how to multiply fraction and fraction, improper fraction, whole number, mixed numbers etc.

To multiply fraction and fraction


To multiply fraction and fraction

To multiply fraction and whole number

To multiply fraction and improper fraction

To multiply fraction and mixed number

To multiply 2 mixed numbers

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how to add fraction and whole number, improper fraction, mixed numbers

Addition of fraction and whole numbers
Examples:
1. 1/3 + 5 = 5 1/3
2. 4/6 + 12 = 12 4/6   or 12 2/3

Addition of fraction and improper fraction
Pre requisite : how to add fractions with different denominators
Examples:
1. 2/5 + 9/4 =

    2(4)  + 9 (5)   =
    5(4)      4(5)

      8   + 45       = 
      20    20     

        53    or 2 13/20
        20

2.    5/7 + 6/5 =

       5(5) + 6(7)  =
       7(5)    5(7)

         25   +  42    =
         35       35 

         67     or  1 32/35
         35



Addition of fraction and mixed numbers
Examples:
1. 4/9 + 3 1/4 =

     4/9 +  13/4                      3  1/4 = (4)(3)+1 =13/4   changed to improper fraction

      4(4)  + 13 (9) =
     9(4)        4(9)

      16     +   117  =
      36           36

      133    or  3 25/36
       36
Addition of mixed numbers and mixed numbers

2.  2 2/3  + 4 1/5  =       both mixed numbers

     8/3   +  21/5   =           changed to improper fractions

     8(5)   + 21(3)  =
     3(5)       5(3)

      40    +   63  =
      15         15

      103  or   6  13/15
       15  

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how to divide integers

To divide integers with like signs, divide the integers and quotient is always positive.

Examples:
1. (-20)/(-5) = 4      positive quotient
2. (100)/(10) = 10   positive quotient

To divide integers with unlike signs, the quotient is always negative

Examples:
1. (-36)/ (6) = -6   negative quotient
2. (81)/ (-9) = -9  negative quotient

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how to find the common difference in arithmetic sequence

Pre requisite : how to add of integers, how to subtract  integers
To get the common difference in arithmetic sequence, subtract  the 2nd term to the first term, or  3rd term to 2nd term.....
Examples
1. 2, 4, 6, 8,....
    4 - 2 = 2      so the common difference is 2.. upon evaluating 6-(4), 8-(6), ....    will reveal
                        a common difference of 2 
6-(4)=
(6) +(-4) = 2

8-(6)=
8+(-6) =   2
                       knowing the common difference will give the discovery of succeeding pattern are
                        arithmetic sequence.
                       common difference is used in generating the next pattern as
                       2 + 2 = 4,  4+2 = 6,   6 + 2 = 8       so the arithmetic sequence is 2, 4, 6, 8 ......
                   

        

2. 12, 6, 0, -6, -12....
    (6)-(12) =
     (6) + (-12) = -6    so the common difference  is -6 upon evaluating 0 -(6), (-6)-(0), (-12)-(-6)...
                          will reveal a common difference of -6.
                          exposing....
     0 - (6)=
     0 +(-6)= -6    difference

     (-6)-(0) =
     (-6) + 0  = -6  difference

     (-12) - (-6)=
     (-12) + (6) = -6   difference     therefore the common difference is -6




Exercises:  Find the difference of the following arithmetic sequence.
1. 5, 10, 15, 20, .......
2. 3, 6 , 9, 12, .......
3. 100, 75, 50, 25, ....
4. 1/2, 1, 3/2, 4/2,.....
5. -4, -1, 2, .....
6. 3, 0, -3, -6, .....     







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how to multiply integers

To multiply integers with like signs, the multiply the factors and the product is always positive
 Examples
1. (-5) (-4) = 20
2. (9 ) (2)  = 18    
To multiply integers with unlike signs, the multiply the factors and the product is always negative.
Examples:
1. (-12) (2) = -24
2. (15) (-3) = -45

To multiply multiple products
Examples
1. (2) (-2) (1) = (-4) (1) = -4
2. ( -1) (1) (1) (-2) = (-1) (-2) = 2
3. (1) (2) (-3) (1) (-1) = (2)(-3) (-1) = (-6) (-1) = 6
                                   or
                                   =(2) (3) = 6

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The Odd and Even Numbers

     A  Math IQ test  was given to a 67  applicants of  AZ Security Agency . From the numbers 1, 2, 3, 4, 5, 6, up 100    identify the numbers that are  even and the odd numbers.
      
Name:________________                  Position Applied:_______________
                         
Part I                               Odd                                    Even
                          _______________                 ______________
                          _______________                 ______________
                         ________________                ______________

                        (100 pts.)

Part II   Why are  odd and even  words  appropriate  for its names?(Hint: count the
              letters )   (20 pts.)  

Part III   The total applicants are 67 , find sum of the given two digits? And  identify the sum if
               odd or even.  (20 pts.)
-------------------------------------------------------------------------------------------------------
The questions may become difficult if odd and even numbers could not be identified.


So odd numbers are whole numbers that are not divisible by 2 or can not  be divided exactly by 2
examples
1. 1, 3, 5, 7, 9, 11, ......
While even numbers are whole numbers that are divisible by 2 or can be divided by 2 exactly without remainder.
examples
2, 4, 6, 8, 10, 12,........ it is  multiple of 2

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how to subtract integers

to subtract the two integers , change the sign of the subtrahend and proceed to the law of addition of integers.
Examples
1. (-9) - (5)
    (-9) + (-5)  = -14    change the sign of the subtrahend integer and proceed to the law of
                                   addition of integers   ( see addition of integers with like signs)

2. (20) - (-45)
    (20) + (45)  =  65    change the sign of the subtrahend integer and proceed to the law of
                                   addition of integers   ( see addition of integers with like signs)     

3. (20) - (45)
    (20) +(-45)  =  -25 change the sign of the subtrahend integer and proceed to the law of
                                   addition of integers   ( see addition of integers with unlike signs)
4. (-55)- (-40)
    (-55) + (40)  = -15 change the sign of the subtrahend integer and proceed to the law of
                                   addition of integers   ( see addition of integers with like signs)
to be continued.....
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