CHAPTER-7
Fractions
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 ½ .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 ¼ roti.
Mukta: And which is more ½
roti or ¼
roti?
Shabnam: When 2 children share 1 roti
equally, each child gets½
roti. When
4 children share 1 roti equally, each
child gets
¼roti. Since, in the second
group more children share the same one roti, each child gets a smaller share. So, 1/ 2 roti is
more than
¼ roti.
1/2>1/4
7.1 Fractional Units and Equal Shares
Beni: Which fraction is greater? 1/5or 1/9?
Arvin: 9 is bigger than 5. So I would guess that 1\9 is greater than 1\5 . 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 1\5 roti. If one roti is shared among 9 children, each one gets a sharof 1\9 roti?
Beni: Exactly! Now think again - which share is higher?
Arvin: If I share with more people, I will get less. So 1\9 < 1\5
Beni: You got it!
Oh, so 1 /100 is bigger than 1/ 200!
When one unit is divided into several equal parts, each part is
called a fractional unit. These are all fractional units:
1 /2 , 1/ 3 , 1/ 4 , 1 /5 , 1 /6 , …, 1/ 10 , …, 1/ 50 , …, 1/ 100 , etc
We also sometimes refer to fractional units as ‘unit fractions.’
- Figure it Out
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.
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 0/2 kg. The small one weighs 1 /4 kg.
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 3
4 is referred to as tripada. This has the same meaning as the words for 3
4 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.
Below, a picture of the chikki broken
into 2 pieces is shown. How much of
the original chikki is each piece?
We can see that the bigger piece has 3 pieces of 1/ 4 chikki in it. So, we
can measure the bigger piece using the fractional unit 1/ 4. We see that
the bigger piece is 3/ 4 chikki.
- 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?
Figure it Out
The figures below show different fractional units of a whole chikki.
How much of a whole chikki is each piece
7.3 Measuring Using Fractional Units
Take a strip of paper. We consider this paper strip to be one unit long.
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?
What will you get if you fold the previously-folded strip again
into two equal parts? You will now get four equal parts
Do it once more! Fill in the blank boxes
Fractional quantities can be measured using fractional units
Let us look at another example
We can describe how much the quantity is by collecting together the
fractional units .
Figure it Out
1. Continue this table of 1/ 2 for 2 more steps.
2. Can you create a similar table for 1/ 4 ?
3. Make 1 /3 using a paper strip. Can you use this to also make 1/ 6 ?
4. Draw a picture and write an addition statement as above to show:
a. 5 times 1
4 of a roti b. 9 times 1
4 of a roti
5. Match each fractional unit with the correct picture:
1/3 1/5 1/8 1/6
Reading Fractions
We usually read the fraction 3 /4 as ‘three quarters’ or ‘three upon four’, but reading it as ‘3 times 1/ 4’ helps us to understand the size of the fraction because it clearly shows what the fractional unit is (1/ 4) 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 5 /6 , 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?
The distance between 0 and 1 is one unit long. It is divided into
two equal parts. So, the length of each part is 1 /2 unit. So, this blue line
is 1/ 2 unit long.
- Now, can you find the lengths of the various blue lines shown
below? Fill in the boxes as well.
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
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 1/ 10 , 3 /10 , and 4 /5 .
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 1/ 2
. So the blue line is 1/ 2
units
long. Write the fraction that gives the length of the black line in the
box.
5. Write the fraction that gives the lengths of the black lines in the
respective boxes .
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 one
You 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 unit |
Lengths more than 1 unit |
|---|---|
- Did you notice something common between the fractions that
are greater than 1?
We know that 3 /2 , 5 /2 and 7 /2 are all greater than 1 unit. But can we
see how many whole units they contain?
3/ 2 = 1/ 2 + 1/ 2 + 1 /2 = 1 + 1/ 2
5/ 2 = 1 /2 + 1 /2 + 1 /2 + 1 /2 + 1 /2 = 2 + 1 /2
I know that 1/ 3 + 1 /3 + 1 /3 = 3 /3 = 1. If I add one more 1 /3,
I will get more than 1 unit! So, 4 /3 > 1.
- Figure it Out
2. How many whole units are there in 4 /3 and in 7/ 3 ?
Writing fractions greater than one as mixed numbers
We saw that: 3/ 2 = 1 + 1 /2 .
