Subhash had learnt about fractions in Classes IV and V, so whenever possible he would try to use fractions. One occasion was when he forgot his lunch at home. His friend Farida invited him to share her lunch. She had five pooris in her lunch box. So, Subhash and Farida took two pooris each. Then Farida made two equal halves of the fifth poori and gave one-half to Subhash and took the other half herself. Thus, both Subhash and Farida had 2 full pooris and one-half poori.

Where do you come across situations with fractions in your life?

Subhash knew that one-half is written as . While eating he further divided his half poori into two equal
parts and asked Farida what fraction of the whole poori was that piece? (Fig 7.1)

Without answering, Farida also divided her portion of the half puri into two equal parts and kept them beside Subhash’s shares. She said that these four equal parts together make one whole (Fig 7.2). So, each equal part is one-fourth of one whole poori and 4 parts together will be or 1 whole poori.

When they ate, they discussed what they had learnt earlier. Three parts out of 4 equal parts is . Similarly, is obtained when we divide a whole into seven equal parts and take three parts (Fig 7.3). For , we divide a whole into eight equal parts and take one part out of it (Fig 7.4).

7.2 A Fraction

Let us recapitulate the discussion.

A fraction means a part of a group or of a region.

is a fraction. We read it as “five-twelfths”.

What does “12” stand for? It is the number of equal parts into which the whole has been divided.

What does “5” stand for? It is the number of equal parts which have been taken out.

Here 5 is called the numerator and 12 is called the denominator.

Play this Game

You can play this game with your friends.

Take many copies of the grid as shown here.

Consider any fraction, say .

Each one of you should shade of the grid.

Exercise 7.1

2. Colour the part according to the given fraction.

3. Identify the error, if any.

4. What fraction of a day is 8 hours?

(b) What part of a sandwich will each boy receive?

7. Kanchan dyes dresses. She had to dye 30 dresses. She has so far finished 20 dresses. What fraction of dresses has she finished?

8. Write the natural numbers from 2 to 12. What fraction of them are prime numbers?

9. Write the natural numbers from 102 to 113. What fraction of them are prime numbers?

10. What fraction of these circles have X’s in them?

11. Kristin received a CD player for her birthday. She bought 3 CDs and received 5 others as gifts. What fraction of her total CDs did she buy and what fraction did she receive as gifts?

7.3 Fraction on the Number Line

You have learnt to show whole numbers like 0,1,2... on a number line.

We know that is greater than 0 and less than 1, so it should lie between 0 and 1.

Since we have to show , we divide the gap between 0 and 1 into two equal parts and show 1 part as (as shown in the Fig 7.5).

Suppose we want to show on a number line. Into how many equal parts should the length between 0 and 1 be divided? We divide the length between 0 and 1 into 3 equal parts and show one part as (as shown in the Fig 7.6)

Can we show on this number line? means 2 parts out of 3 parts as shown (Fig 7.7).

is the point zero whereas since is 1 whole, it can be shown by the point 1 (as shown in Fig 7.7)

So if we have to show on a number line, then, into how many equal parts should the length between 0 and 1 be divided? If P shows then how many equal divisions lie between 0 and P? Where do and lie?

Does any one of the fractions lie beyond 1?

All these fractions lie to the left of 1as they are less than 1.

In fact, all the fractions we have learnt so far are less than 1. These are proper fractions. A proper fraction as Farida said (Sec. 7.1), is a number representing part of a whole. In a proper fraction the denominator shows the number of parts into which the whole is divided and the numerator shows the number of parts which have been considered. Therefore, in a proper fraction the numerator is always less than the denominator.

7.5 Improper and Mixed Fractions

Anagha, Ravi, Reshma and John shared their tiffin. Along with their food, they had also, brought 5 apples. After eating the other food, the four friends wanted to eat apples.

How can they share five apples among four of them?

Anagha said, ‘Let each of us have one full apple and a quarter of the fifth apple.’

Reshma said, ‘That is fine, but we can also divide each of the five apples into 4 equal parts and take one-quarter from each apple.’

