Simple Fractions

simple-fractions

In a fraction :
     → is the numerator
2      → is the denominator

Anything divided by zero is infinity:

X = ∞
0

Here:
X ² → is the exponent

is the base
Anything to the power zero is always one

X º = 1

Solving parenthesis, brackets and braces:

First solve the parenthesis, then the brackets and finally braces.

second square brackets
 ‾‾‾‾↓
{     [    ( )    ]     }→third curly braces
            ↓
first parenthesis

example i):

{ 8 [ 10 ( 3 + 2 ) ] }

= { 8 [10 X 5] }

=8 X 50

= 400

example ii):

{ 5 [ 12 ( 10 – 5 ) ] }

= { 5 [ 12 (5) ] }

={ 5 [ 60 ] }

= 300

Whenever a parenthesis or bracket or brace is preceded by a “minus” ( negative sign) then whatever is inside the parenthesis or bracket or brace, it’s sign is inversed.

example:

− 7 − ( 3 − 1 + 7x − 2y ) =

− 7 − 3 + 1 − 7x + 2y =

 10 + 1 − 7x + 2y =

  11 − 7x  + 2y

Always while solving an equation there are few things to remember:

two negative signs make a positive

“−”   “−”  =   ” +”

one negative and one positive make a negative sign

“−”   “+”   =  “−”

two positives make a positive sign

“+”   “+”  =  “+”

Multiplying fractions :

example i):

5 x 2 = 10
8    3     24

example ii):

9 x 12 = 108
10   13   130

example iii):

112 x 214 = 23968
502    100    50200

Dividing fractions :

First we inverse the right hand side fraction then reduce it and solve the fraction further.

example i) :

2  ÷    =
8      27

2   x   27  =
8          9

2 ¹ x   27  3 =   3
 4      9  1       4

example ii) :

8   ÷   12   =
7          9

8   x     9   =
7          12

 8   x    3   =     24      =     6
7         12   4        28              7

example iii) :

5   ÷   205   =
6         500

5     x    500    =
6           205

5     x    500   100  50   =        250  
6         205    41                   123

Adding fractions :

example i) :

4   +   5  =
3        6

24    +   15    =       39
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾          ‾‾‾‾‾‾‾
18                           18

 We could do the above example by following either of the two methods:

conventional method:

1) First  we multiply the denominators, in this case 3×6=18. Write the product of the two as common denominator.

2) Then we figure out, how the first  denominator 3, can be multiplied with some number, so as to get 18.

3) That same number, by which we multiplied 3 to get 18, in this case 6, has to be multiplied by the first numerator 4 → 6×4 that gives us 24.

4) The same way we solve the other half of the problem  and add the two to get 39/18.

easier unconventional method :

Here we get common denominator by multiplying the two denominators 3 x 6=18. Leave it as it is.

1) We criss-cross multiply the two fractions together.

2) 4 X 6 and 3 X 5.

3) We get 24 and 15.

4) Add  the two and divide them by the common denominator that is (24+15) / 18.

n1   +   n2  =
d1         d2

n1.d2 +  n2.d1
d1.d2

example ii) :

  +   6   =
3         7

49   +  18  =   67
21                   21

 example iii) :

2    +   8   =
7         6

12   +   56   =   68
42                     42

example iv) :

12   +    =
7          54

648    +   56    =    704
378                        378

 example v) :

856   +   293   =
899        453

387768  +  263407      =
407247

651175
407247

Subtracting fractions :

We follow the same above mentioned method for subtracting fractions.

example i)

8    −     =
9         3

 24     −   18    =    6   =   2
27                         27       9

example ii)

  −     =
2           7

42    −   6    =          36    =     18
14                            14             7

Scientific notation:

If we have to express very large number in a concise manner, we resort to scientific notation.

Below are examples of some numbers expressed in scientific notation with positive and negative exponents .

♦  724000000 = 7.24  x 10 8

♦  12785600000 = 12.78  x  10 7

♦   290  =  2.9  x 10 2

♦  15000000000  =  1.5  x  10 10

♦   7000000000000  = 7 x 10 12

 0.074 = 7.4 x [latex size="-2"]10^{-2}[/latex]

♦  0.00000000278 = 2.78 x [latex size=”-1″]10^{-9}[/latex]

♦   0.00000670 = 6.7 x  10 – 6

♦  0.00000000000000456 = 4.56  x  10 – 15

♦  0.0000000000000000000034 = 3.4  x  10  – 21

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