We can write other fractions in a similar way. For example,
4/ 3 = 1 /3 + 1 /3 + 1 /3 + 1 /3 = 1 + 1/ 3 .
3 × 1/ 3 = 1
- Figure it Out
a. 8/ 3 b. 11 /5 c . 9 /4
We saw that
8/ 3 = 2 + 2/ 3
Fraction Mixed number
- 8/ 3 ------> Fraction
- 2 + 2/ 3 ------->
Mixed number
- This number is thus also called ‘two
and two thirds’. We also write it as 2 ⅔
.
2. Can all fractions greater than 1 be written as such mixed numbers?
A mixed number / 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., 9/ 2 = 4 ½
):
a. 9/2 b. 9/5 c. 21/ 19 d. 47/ 9 e. 12 /11 f. 19/6
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 + 3/4 , this means 1 + 1 + 1 + 3 /4 . I know
1 = 1/ 4 + 1 /4 + 1/ 4 + 1/ 4 .
So I get
(1/ 4 + 1/ 4 + 1 /4 + 1 /4 ) + ( 1 /4 + 1 /4 + 1/ 4 + 1/ 4 ) + ( 1 /4 + 1 /4 + 1 /4 + 1 /4 ) + ( 1 /4 + 1/ 4 + 1 /4 ) = 15 /4 .
Therefore, (4 × 1 /4 ) + (4 × 1/ 4 ) + (4 × 1/ 4 ) + (3 × 1 /4 ) = 15 /4 .
- Figure it Out
a. 3 ¼ (b7)⅔ (c) 94⁄9
(d)3⅙ (e)23⁄11
(f)39⁄10
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?
• Are the lengths 1/2 and 2/4 equal?
• Are the lengths 2/4 and 4/8 equal?
We can say that 1/2 = 2/4 = 4/8 .
These are ‘equivalent fractions’ that denote the same length, but
they are expressed in terms of different fractional units.
Now, check whether 1 /3 and 2/ 6 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:
2. Are 2/ 3 and 4 /6 equivalent
fractions? Why?
3. How many pieces of
length 1/ 6 will make a
length of 1/ 2 ?
4. How many pieces of length 1/ 6 will make a length of 1/ 3 ?
We can extend this idea to make a fraction wall up to the fractional
unit 1 /10 . (This fraction wall is given at the end of the book.)
- Figure it Out
2. Write two equivalent fractions for 2/ 6 .
3. 4/ 6 =...... =........... =............... = ............ (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 1 /4
.
You can also express this event through division facts, addition
facts, and multiplication facts.
The division fact is 1 ÷ 4 = 1/4 .
The addition fact is 1 = 1/4 + 1/4 + 1/4 + 1/4 .
The multiplication fact is 1 = 4 × 1/4 .
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?
Now, if there are 10 children
in my group, how many
cakes will I need so that they
get same amount of cake as
Anil?
What if we put two such groups
together? one group where
2 cakes are divided equally
between 5 children, and another
group again with 4 cakes and 10
children.
Group1
Group 2
So, the share of each child is the
same in both these situations!
So, 2/ 5 = 4 /10 !
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 1/2 = 2/4 = 3/6
1 roti is divided equally between 2
1/2 1/2
2 rotis are divided
equally among 4
2 /4 2 /4
2/ 4 2/ 4
3 rotis are divided equally among 6
3 /6 3 /6 3 /6 3 /6 3/ 6 3/ 6
Fractions where the shares are equal are called ‘equivalent fractions’.
So, 1 /2 , 2/ 4 , and 3/ 6 are all equivalent fractions .
Find some more fractions equivalent to 1/ 2 . Write them in the ............. ............. ...............
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?
2 /3 is also called the simplest form of 4/ 6 . It is also the simplest form of 6/ 9 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.
So, 5/4 =..../ 8 .
b. 4 kg of potatoes divided equally in 3 bags is the same as
12 kgs of potatoes divided equally in ___ bags.
So, 4/3 = 12/.....
c. 7 rotis divided among 5 children is the same as____rotis divided
among _____ children.
So, 7 /5 = ...../......
- In which group will each child get more chikki?
