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Fractions

- There is a birthday party in Meenu’s house. 10 children were attending the party. At the end of the party, there was a bar chocolate left. They decided to share the chocolate among themselves. Let’s see how they share…….

Solution: total no. of children = 10

total pieces of the bar chocolate = 15.

1.5

10 15

The portion coloured represents the piece of chocolate given to each child.

Fractions

- There are 10 balls in the collection. If we have to colour 2/5 of the collection, we will divide the collection into 5 equal parts. Then each part will contain 10/ 5= 2 balls. Now to colour 2/5, we will have to colour 2+2= 4 balls.

Fraction is a number representing a part of a whole. The whole maybe a single object or a collection of objects.

A fraction has two parts – numerator and denominator.

3

4

I am known as the numerator. I shows the number of parts picked.

I am known as the denominator. I shows the equal parts made.

From a basket containing 20 oranges, 2/5 were to be taken out. How many oranges were to be taken out?

Solution:

The fraction 2/5 means 2 parts out of 5 parts. So let us first arrange the oranges in 5 groups. Each group will have 20/5= 4 oranges. We need to take out 2 groups i.e. 4+4= 8 oranges.

Ans: 8 oranges were to be taken out.

Proper and improper fractions

- So far we were looking at fractions where the numerator was smaller than the denominator e.g. 2/5, 7/12, 15/23 etc. Here, 2<5, 7<12 and 15<23. These type of fractions are called Proper fractions.
- Now consider the following examples.

5 apples were to be shared equally among 3 friends. We can show this

as 5/3. similarly, if 6 bananas were shared equally among 4 friends, each would get 6/4 bananas.

- In such cases we see that the numerator is greater than the denominator. Such fractions are called Improper fractions. For example, fractions like 17/9, 14/5, 23/11 are all improper fractions as 17>9, 14>5, and 23/11.

Mixed fractions or mixed numbers

- A combination of a whole number and a proper fraction is called Mixed fraction or Mixed number.

consider the following:

This is 2+ ½ and is written as

2 ½.

Now consider This is 1+ ¼

and written

as 1 ¼.

Some more examples of mixed fractions are

3 ½, 5 ¾, and 4 ¼.

Conversion of mixed fraction into improper fraction

- Note the mixed fraction e.g. 2 ¼.
- Multiply the whole number by the denominator of the proper fraction. In this case it is 2*4= 8.
- To the product add the numerator of the proper fraction. Here, it is 1. So, 8+1= 9.

Hence, 2 ¼= [ 2*4] +1 = 8+1 = 9/4.

4 4

Improper fraction= [ whole number* denominator] + numerator

denominator

Conversion of improper fraction into mixed fraction

- Note down the improper fraction.
- Divide the numerator by the denominator and get the quotient and remainder.
- The quotient will be the whole number and the remainder will be the numerator of the proper fraction.

Therefore,

For example, to write the improper fraction 23/7 as a mixed fraction, we divide 23 by 7 and get 3 as the quotient and 2 as the remainder. So, the mixed fraction= 3 2/7.

Mixed fraction= quotient remainder

denominator

Fractions on the number line

- We have already learnt how to represent whole numbers on the number line. Let us now see how to represent fractions on the number line.

Draw a line and mark O on it. Mark a point A at unit distance from it. O represents zero and A represents 1.

0 1

O

A

Unit portion

Let us try to represent the fractions ½ and ¼ on the number line.

Divide the distance between O and A into 2 equal parts. Say M is the point of division. Then, M represents the fraction ½.

0 ½ 1

O M A

Now, let us divide the distance OA into 4 equal parts.

0 ¼ 2/4 ¾ 1

O P M N A

Points P, M, and N divide the distance OA into 4 equal parts. Then, P represents ¼, M will represent 2/4 and N will represent ¾. Point A can also be said to represent 4/4 which means 1.

Equivalent Fractions

- Look at the figures below.

½ 2/4 3/6 4/8

We observe that, though different fractions represent the shaded portion in each case, the same part of the whole has been shaded in all the figures.

Hence, we conclude that the fraction ½, 2/4, 3/6, and 4/8 actually represent the same parts and have the same value. Such fractions which have the same value are called Equivalent fractions.

Let us take ½ and multiply both the numerator and denominator by 2, 3 and 4 in turn.

We get 1*2 = 2/4, 1*3= 3/6 and 1*4= 4/8.

2*2 2*3 2*4

All these are equivalent fractions of ½.

To get equivalent fractions of a fraction, we multiply the numerator and denominator by the same number. If we divide the numerator and denominator of a fraction by the same number, we again get equivalent fractions of that fraction.

All these are equivalent fractions.

Fractions in lowest terms

- A fraction is said to be in its lowest terms if the numerator and the denominator do not have any common factor except 1. fractions in lowest terms are also known as fractions in the Simplest form.

Let us look at the fractions 12/16 and 40/50.

