FRACTION

Fraction - Introduction

Fraction :

fraction represents a part of a whole or, more generally, any number of equal parts.  a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters.

In fractions (for example 17/23), the number above the line (17) is called numerator and the number below the line (23) is called denominator. 

The numerator represents a number of equal parts, and the denominator, which cannot be zero, indicates how many of those parts make up a unit or a whole.

For example, in the fraction 3/4, the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole. 

The picture given below illustrates the fraction 3/4. 

Different kinds of fractions

Now, let us look at some different kinds of fractions.

Proper fraction :

A fraction is called a proper fraction if its

Denominator > Numerator.

Example : 3/4, 1/2, 9/10, 5/6

Improper fraction :

A fraction is called an improper fraction if its

Numerator > Denominator.

Example : 5/4, 6/5, 41/30, 51/25

Mixed fraction :

A fraction consisting of a natural number and a proper fraction is called a mixed fractions.

Example : 2 3/4, 1 4/5, 5 1/7

Think it : 

Mixed fraction  =  Natural number + Proper fraction

Let us look at next stuff on "Fractions in mathematics"

Addition and subtraction of fractions with same denominator

Example 1 : 

Simplify : 2/5 + 3/5

Solution : 

Here, for both the fractions, we have the same denominator, we have to take only one denominator and add the numerators. 

Then, we get 

2/5 + 3/5  =  (2+3) / 5  =  5/5  =  1

Example 2 :

Simplify : 7/5 - 3/5

Solution : 

Here, for both the fractions, we have the same denominator, we have to take only one denominator and subtract the numerators. 

Then, we get 

7/5 - 3/5  =  (7-3) / 5  =  4/5

Addition and subtraction of fractions with different denominators

Here, we explain two methods to add two fractions with different denominators.

1)  Cross multiplication method

2)  L.C.M method

Cross multiplication method :

If the denominators of the fractions are co-prime or relatively prime, we have to apply this method.

Fro example, let us consider the two fractions 1/8,  1/3.

In the above two fractions, denominators are 8 and 3.

For 8 and 3, there is no common divisor other than 1. So 8 and 3 are co-prime.

Here we have to apply cross-multiplication method to add the two fractions 1/8 and 1/3 as given below. 

L.C.M method :

If the denominators of the fractions are not co-prime (there is a common divisor other than 1), we have to apply this method.

Fro example, let us consider the two fractions 5/12,  1/20.

In the above two fractions, denominators are 12 and 20.

For 12 and 20, if there is at least one common divisor other than 1, then 12 and 20 are not co-prime.

For 12 & 20, we have the following common divisors other than 1.

2 & 4

So 12 and 20 are not co-prime.

In the next step, we have to find the L.C.M (Least common multiple) of 12 and 20.

12 =  2² x 3

20 = 2² x 5

When we decompose 12 and 20 in to prime numbers, we find 2, 3 and 5 as prime factors for 12 and 20. 

To get L.C.M of 12 and 20, we have to take 2, 3 and 5 with maximum powers found above.

So, L.C.M of 12 and 20 = 2² x 3 x 5

                                        = 4 x 3 x 5

                                        = 60

Now we have to make the denominators of both the fractions to be 60 and add the two fractions 5/12 and 1/20 as given below.

Note :

We have to do the same process for subtraction of two fractions with different denominators. 

Let us look at next stuff on "Fractions in mathematics"

Multiplication of a fraction by a whole number

To multiply a proper or improper fraction with the whole number,first, we have to multiply the whole number with the numerator of the fraction, keeping the denominator same.

For example, 

2 x 3/5   =  6/5

3 x  7/11  =  21/11

To multiply a mixed fraction by a whole number, first convert the mixed fraction to an improper fraction and then multiply.

For example, 

4 x 3 4/7   =  4 x 25/7  =  100/7  =  14 2/7

Let us look at next stuff on "Fractions in mathematics"

Converting improper fractions to mixed numbers

The picture given below clearly illustrates, how to convert improper fractions in to mixed numbers 

Multiplication of a fraction by a fraction

To multiply a proper or improper fraction by another proper or improper fraction, we have to multiply the numerators and denominators. 

For example, 

2/3 x 4/5   =  8/15

1/3 x  7/11  =  7/33

The reciprocal of a fraction

If the product of two non-zero numbers is equal to one, then the two numbers are reciprocal to each other. 

That is, the reciprocal of 3/5 is 5/3 and the reciprocal 5/3 is 3/5.

Moreover, 

(5/3)x(3/5)  =  1

Note:

Reciprocal of 1 is 1 itself. 0 does not have a reciprocal.

Division of a whole number by a fraction

To divide a whole number by any fraction, multiply that whole number by the reciprocal of that fraction.

For example, 

÷  2/5  =  6 x 5/2  =  30/2  =  15

While dividing a whole number by a mixed fraction, first convert the mixed fraction into improper fraction and then solve it.

÷  3 4/5  =  6 ÷ 19/5  =  6 x 5/19  =  30/19  =  1 11/19

Division of a fraction by another fraction

To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.

For example,

1/5 ÷  3/7  =  1/5 x 7/3  =  7/15

After having gone through the stuff given above, we hope that the students would have understood "Fractions in mathematics". 

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ALGEBRA

Variables and constants

Writing and evaluating expressions

Solving linear equations using elimination method

Solving linear equations using substitution method

Solving linear equations using cross multiplication method

Solving one step equations

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Solving quadratic equations by quadratic formula

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Nature of the roots of a quadratic equations

Sum and product of the roots of a quadratic equations 

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Combining like terms

Square root of polynomials 

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Remainder theorem

Synthetic division

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Negative exponents rules

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Domain and range of rational functions

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Decimal representation of rational numbers

Finding square root using long division

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Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6