Mukta: So, we must compare 1 /2 and 5
8 . Which is more?
Shabnam: Well, we have seen that 1 /2 = 4/ 8 ; and clearly 4/ 8 < 5/ 8 . 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?
Shabnam: The children of which group will get more chikki this
time?
Mukta: We must compare 1 /7 and 4 /7 .
Now
1 × 4 /2 × 4 = 4 /8 so, 1 /2 = 4 /8 .
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 4/ 7 chikki. When 4 chikkis are divided equally
among 8 children, each one will get 4/ 8 chikki. So 4 /7 > 4/8.
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, 4/ 7
> 4/ 8
and 4/ 8 = 1/ 2
, so 4/ 7
> 1/ 2
.
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
1 /5 < 2 /5 , 3/ 7 < 4/ 7 , and 1/ 2 < 5 /8 .
- Now, decide in which of the two groups will each child get a
larger share:
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?
When the number of
children is the same, it is
easier to compare, isn’t it?
Shabnam: To compare the first two groups,
we have to find fractions
equivalent to the fractions
3 /4 and 7 /10.
Mukta: How about 6 /8 = 3/ 4 and 21 /30 = 7 /10 ?
Shabnam: There is a condition. The fractional unit used for the two
fractions have to be the same! Like 2
6 and 3
6 both use the
same fractional unit 1
6 (i.e., the denominators are the same).
But 6
8 and 21
30 do not use the same fractional units (they have
different denominators).
Mukta: Okay, so let us start making equivalent fractions then:
3
4 = 6
8 = 9
12 = 12
16 = 15
20 … 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 3/ 4 and 7 /10 . The product of the two denominators (4
and 10) is 40.
3/ 4 = 6 /8 = 9 /12 = 12 /16 = 15 /20 = 18/ 24 = … = 27 /36 = 30/40.
7/10 = 14/ 20 = 21/ 30 = 28/ 40 .
Go till we reach the
denominator 40.
But notice that
15
20 and
14
20 also had
the same denominator!
Yes! We just needed to get the
same fractional units for each
fraction.
Shabnam: So, fractions equivalent to 3 /4 and 7 /10 with the same fractional
unit (same denominators) are 30/ 40 and 28/ 40 , or 15 /20 and 14/ 20.
Since clearly 30 /40 > 28 /40 , we conclude that 3 /4 > 7 /10.
- Find equivalent fractions for the given pairs of fractions such that the fractional units are the same .
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 16/ 20 in lowest terms? No, 4 is a common factor
of 16 and 20. Let us reduce 16/ 20 to lowest terms.
We know that both 16 (numerator) and 20 (denominator) are
divisible by 4.
So, 16 ÷ 4 /20 ÷ 4 = 4/ 5 .
Now, there is no common factor between 4 and 5. Hence, 16
20
expressed in lowest terms is 4
5. So, 4
5 is called the simplest form of 16
20 ,
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 36 /60 in lowest terms. First, we notice
that both the numerator and denominator are even. So, we divide
both by 2, and see that 36 /60 = 18/ 30.
Both the numerator and denominator are even again, so we can
divide them each by 2 again; we get 18 /30 = 9 /15.
Now, 3 and 5 have no common factor other than 1, so, 36 /60 in lowest
terms is 3 /5.
Alternatively, we could have noticed that in 36/ 60 , 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 36/ 60 = 3/ 5 straight away.
Either method works and will give the same answer! But
sometimes it can be easier to go in steps.
- Figure it Out
7.7 Comparing Fractions
Which is greater, 4
5 or 7
9 ? It can be difficult to compare two such
fractionsdirectly. However,weknowhowtofindfractions equivalent
to two fractions with the same denominator. Let us see how we can
use it:
7/9=7*5/8*5=35/45
so, 4⁄5>7⁄9 !
45 is a common multiple
of 5 and 9, so we can
use 45 as a common
denominator.
Clearly36⁄45>35⁄45
Let us try this for another pair: 7 /9 and 17/ 21 .
63 is a common multiple of 9 and 21. We can then write
7/ 9 = 7×7 /9×7 = 49 /63 , 17/ 21 = 17×3/ 21×3 = 51 /63 .
Clearly, 49 /63 < 51/ 63 . So, 7 /9 < 17 /21!