Ravi said, ‘In both the ways of sharing each of us would get the same share, i.e., 5 quarters. Since 4 quarters make one whole, we can also say that each of us would get 1 whole and one quarter. The value of each share would be five divided by four. Is it written as 5 ÷ 4?’ John said, ‘Yes the same as ’. Reshma added that in , the numerator is bigger than the denominator. The fractions, where the numerator is bigger than the denominator are called improper fractions.

1. Write five improper fractions with denominator 7.

2. Write five improper fractions with numerator 11.

Ravi reminded John, ‘What is the other way of writing the share? Does it follow from Anagha’s way of dividing 5 apples?’

John nodded, ‘Yes, It indeed follows from Anagha’s way. In her way, each share is one whole and one quarter. It is and written in short as . Remember, is the same as ’.

Recall the pooris eaten by Farida. She got poories (Fig 7.9), i.e.

How many shaded halves are there in ? There are 5 shaded halves.

So, the fraction can also be written as . is the same as .

Fractions such as and are called Mixed Fractions. A mixed fraction has a combination of a whole and a part.

Example 1 : Express the following as mixed fractions :

(a) (b) (c) (d)

Solution : (

Try (c) and (d) using both the methods for yourself.

Thus, we can express an improper fraction as a mixed fraction by dividing the numerator by denominator to obtain the quotient and the remainder. Then the mixed fraction will be written as .

Example 2 : Express the following mixed fractions as improper fractions:

(a) (b) (c)

Solution :

 

  Thus, we can express a mixed fraction as an improper fraction as

.

Exercise 7.2

1. Draw number lines and locate the points on them :

(a) (b) (c)

2. Express the following as mixed fractions :

(a) (b) (c)

(d) (e) (f)

3. Express the following as improper fractions :

(a) (b) (c) (d) (e) (f)

7.6 Equivalent Fractions

Look at all these representations of fraction (Fig 7.10).

These fractions are , representing the parts taken from the total number of parts. If we place the pictorial representation of one over the other they are found to be equal. Do you agree?

These fractions are called equivalent fractions. Think of three more fractions that are equivalent to the above fractions.

Understanding equivalent fractions

are all equivalent fractions. They represent the same part of a whole.

Think, discuss and write

Why do the equivalent fractions represent the same part of a whole? How can we obtain one from the other?

We note . Similarly, and

Rajni says that equivalent fractions of are :

 and many more.

Do you agree with her? Explain.

Is there any other way to obtain equivalent fractions? Look at Fig 7.11.

These include equal number of shaded things i.e. =

To find an equivalent fraction, we may divide both the numerator and the denominator by the same number.

One equivalent fraction of is

Can you find an equivalent fraction of having denominator 5 ?

Example 3 : Find the equivalent fraction of with numerator 6.

Solution : We know 2 × 3 = 6. This means we need to multiply both the numerator and the denominator by 3 to get the equivalent fraction.

Hence, ; is the required equivalent fraction.

Can you show this pictorially?

Solution : We have

We observe the denominator and find 35 ÷ 5 = 7. We, therefore, divide both the numerator and the denominator of by 5.

Thus, .

An interesting fact

Let us now note an interesting fact about equivalent fractions. For this, complete the given table. The first two rows have already been completed for you.

What do we infer? The product of the numerator of the first and the denominator of the second is equal to the product of denominator of the first and the numerator of the second in all these cases. These two products are called cross products. Work out the cross products for other pairs of equivalent fractions. Do you find any pair of fractions for which cross products are not equal? This rule is helpful in finding equivalent fractions.

Example 5 : Find the equivalent fraction of with denominator 63.

Solution : We have 

For this, we should have, 9 ×  = 2 × 63.

But 63 = 7 × 9, so 9 ×  = 2 × 7 × 9 = 14 × 9 = 9 × 14

or 9 ×  = 9 × 14

By comparison,  = 14. Therefore, .

Given the fraction , let us try to get an equivalent fraction in which the numerator and the denominator have no common factor except 1.

How do we do it? We see that both 36 and 54 are divisible by 2.

But 18 and 27 also have common factors other than one.

The common factors are 1, 3, 9; the highest is 9.

Therefore,

Now 2 and 3 have no common factor except 1; we say that the fraction is in the simplest form.

A fraction is said to be in the simplest (or lowest) form if its numerator and denominator have no common factor except 1.

