Solution - Adding, subtracting and finding the least common multiple

Step  1  :

             3
 Simplify   ——
            16

Equation at the end of step  1  :

        3  3  3   3
  (((-3-—)-—)-—)-——
        2  4  8  16

Step  2  :

            3
 Simplify   —
            8

Equation at the end of step  2  :

        3  3  3   3
  (((-3-—)-—)-—)-——
        2  4  8  16

Step  3  :

            3
 Simplify   —
            4

Equation at the end of step  3  :

           3     3     3      3
  (((-3 -  —) -  —) -  —) -  ——
           2     4     8     16

Step  4  :

            3
 Simplify   —
            2

Equation at the end of step  4  :

           3     3     3      3
  (((-3 -  —) -  —) -  —) -  ——
           2     4     8     16

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  2  as the denominator :

           -3     -3 • 2
     -3 =  ——  =  ——————
           1        2   

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 5.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 -3 • 2 - (3)     -9
 ————————————  =  ——
      2           2 

Equation at the end of step  5  :

    -9    3     3      3
  ((—— -  —) -  —) -  ——
    2     4     8     16

Step  6  :

Calculating the Least Common Multiple :

 6.1    Find the Least Common Multiple

      The left denominator is :       2 

      The right denominator is :       4 

      Least Common Multiple:
      4 

Calculating Multipliers :

 6.2    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 2

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 6.3      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)²   and  (y²+y)/(y+1)³  are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

   L. Mult. • L. Num.      -9 • 2
   ——————————————————  =   ——————
         L.C.M               4   

   R. Mult. • R. Num.      3
   ——————————————————  =   —
         L.C.M             4

Adding fractions that have a common denominator :

 6.4       Adding up the two equivalent fractions

 -9 • 2 - (3)     -21
 ————————————  =  ———
      4            4 

Equation at the end of step  6  :

   -21    3      3
  (——— -  —) -  ——
    4     8     16

Step  7  :

Calculating the Least Common Multiple :

 7.1    Find the Least Common Multiple

      The left denominator is :       4 

      The right denominator is :       8 

      Least Common Multiple:
      8 

Calculating Multipliers :

 7.2    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 2

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 7.3      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      -21 • 2
   ——————————————————  =   ———————
         L.C.M                8   

   R. Mult. • R. Num.      3
   ——————————————————  =   —
         L.C.M             8

Adding fractions that have a common denominator :

 7.4       Adding up the two equivalent fractions

 -21 • 2 - (3)     -45
 —————————————  =  ———
       8            8 

Equation at the end of step  7  :

  -45     3
  ——— -  ——
   8     16

Step  8  :

Calculating the Least Common Multiple :

 8.1    Find the Least Common Multiple

      The left denominator is :       8 

      The right denominator is :       16 

      Least Common Multiple:
      16 

Calculating Multipliers :

 8.2    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 2

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 8.3      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      -45 • 2
   ——————————————————  =   ———————
         L.C.M               16   

   R. Mult. • R. Num.       3
   ——————————————————  =   ——
         L.C.M             16

Adding fractions that have a common denominator :

 8.4       Adding up the two equivalent fractions

 -45 • 2 - (3)     -93
 —————————————  =  ———
      16           16 

Final result :

  -93            
  ——— = -5.81250 
  16