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 / 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.) 8⁄3 , 5⁄2 (b) 4⁄9 ,3⁄7 (c) 7⁄10 , 9⁄14 (d.)12⁄5 ,8⁄5 (e) 9⁄4 ,5⁄2
2. Write the following fractions in ascending order.
(a) 7⁄10
, 11⁄15 2⁄5 (
b). 19⁄24
, 5⁄6
, 7⁄12
3. Write the following fractions in descending order.
(a). ,25⁄16
, 7⁄8
, 13⁄4 , 17⁄32 ( b.) 3⁄4
, 12⁄
5 , 7⁄
12 , 5⁄4
7.8 Addition and Subtraction of Fractions
Meena’s father made some chikki. Meena ate 1/ 2
of it and her younger brother ate 1/ 4 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 1/ 2 of it as
shown in the picture.
Let us now divide the
remaining half into two further halves as shown. Each of these
pieces is 1/ 4 of the whole chikki.
Meena’s brother ate 1 /4
of the whole chikki, as is
shown in the picture.
The total chikki eaten
is 1/ 2 (by Meena) and 1 /4 (by her
brother)
The total chikki eaten
is
= 1 /2 + 1/ 4
= 1/ 4 + 1/ 4 + 1/ 4
= 3 × 1/ 4
= 3 /4 .
How much of the total chikki is remaining?
Adding Fractions with the Same Fractional Unit or
Denominator
Example: Find the sum of 2 /5 and 1/ 5 .
Let us represent both using the rectangular strips. In both fractions,
the fractional unit is the same 1
5
, so, each strip will be divided into 5
equal parts.
So 2 /5 will be represented as—
And 1/ 5 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
1 /5
.
In this case, the total number of shaded parts is 3. Since, each
shaded part represents the fractional unit 1/ 5 , we see that the 3 shaded
parts together represent the fraction 3 /5 .
Therefore, 2
5 + 1
5 = 3
5 !
Example: Find the sum of 4/ 7 and 6 /7 .
Let us represent both again using the rectangular strip model. Here in
both fractions, the fractional unit is the same, i.e., 1
7 , so each strip will
be divided into 7 equal parts.
Then 4/ 7 will be represented as—
and 6/ 7 will be represented as—
In this case, the total number of
shaded parts is 10, and each shaded part
represents the fractional unit 1 /7 , so, the
10 shaded parts together represent the
fraction 10 /7 as seen here.
While adding fractions
with the same fractional
unit, just add the number of
fractional units from each
fraction.
Therefore, 4/ 7 + 6 /7 = 10 /7
= 1 + 3 /7
= 1 3⁄7
.
- Try adding 4/ 7 + 6/ 7 using a number line. Do you get the same answer?
Adding Fractions with Different Fractional Units or
Denominators
Example: Find the sum of 1/ 4 and 1/ 3 .
To add fractions with different fractional units, first convert the
fractions into equivalent fractions with the same denominator/ fractional unit. In this case, the common denominator can be made
3 × 4 = 12, i.e., we can find equivalent fractions with fractional unit 1 /12 .
Let us write the equivalent fraction for each given fraction.
1 /4 = 1 × 3 /4 × 3 = 3 /12 , 1/ 3 = 1 × 4 /3 × 4 = 4 /12 .
Now, 3 /12 and 4 /12 have the same fractional unit, i.e., 1 /12 .
Therefore, 1 /4 + 1 /3 = 3 /12 + 4 /12 = 7 /12 .
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 2 /3 and 1 /5 .
The denominators of the given fractions are 3 and 5. The lowest
common multiple of 3 and 5 is 15. Then we see that
2/ 3 = 2 × 5 /3 × 5 = 10 /15 , 1/ 5 = 1 × 3 /5 × 3 = 3 /15 .
Therefore, 2 /3 + 1 /5 = 10 /15 + 3 /15 = 13 /15 .
Example: Find the sum of 1 /6 and 1 /3 .
The smallest common multiple of 6 and 3 is 6.
1/ 6 will remain 1 /6 .
1/ 3 = 1 × 2 /3 × 2 = 2 /6
Therefore, 1 /6 + 1 /3 = 1 /6 + 2/ 6 = 3 /6 .