The shortest way

The shortest way to find the equivalent fraction in the simplest form is to find the HCF of the numerator and denominator, and then divide both of them by the HCF.

Consider .

The HCF of 36 and 24 is 12.

Therefore, =. The fraction is in the lowest form.

Thus, HCF helps us to reduce a fraction to its lowest form.

Exercise 7.3

1. Write the fractions. Are all these fractions equivalent?

(a)

(b)

2. Write the fractions and pair up the equivalent fractions from each row.

(a) (b) (c) (d) (e)

4. Find the equivalent fraction of having

(a) denominator 20 (b) numerator 9

(c) denominator 30 (d) numerator 27

(a) numerator 9 (b) denominator 4

6. Check whether the given fractions are equivalent :

7. Reduce the following fractions to simplest form :

(a) (b) (c) (d) (e)

8. Ramesh had 20 pencils, Sheelu had 50 pencils and Jamaal had 80 pencils. After 4 months, Ramesh used up 10 pencils, Sheelu used up 25 pencils and Jamaal used up 40 pencils. What fraction did each use up? Check if each has used up an equal fraction of her/his pencils?

9. Match the equivalent fractions and write two more for each.

7.8 Like Fractions

Fractions with same denominators are called like fractions.

Thus, are all like fractions. Are and like fractions?

Their denominators are different. Therefore, they are not like fractions. They are called unlike fractions.

Write five pairs of like fractions and five pairs of unlike fractions.

7.9 Comparing Fractions

Sohni has rotis in her plate and Rita has rotis in her plate. Who has more rotis in her plate? Clearly, Sohni has 3 full rotis and more and Rita has less than 3 rotis. So, Sohni has more rotis.

Consider and as shown in Fig. 7.12. The portion of the whole corresponding to is clearly larger than the portion of the same whole corresponding to .

So is greater than .

But often it is not easy to say which one out of a pair of fractions is larger. For example, which is greater, or ? For this, we may wish to show the fractions using figures (as in fig. 7.12), but drawing figures may not be easy especially with denominators like 13. We should therefore like to have a systematic procedure to compare fractions. It is particularly easy to compare like fractions. We do this first.

7.9.1 Comparing like fractions

Like fractions are fractions with the same denominator. Which of these are like fractions?

Let us compare two like fractions: and .

In both the fractions the whole is divided into 8 equal parts. For and , we take 3 and 5 parts respectively out of the 8 equal parts. Clearly, out of 8 equal parts, the portion corresponding to 5 parts is larger than the portion corresponding to 3 parts. Hence, >. Note the number of the parts taken is given by the numerator. It is, therefore, clear that for two fractions with the same denominator, the fraction with the greater numerator is greater. Between and , is greater. Between and , is greater and so on.

7.9.2 Comparing unlike fractions

Two fractions are unlike if they have different denominators. For example,
and are unlike fractions. So are and .

Consider a pair of unlike fractions and , in which the numerator is the same.

Which is greater or ?

In , we divide the whole into 3 equal parts and take one. In , we divide the whole into 5 equal parts and take one. Note that in , the whole is divided into a smaller number of parts than in . The equal part that we get in is, therefore, larger than the equal part we get in . Since in both cases we take the same number of parts (i.e. one), the portion of the whole showing is larger than the portion showing , and therfore > .

the two.

Thus, and so on.

Let us arrange in increasing order. All these fractions are unlike, but their numerator is the same. Hence, in such case, the larger the denominator, the smaller is the fraction. The smallest is , as it has the largest denominator. The next three fractions in order are . The greatest fraction is (It is with the smallest denominator). The arrangement in increasing order, therefore, is .

Suppose we want to compare and . Their numerators are different and so are their denominators. We know how to compare like fractions, i.e. fractions with the same denominator. We should, therefore, try to change the denominators of the given fractions, so that they become equal. For this purpose, we can use the method of equivalent fractions which we already know. Using this method we can change the denominator of a fraction without changing its value.

Let us find equivalent fractions of both and .

= .... Similarly, = ....

The equivalent fractions of and with the same denominator 12 are repectively.

i.e. and . Since, we have, >.

Example 6 : Compare and .

