Fractions simplification

The essence of mathematics is its freedom --Cantor G

Ex1.   1.9 x 1.9 - 1.1 x 1.1 = ?
          = (1.9)² - (1.1) ² = 3.61 - 1.21 = 2.40 ( Its simple as you know squares of 19 and 11 )

Ex2.   2.999 x 2.999 - 0.001 x 0.001 = ?
         Here one algebra formula is there to simplify this i.e. a² - b² = (a+b) (a-b)
         Here a = 2.999, b = 0.001
         So 2.999² - 0.001² = (2.999+0.001)(2.999-0.001)
                                        =  (3)(2.998)
                                        = 8.994

Ex3. 5.9x5.9 - 4.1x4.1 = ?   ( try yourself)

Ex4. 56.67 /100 = 5.667/?
        =56.67x100/100x100 = 5.667x1000/1000x?
        =5667/10000 = 5667/1000x?
        =x = 10

Ex5. 1.765/10 = 17.65/?   ( try yourself)

Subtraction of fractions

Go down deep enough into anything and you will find mathematics.  ~Dean Schlicter

Fraction subtraction is similar to addition only. Every fraction simplification involves making denominator equal.

Let us try one example.

Ex1 : 1/2 - 1/4 = ?

           = (1x2)/(2x2) - 1/4 ( now both denominators are same i.e. 4)

           = 2/4 - 1/4 = 1/4

Ex2 : 1/3 - 1/4 - 1/6 = ?

           = (1x4)/(3x4) - (1x3)/(4x3) - (1x2)/(2x6) ( now all denominators are same)

           =      4/12 - 3/12 - 2/12 = -1/12

In the above example, how to conclude that we need to make all denominators to 12.

LCM of 3, 4 and 6 is 12. 

As I mentioned in LCM post, you can find this very easily. Please look into LCM post to know more details.

In next post we will do fraction simplification. We will try more examples...

 

Addition of fractions

Pure mathematics is, in its way, the poetry of logical ideas.  ~Albert Einstein

 

Fraction addition

Thumb rule in fractions: Simplification is easy if all denominators in fractions are equal. So try to make all denominators same. The purpose of finding LCM is making all denominators same.

Ex : 1/2 + 1/3

1/2 means 1 part out of 2 parts

1/3 means 1 part out of 3 parts

Total no of parts should be same to do addition or subtraction

so what we will do is

 1/2 = 1x3/2x3 = 3/6 means 3 parts out of 6 parts

 1/3 = 1x2/3x2 = 2/6 means 2 parts out of 6 parts

As total parts are same for both fractions, result is addition of two numerators 

result = 1/2 + 1/3 = 3/6 + 2/6 = 5/6

Conclusion is 1/a + 1/b = (a+b)/ axb

We can do this by simply memorizing above formula. But how many formulas you can remember. Try to learn simplification through mental calculation.

Let us try one more example.

Ex 2: 5/6 + 2/3 + 1/2

5/6 + 2/3 + 1/2 = 5/6 + (2x2)/(3x2) + (1x3)/(2x3)       -- making all denominators same

 = 5/6 + 4/6 + 3/6 = 12/6 = 2

Please remember if you multiply numerator and denominator with the same number, there is no change in value of fraction.

Try following examples: 

Ex 3 : 1/3 + 1/7 = ?

Ex 4 : 2/3 + 1/4 + 1/2 = ?

LCM of three numbers

Always substitute mentally the definitions in place of the defined.

                                                                                            --- Blaise Pascal

Ex 1: Find LCM of 2, 3 and 4

LCM of 2 and 4 : 4

LCM of 3 and 4 : 3x4= 12 (as no common factor for 3 and 4)

LCM of 2, 3 and 4 = 12

Ex2 : Find LCM of 3, 4, 5 and 6

LCM of 3, 4 and 5 : 3x4x5= 60 (as no common factor for 3, 4 and 5)

60 is divisible by  6.

LCM of 3, 4, 5 and 6 : 60

Ex3 : Find LCM of 12, 20, 25 and 30

LCM of 20 and 30 = 2x3x10= 60 ( no common factor for 2 and 3, 10 is common divisor for 20 and 30)
60 is divisible by 12 but not divisible by 25
LCM of 60 and 25 = 60x5 = 300 (as 300 is divisible by 12, 20, 25 and 30 )

LCM - Least Common Multiple

Quote: Mathematics is the door and key to the sciences.   ---Roger Bacon

Ex1. Find LCM 2 and 4?