The fraction 3 /6 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) :
3/ 6 = 3 ÷ 3 /6 ÷ 3 = 1 /2 .
Therefore, 1 /6 + 1 /3 = 1 /2 .
Figure it Out
1. Add the following fractions using Brahmagupta’s method:
(a) 2/ 7 + 5/ 7 + 6/ 7
(b) 3 /4 + 1/ 3
(c) 2/ 3 + 5 /6
(d.) 2 /3 + 2/ 7
(e) 3/ 4 + 1/3 + 1 /5
(f) 2 /3 + 4/ 5
(g) 4/ 5 + 2 /3
(h) 3 /5 + 5/ 8
(i) 9 /2 + 5/ 4
(j) 8 /3 + 2 /7
(k) 3 /4 + 1 /3 + 1/ 5
(l) 2/ 3 + 4/ 5 + 3/ 7
(m) 9/ 2 + 5/ 4 + 7/6
2. Rahim mixes 2 /3 litres of yellow paint with 3/ 4 litres of blue paint to
make green paint. What is the volume of green paint he has made?
3. Geeta bought 2/ 5
meter of lace and Shamim bought 3/ 4 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 4/ 7 from 6 /7 , i.e., what is
6 /7 – 4 /7 ?
To solve this problem, we can again use the rectangular strips.
In both fractions, the fractional unit is the same i.e. 1/ 7. Let us first
represent the bigger fraction using a rectangular strip model as
shown:
Each shaded part represents 1 /7 . Now, we need to subtract 4 /7 . 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., 6 /7 – 4 /7 = 2 /7 .
Try doing this same exercise using the number line .
Figure it Out
(1) 5/ 8 – 3/ 8 (2) 7/ 9 – 5 /9 (3.) 10 /27 – 1 /27
Subtraction of Fractions with Different Fractional Units or
Denominators
Example: What is 3 /4 – 2 /3 ?
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.
3 /4 = (3×3)/ (4×3) = 9 /12
Yes! By doing this we can easily
subtract the two fractions
Think! Why did we choose to
multiply both the numerator and
denominator by 3?
and similarly,
2/ 3 = (2×4)/ (3×4) = 8 /12 .
Again! Why did we choose to multiply
both the numerator and denominator
here by 4
Therefore, 3 /4 – 2/ 3 = 9 /12 – 8/ 12 = 1 /12 .
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. 8 /15 – 3 /15 b. 2/ 5 – 4 /15 c. 5/ 6 – 4 /9 d. 2/ 3 – 1/ 2
2. Subtract as indicated:
a. 13 /4 from 10 /3 b. 18/ 5 from 23 /3 c. 29/ 7 from 45/ 7
3. Solve the following problems:
a. Jaya’s school is 7/ 10 km from her home. She takes an auto for
1/ 2 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 10/ 3 minutes to take a complete round of the
park and her friend Namit takes 13 /4 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 1/ 2, they
wrote it as 1/ 2 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 ‘1 /2’ 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., 19/ 24 = 1 /2 + 1/ 6 + 1/ 8, 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 ‘Sulbasutras’ 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, e.g.,
1 /2 + 1/ 2 = 1, 1/ 3 + 1 /3 + 1/ 3 = 1, 1 /4 + 1/ 4 + 1 /4 + 1/ 4 = 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 ½ is the largest fractional unit, and 1/ 2 + 1 /2 = 1.
To get different fractional units, we would have to replace at
least one of the 1 /2’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
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
1 /3 + 1 /3 + 1 /3 = 1. To get the fractional units to be different, we will
have to increase at least one of the 1
3’s, and decrease at least
one of the other 1 /3’s to compensate for that increase. The only
way to increase 1 /3 to another fractional unit is to replace it by
1/ 2. So 1 /2 must be one of the fractional units.
Now 1/ 2 + 1/ 4 + 1/ 4 = 1. To get the fractional units to be different, we
will have to increase one of the 1/ 4’s and decrease the other 1/ 4 to
compensate for that increase. Now the only way to increase 1 /4 to another fractional unit, that is different from 1 /2, is to replace
it by 1/ 3. So two of the fractions must be 1 /2 and 1/ 3! 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.
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?
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 visualize it!
SUMMARY
- 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 5 6, 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/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