For 12/16, we can divide the numerator and the denominator by 2 to get 6/8 or we can divide 12/16 by 4 to get ¾.It means that 12 and 16 have the common factors 2 and 4. So, 12/16 is not in its lowest terms.

For 40/50, we can divide the numerator and the denominator by 2 to get 20/25 or we can divide

40/50 by 5 to get 8/10 or we can divide 40/50 by 10 to get 4/5. it means that 40 and 50 have the common factors 2, 5 and 10. So, 40/50 is not in its lowest terms.

- In the first case, after dividing 12/16 by 4 we got ¾. Here, 3 and 4 have no common factor but 1. in the second case, after dividing 40/50 by 10 we got 4/5. Here, 4 and 5 have no common factor but 1. Hence, ¾ and 4/5 are said to be fractions in lowest terms.

Reducing fractions to their lowest terms

- Find the HCF of the numerator and the denominator.
- Divide the numerator and the denominator by this HCF.
- The resulting equivalent fraction will be in the lowest terms.
- Fractions which have 1 as the numerator are called Unit fractions. For example, 1/5, 1/10, 1/17, 1/28, 1/56, 1/100 and 1/912 are unit fractions.

Unit fractions

Like and unlike fractions

- Fractions with the same denominators are called Like fractions. For example, 2/7, 1/7, 9/7, 5/7 are like fractions and 1/11, 5/11, 8/11, 10/11 are also like fractions.
- Fractions with different denominators are called Unlike fractions. For example, 2/7, 3/9, 2/15, 6/17 are unlike fractions and 4/5, 6/11, 10/17, 15/31 are also unlike fractions.

Comparing and ordering of fractions

- Comparing Like Fractions

Like fractions have the same denominator, and so to find out which fraction is larger, we have to look at the numerator only. For example, to find out which is larger 3/4 or 2/4, we see that 4 is the denominator i.e. 12 equal parts are made.

2/4 3/4

In the first case 2 parts are shaded, while in the second case 3 parts are shaded. We can easily see that the first fraction is larger. So ¾> 2/4.

Let us look at three like fractions.

2/6 1/6 4/6

We can easily see that 4/6 is the largest fraction and 1/6 is the smallest.

Also, 1/6< 2/6<4/6.

- In like fractions, the fraction with the larger numerator is the larger fraction and the one with the smaller numerator is the smaller fraction.

Comparing unlike fractions

We will study the comparison of unlike fractions in two parts.

- Unlike fractions with same numerators
- Unlike fractions with different numerators.
- Unlike fractions with same numerators

Let us look at the fractions ¼, ½, and 1/3.

¼ 1/3 ½

We can easily see that ½ is the largest fraction and ¼ is the smallest fraction.

We also see that ½> 1/3> ¼.

Let us look at the fractions 2/4, 2/3 and 2/5.

2/5 2/3 2/4

We can see that 2/3 is the largest fraction and 2/5 is the smallest fraction.

Also, 2/3>2/4>2/5.

- In unlike fractions with the same numerators, the fraction will the smallest denominator is the largest fraction and the fraction with the largest denominator is the smallest fraction.

Unlike fractions with different denominators

- Here, we will be dealing with fractions with different numerators and denominators e.g. 2/7, 3/11, 4/9, 1/5, etc.

Here, since both the numerator and denominator are different, it would be very difficult to decide which is larger.

So, we will convert then into Like fractions and will find equivalent fractions for the given fractions. All these equivalent fractions will have the same denominator.

Suppose we have to compare 2/7 and 3/5. We need to find equivalent fractions for both these fractions which have the same denominator.

But, what should this denominator be? How can we decide?

For this we find the LCM of the denominators of the given fractions.

In this case we will find the LCM of 7 and 5 which will be 7*5= 35 as 7 and 5 are co- primes.

Thus we write 2 = and 3 =

7 35 5 35

Since 7*5= 35, so the numerator will be 2*5= 10. Hence, 3 = 21

5 35

Now, instead of comparing 2 and 3 , we

7 5

compare 10/35 and 21/35. We see that 21/35> 10/35 as 21> 10. Therefore, 3/5 >2/7.

Addition and subtraction of like fractions

- Since like fractions have the same denominator, it is easy to add or subtract them. In addition sums, we add the numerators and write the answer with the same denominator as of the given fractions.

For example, to add 2 + 3 , we add 2+3= 5

7 7

and write the answer as 5/7. Similarly, if we have to add 1 + 3 + 2 , we add 1+3+2=6 and

8 8 8

write the answer as 6/8. Normally, answers are written in the lowest terms, so we write 6/8 as ¾.

- To subtract like fractions also, we just subtract the numerators and write the answer as the difference of the numerators with the same denominator as of the given fractions. For example, to subtract 7 _ 3, we

8 8

subtract the numerators 7 – 3= 4and write the answer as 4/8. Normally answers are written in the lowest terms, so we write 4/8 as ½.

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