Solution : The fractions are unlike fractions. Their numerators are different too. Let us write their equivalent fractions.

and

The equivalent fractions with the same denominator are :

and

Since, so,

Note that the common denominator of the equivalent fractions is 30 which is 5 × 6. It is a common multiple of both 5 and 6.

So, when we compare two unlike fractions, we first get their equivalent fractions with a denominator which is a common multiple of the denominators of both the fractions.

Example 7 : Compare and .

Solution : The fractions are unlike. We should first get their equivalent fractions with a denominator which is a common multiple of 6 and 15.

Now, ,

Since we have .

Why LCM?

The product of 6 and 15 is 90; obviously 90 is also a common multiple of
6 and 15. We may use 90 instead of 30; it will not be wrong. But we know that it is easier and more convenient to work with smaller numbers. So the common multiple that we take is as small as possible. This is why the LCM of the denominators of the fractions is preferred as the common denominator.

Exercise 7.4

1. Write shaded portion as fraction. Arrange them in ascending and descending order using correct sign ‘<’, ‘=’, ‘>’ between the fractions:

(a)

(b)

(c) Show and on the number line. Put appropriate signs between the fractions given.

, 0, ,

2. Compare the fractions and put an appropriate sign.

(a) (b) (c) (d)

3. Make five more such pairs and put appropriate signs.

4. Look at the figures and write ‘<’ or ‘>’, ‘=’ between the given pairs of fractions.

(a) (b) (c) (d) (e)

Make five more such problems and solve them with your friends.

5. How quickly can you do this? Fill appropriate sign. ( ‘<’, ‘=’, ‘>’)

(a)    (b)    (c)  

(d)    (e)    (f)  

(g)    (h)    (i)   

(j)    (k)   

6. The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form.

7. Find answers to the following. Write and indicate how you solved them.

(a) Is equal to ? (b) Is equal to ?

(c) Is equal to ? (d) Is equal to ?

8. Ila read 25 pages of a book containing 100 pages. Lalita read of the same book. Who read less?

9. Rafiq exercised for of an hour, while Rohit exercised for 

Who exercised for a longer time?

10. In a class A of 25 students, 20 passed with 60% or more marks; in another class B of 30 students, 24 passed with 60% or more marks. In which class was a greater fraction of students getting with 60% or more marks?

7.10 Addition and Subtraction of Fractions

So far in our study we have learnt about natural numbers, whole numbers and then integers. In the present chapter, we are learning about fractions, a different type of numbers.

Whenever we come across new type of numbers, we want to know how to operate with them. Can we combine and add them? If so, how? Can we take away some number from another? i.e., can we subtract one from the other? and so on. Which of the properties learnt earlier about the numbers hold now? Which are the new properties? We also see how these help us deal with our daily life situations.

Look at the following examples: A tea stall owner consumes in her shop litres of milk in the morning and litres of milk in the evening in preparing tea. What is the total amount of milk she uses in the stall?

Or Shekhar ate 2 chapatis for lunch and chapatis for dinner. What is the total number of chapatis he ate?

Clearly, both the situations require the fractions to be added. Some of these additions can be done orally and the sum can be found quite easily.

Make five such problems with your friends and solve them.

7.10.1 Adding or subtracting like fractions

All fractions cannot be added orally. We need to know how they can be added in different situations and learn the procedure for it. We begin by looking at addition of like fractions.

Take a 7 × 4 grid sheet (Fig 7.13). The sheet has seven boxes in each row and four boxes in each column.

How many boxes are there in total?

Colour five of its boxes in green.

What fraction of the whole is the green region?

Now colour another four of its boxes in yellow.

What fraction of the whole is this yellow region?

What fraction of the whole is coloured altogether?

Does this explain that ?

Look at more examples

In Fig 7.14 (i) we have 2 quarter parts of the figure shaded. This means we have 2 parts out of 4 shaded or of the figure shaded.

That is,

Look at Fig 7.14 (ii)

Fig 7.14 (ii) demonstrates .

What do we learn from the above examples? The sum of two or more like fractions can be obtained as follows :

Step 1 Add the numerators.

Step 2 Retain the (common) denominator.

