Solution - Adding, subtracting and finding the least common multiple

Step  1  :

             1 
 Simplify   ———
            625

Equation at the end of step  1  :

       1   1   1    1 
  (((1+—)+——)+———)+———
       5  25  125  625

Step  2  :

             1 
 Simplify   ———
            125

Equation at the end of step  2  :

       1   1   1    1 
  (((1+—)+——)+———)+———
       5  25  125  625

Step  3  :

             1
 Simplify   ——
            25

Equation at the end of step  3  :

          1      1      1       1 
  (((1 +  —) +  ——) +  ———) +  ———
          5     25     125     625

Step  4  :

            1
 Simplify   —
            5

Equation at the end of step  4  :

          1      1      1       1 
  (((1 +  —) +  ——) +  ———) +  ———
          5     25     125     625

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  5  as the denominator :

          1     1 • 5
     1 =  —  =  —————
          1       5  

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:

 5 + 1     6
 —————  =  —
   5       5

Equation at the end of step  5  :

    6     1      1       1 
  ((— +  ——) +  ———) +  ———
    5    25     125     625

Step  6  :

Calculating the Least Common Multiple :

 6.1    Find the Least Common Multiple

      The left denominator is :       5 

      The right denominator is :       25 

      Least Common Multiple:
      25 

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 = 5

   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.      6 • 5
   ——————————————————  =   —————
         L.C.M              25  

   R. Mult. • R. Num.       1
   ——————————————————  =   ——
         L.C.M             25

Adding fractions that have a common denominator :

 6.4       Adding up the two equivalent fractions

 6 • 5 + 1     31
 —————————  =  ——
    25         25

Equation at the end of step  6  :

   31     1       1 
  (—— +  ———) +  ———
   25    125     625

Step  7  :

Calculating the Least Common Multiple :

 7.1    Find the Least Common Multiple

      The left denominator is :       25 

      The right denominator is :       125 

      Least Common Multiple:
      125 

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 = 5

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 7.3      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      31 • 5
   ——————————————————  =   ——————
         L.C.M              125  

   R. Mult. • R. Num.       1 
   ——————————————————  =   ———
         L.C.M             125

Adding fractions that have a common denominator :

 7.4       Adding up the two equivalent fractions

 31 • 5 + 1     156
 ——————————  =  ———
    125         125

Equation at the end of step  7  :

  156     1 
  ——— +  ———
  125    625

Step  8  :

Calculating the Least Common Multiple :

 8.1    Find the Least Common Multiple

      The left denominator is :       125 

      The right denominator is :       625 

      Least Common Multiple:
      625 

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 = 5

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 8.3      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      156 • 5
   ——————————————————  =   ———————
         L.C.M               625  

   R. Mult. • R. Num.       1 
   ——————————————————  =   ———
         L.C.M             625

Adding fractions that have a common denominator :

 8.4       Adding up the two equivalent fractions

 156 • 5 + 1     781
 ———————————  =  ———
     625         625

Final result :

  781           
  ——— = 1.24960 
  625