
Q. Express the following fractions in lowest terms:
a.
𝟓𝟏 b.
𝟏𝟒𝟒 c.
𝟏𝟒𝟕 d.
Ans. a.
𝑏.
c.
d.
Page no. 174
Section 7.7
Figure it out
Q1. Compare the following fractions and justify your answers:
a.
𝟑,
𝟐 b.
𝟗 ,
𝟕 c.
𝟏𝟎 ,
d.
𝟓 ,
𝟓 e.
𝟒 ,
Ans.
a.
3 >
b.
9 >
10 >
𝑐.
d.
5 >
4 <
𝑒.
Q2. Write the following fractions in ascending order.
a.
𝟏𝟎,
𝟏𝟓,
𝟓 b.
𝟔,
𝟐𝟒,
Ans. a.
5 <
10 <
b.
12 <
24 <
Q3. Write the following fractions in descending order.
a.
𝟏𝟔,
𝟖,
𝟒,
𝟑𝟐 b.
𝟒,
𝟓,
𝟏𝟐,
Ans. a.
4 >
16 >
8 >
b.
5 >
4 >
4 >
Section 7.8
Page no. 177
𝟕 +
𝟕 using a number line. Do you get the same answer?
Q. Try adding
Ans.
7 +
7 +
7 =
7 = 1 +
Yes, using a number line, the answer is same.
Page no. 179
Section 7.8
Figure it out
Q1. Add the following fractions using Brahmagupta’s method:
Ans.
a.
b.
93
c.
6 =
6 3 =
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
𝟑 litres of yellow paint with
𝟒 litres of blue paint to make green paint.
Q2. Rahim mixes
What is the volume of green paint he has made?
Ans. 1
12 litres
𝟓 meter of lace and Shamim bought
𝟒 meter of the same lace to put a
Q3. Geeta bought
complete border on a table cloth whose perimeter is 1 meter long. Find the total length of the lace they both have bought. Will the lace be sufficient to cover the whole border?
Ans. Total length of the lace = 1
20 m
Yes.
Section 7.8
Figure it out (Page no. 181)
Q1.
8 -
4, in lowest terms)
Ans.
8 (=
Q2.
𝟗 -
Ans.
𝟐𝟕 -
Q3.
27 (=
Ans.
3, in lowest terms)
Page no. 181
Section 7.8
Figure it out
Ans. 1.
4 2.
9 3
Figure it out
Q1. Carry out the following subtractions using Brahmagupta’s method:
a.
15 -
15 b.
5 -
15 c.
6 -
9 d.
3 -
Ans. a.
b.
c.
d.
Q2. Subtract as indicated:
a.
𝟒 from
𝟑 b.
𝟓 from
𝟑 c.
𝟕 from
Ans. a.
b.
c.
Q3. Solve the following problems:
𝟏𝟎 km from her home. She takes an auto for
𝟐 km from her home
a. Jaya’s school is
daily, and then walks the remaining distance to reach her school. How much does she walk daily to reach the school?
𝟑 minutes to take a complete round of the park and her friend
b. Jeevika takes
𝟒 minutes to do the same. Who takes less time and by how much?
Namit takes
Ans. a.
5 Km
12 minutes
b. Namit takes less time than Jeevika, by