Multiples of 2 : 2, 4, 6, 8,...
Multiples of 4 : 4, 8, 12, 16,...
Common multiples of 2 and 4 : 4, 8, ...
Least common multiple : 4

Rules from above example:

1. If one number is multiple of other number, the bigger number is LCM

2. To find LCM easily take the small number multiples and check multiple is divisible by bigger number. If yes, then the multiple is LCM

Ex2 : find LCM of 3 and 9?

LCM is 9 as 9 is divisible by 3.

Ex3. Find LCM of 4 and 6?

Small number is 4
Multiples of 4 : 4, 8, 12, 16, ...
Is 4 divisible by 6? No
Is 8 divisible by 6? No
Is 12 divisible by 6? Yes. So LCM is12

Ex4. Find LCM of 3 and 4?

multiples of 3: 3, 6, 9, 12, 15, 18, 21, ...
multiples of 4: 4, 8, 12, 16, 20, 24, 28, ...
So LCM is 12

Rule: if both the numbers doesn't have common factors, then LCM is product of two numbers.

Lets see how to find LCM  for more than two numbers in next post

Comparision of fractions

Quote: The essence of mathematics is not to make simple things complicated, but to make complicated things simple.    --S.Gudder

Let me take one example
1. compare 1/4 and 3/4 
3/4 > 1/4

Rule 1 : If denominators of the both fractions are equal, compare numerators.

In the above example denominator of the both fractions is 4.
So compare 1 and 3.
As 3 > 1, 3/4 > 1/4

2. compare 3/7 and 3/9
 3/7 > 3/9

Rule 2 : if numerators of the both fractions are equal, denominator with more value is lesser.

 In the above example 3/7 and 3/9 having same numerator. So denominator with more value is lesser.
 In this case denominator 9 is more. So 3/9 is lesser than 3/7

3. compare 2/3 and 4/5 
    step 1 : 2/3   ?  4/5
    step 2 : 2x2/3x2 ? 4/5 ( multiply denominator and numerator with same value i.e. 2)
    Multiplying with 2 to make both the numerator will become same.

4. compare  2/7 and 5/9

    This is generic method and can be applied every example above. 

    Known as Cross multiplication method.

    step 1 : 2/7 , 5/9  ( cross multiplication : multiply numerator and denominator )

                2x9  ?  5x7

                18    ?  35

                18    <  35

          So  2/7 < 5/9

    In traditional method, need to find Least Common Multiple(L C M).
    LCM of 9 and 7 is 63 (9 x 7) as both are prime numbers. 

    So 2/7 x 9/9 and 5/9x7/7
          18/63 ? 35/63
          As denominators are equal 18/63 < 35/63 
          Hence 2/7 < 5/9

Lets look how to find LCM in next post...
                                                                                                                                         

Fractions -- Introduction

Quote : Do not worry about your difficulties in mathematics, I aessure you that mine are greater.
                                                                              --Albert Einstein  

What is fraction?

Dividing into equal pieces or parts.

When  fractions are used?

Let me take one example. Two friends want to share 10 rupees. how much each one of them get?

Answer is 5. But how did you get that. By dividing 10 into 2 parts. So by division whole thing divided into equal parts

In the above example each part is Rs. 5/- i.e 10 x 1 / 2

In 1/2
1 is numerator
2 is denominator

Let me ask you one question.
If the two friends divide Rs. 10/- into 4 parts and each one takes two parts. Then each one will get how many rupees? Then also Rs.5/- only

So 10 x1/2 or 10 x 2/4 or 10 x 3/6..... all are equal.

Rule : if you multiply numerator and denominator with the same number, there is no change in the value of fraction.

1/2 = 2/4 = 3/6

Fractions can be expressed as decimal values. For example
1/2 = 0.5
1/4= 0.25


In the above example whole amount i.e. Rs. 10/-
how many friends sharing i.e. denominator


if 3 friends divide Rs. 10/-  then one part value is 10 x 1 / 3
if 4 friends divide Rs. 10/-  then one part value is 10 x 1 / 4
if 5 friends divide Rs. 10/-  then one part value is 10  x 1 / 5

each part value = whole value / no. of friends sharing