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**Introduction to Fractions**

Fractions are types of integers and are used to represent a part of the whole. For example, if I have to say I have one half of a cake, it means that I divided 1 cake into 2 equal and smaller parts and I possess one of the 2 parts. To put it simply, I have 1/2 part of the cake.

They are represented in the format of “p/q” where the “p” part is called the Numerator and the “q” part is called the Denominator. P/q means ‘p’ parts out of ‘q’ number of total parts. The value of a fraction in decimal form can be obtained by dividing the numerator by denominator.

For example ‘3/5’ means 3 parts out of 5, 1/3 means 1 part out of 3. Fractions are also used to denote division or ratios. For example, 3/5 can be read as ‘3 divided by 5’ or as a ratio of ‘3 is to 5’.

**Equivalence**

2 sets of fractions can be the same although their numerators and denominators do not match. Consider the fractions 3/6, 1/2 and 9/18. In all the 3 fractions we see that the numerator is one half or 1/2 of the denominator. Thus all the fractions are easily reducible to the common form 1/2. This method is called Simplifying or reducing the fraction.

If we observe the numerator and denominator to have a common factor we eliminate that factor to reduce the fraction to a simpler form. Some of the important ones are given below.**Addition**

Addition of 2 fractions can take place only when they have the same denominator. For example, if we add 3/6 and 5/6 then this is possible as both the fractions have the same denominator i.e. 6. in such additions we add only the numerators and the denominator stays the same. Consider this as adding 3 parts out of 6 to another 5 parts out of 6. So,

3/6 + 5/6 = (3 + 5)/6 = 8/6

Consider the case when denominators of the fractions are not the same. In such a case we make the denominators of both the fractions same by multiplying the denominators with a suitable factor to form an l.c.m of the denominators and then add them. Consider the fractions 3/6 and 1/2

3/6 + 1/2

In this case, 2 and 6 have common LCM as 6. So, we make both the denominators as 6. To do this we multiply the numerator and denominator of 1/2 with 3 so as to not change the value of the fraction.

so 3/6 + 1/2 = 3/ 6 + 3/6 = 6/6 = 1**Subtraction**

Subtraction of fractions follows the same rule as the addition of fractions where it is necessary to have the same denominator for carrying out the operation. Consider the example 5/6 – 3/6= (5 – 3) / 6 = 2/6 = 1/3. Similarly, 5/6 -1/4 = (5 x 4) / (6 x 4) – (1 x 6) / (4 x 6) = 20/24 – 6/24 = (20 – 6) / 24 = 14 / 24 = 7/12.**Multiplication**

Multiplication of 2 fractions is performed by multiplying the numerators of the 2 numbers and multiplying the denominators of the 2 numbers separately and obtaining the final result. For example, 3/9 x 4/5 = (3 x 4) / (9 x 5) = 12/45**Division**

Division of 2 fractions again, is carried out by dividing the 2 numerators and dividing the 2 denominators separately and obtaining the final result. For example, 6/18 divided by 2/9 = (6 divided by 2) / (18 divided by 9) = 3/2**Solved Example****Ques: Obtain the result if (2/3) is added to (4/9) and is multiplied by 3****Sol:** 3x (2/3 + 4/9)

= 3x (6/9 + 4/9)

= 3x ( 10/9)

= 30/9

= 10/3.3333333333333333**Multiplication of Fractions**

Fractions are represented in the “p/q” format and this form is most preferred for carrying out multiplication of 2 or more fractions.

Multiplication of fractions is a simple operation. Multiply the numerator of all the fractions separately and multiply the denominators of all the fractions separately to obtain the final numerator and denominator.

For example,

Consider 3/5 × 4/9 = (3 × 4)/(5 × 9) = 12/45

Now, in the above question, the numerator and the denominator have a common factor i.e. 3.

Reduce the numerator and denominator to their lowest forms to obtain the final result. So,

12/45 = (3 × 4)/(3 × 3 × 5) = 4/15

A shortcut is to cancel the common terms while multiplying initially rather than dividing the answer. But that process requires some practice and can be developed over a period of time.

Now, Consider 2/6 × 3/4 × 5/7

Now here, we see that the first fraction is reducible to a smaller form, so we do it beforehand, i.e. 2/6 = 1/3

So the question changes to

1/3 × 3/4 × 5/7 = (1 × 3× 5)/(3 × 4× 7)

again, we see that the numerator and denominator both have a common factor i.e. 3, which we cancel from both the numerator or denominator. We basically divide 3 by 3. So we get

(1× 5)/(4 × 7) = 5/28

Note that for multiplication it is not necessary to have the same denominator in both the fractions, a rule mandatory in addition and subtraction of fractions.**Need for multiplication**

Multiplication of fractions is particularly important to obtain ratios. For example, if we need 1/3 of 9/5 we multiply 1/3 to 9/5 to obtain the answer. So 1/3 rd of 9/5 is

1/3 × 9/5 = 1/3 × (3 × 3)/5

= 3/5**Multiplication of a Fraction with a Whole number**

Suppose we have to obtain 3/4th of 7 to obtain this we consider the whole number as a fraction itself.

“Any whole number “x” can be represented as x/1″

It basically means that we divide the number by 1. any number divided by 1 is the number itself. Now the multiplication becomes

3/4 × 7/1

Using the rules of multiplication we get

(3 × 7)/(4 × 1)= 21/4**Multiplication of Mixed Fractions**

A mixed fraction is one which has a whole number and a fraction together. For example, 23/5 or 71/ 9. To multiply them we convert each mixed fraction into a simple fraction and then multiply these fractions.

A mixed fraction of form xy/z is converted to normal by multiplying the denominator by the whole number and adding the numerator to obtain the numerator. For example, in the above form, it is (z.x+ y)/z as a normal fraction.

Consider the multiplication, 2 3/5 × 7 1/9

we first convert them to normal fractions,

= (5 × 2 + 3)/5 and (9× 7 +1)/9

= 13/5 and 64/9

now following rules of multiplication of fractions,

13/5 × 64/9

= (13 × 64)/(5 × 9)

= 8^{32}/ 45**Solved Questions****Ques 1: Multiply 21/5 and 15/4****Sol: **Converting the mixed fraction to normal we get 11/5.

Now, 11/5 × 15/4

= (11 × 15)/(5 × 4)

Canceling common terms,

=( 11× 5 × 3)/ (5× 4)

= 33/4**Ques 2: Of the land owned by a farmer, 90 percent was cleared for planting. Of the cleared land, 40 percent was planted with soybeans and 50 percent of the cleared land was planted with wheat. If the remaining 720 acres of cleared land was planted with corn, how many acres did the farmer own?****1.** 5832**2.** 6420**3.** 7200**4.** 8000**Sol:** Land used for planting wheat and soybeans = 90 percent

Remaining land = 10 percent

So the total area of cleared land must be 7200 acres.

So 7200 is 92 percent of the total land.

Therefore the total land area must be 8000 acres.

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