Step 3 Write the fraction as :

Let us, thus, add and . We have

So, what will be the sum of and ?

Finding the balance

A diagram can explain the situation (Fig 7.15). (Note that, here the given fractions are like fractions).

We find that

(Is this not similar to the method of adding like fractions?)

Step 1 Subtract the smaller numerator from the bigger numerator.

Step 2 Retain the (common) denominator.

Step 3 Write the fraction as :

Can we now subtract from ?

1. Write these fractions appropriately as additions or subtractions :

(a) (b) (c) (d) (e)

(f) (g) (h) (i)

3. Shubham painted of the wall space in his room. His sister Madhavi helped and painted of the wall space. How much did they paint together?

4. Fill in the missing fractions.

(a) (b) (c) (d)

5. Javed was given of a basket of oranges. What fraction of oranges was left in the basket?

7.10.2 Adding and subtracting fractions

We have learnt to add and subtract like fractions. It is also not very difficult to add fractions that do not have the same denominator. When we have to add or subtract fractions we first find equivalent fractions with the same denominator and then proceed.

What added to gives ? This means subtract from to get the required number.

Since and are unlike fractions, in order to subtract them, we first find their equivalent fractions with the same denominator. These are and respectively.

This is because

Therefore,

Note that 10 is the least common multiple (LCM) of 2 and 5.

Example 8 : Subtract from .

Solution : We need to find equivalent fractions of and , which have the same denominator. This denominator is given by the LCM of 4 and 6. The required LCM is 12.

Therefore,

Example 9 : Add to .

Solution : The LCM of 5 and 3 is 15.

Therefore, =

Example 10 : Simplify

Solution : The LCM of 5 and 20 is 20.

Therefore, =

How do we add or subtract mixed fractions?

Mixed fractions can be written either as a whole part plus a proper fraction or entirely as an improper fraction. One way to add (or subtract) mixed fractions is to do the operation seperately for the whole parts and the other way is to write the mixed fractions as improper fractions and then directly add (or subtract) them.

Example 11 : Add and

Solution : 

Now 

=  =  = 

Thus, 

And, therefore, 

Think, discuss and write

Can you find the other way of doing this sum?

Example 12 : Find

Solution : The whole numbers 4 and 2 and the fractional numbers can be subtracted separately. (Note that 4 > 2 and )

Example 13 : Simplify:

Solution : Here 8 > 2 but . We proceed as follows:

Exercise 7.6

1. Solve

(a) (b) (c) (d) (e)

(f) (g) (h) (i) (j)

(k) (l) (m) (n)

2. Sarita bought metre of ribbon and Lalita metre of ribbon. What is the total length of the ribbon they bought?

3. Naina was given 1 piece of cake and Najma was given 1 piece of cake. Find the total amount of cake was given to both of them.

4. Fill in the boxes : (a) (b) (c)

5. Complete the addition-subtraction box.

(a) (b)

6. A piece of wire metre long broke into two pieces. One piece was metre long. How long is the other piece?

7. Nandini’s house is km from her school. She walked some distance and then took a bus for km to reach the school. How far did she walk?

8. Asha and Samuel have bookshelves of the same size partly filled with books. Asha’s shelf is th full and Samuel’s shelf is th full. Whose bookshelf is more full? By what fraction?

9. Jaidev takes minutes to walk across the school ground. Rahul takes minutes to do the same. Who takes less time and by what fraction?

What have we discussed?

1. (a) A fraction is a number representing a part of a whole. The whole may be a single object or a group of objects.

(b) When expressing a situation of counting parts to write a fraction, it must be ensured that all parts are equal.

2. In , 5 is called the numerator and 7 is called the denominator.

3. Fractions can be shown on a number line. Every fraction has a point associated with it on the number line.

4. In a proper fraction, the numerator is less than the denominator. The fractions, where the numerator is greater than the denominator are called improper fractions. An improper fraction can be written as a combination of a whole and a part, and such fraction then called mixed fractions.

5. Each proper or improper fraction has many equivalent fractions. To find an equivalent fraction of a given fraction, we may multiply or divide both the numerator and the denominator of the given fraction by the same number.

6. A fraction is said to be in the simplest (or lowest) form if its numerator and the denominator have no common factor except 1.