Solution - Adding, subtracting and finding the least common multiple

Step  1  :

            r4
 Simplify   ——
            4 

Equation at the end of step  1  :

  (r3)    r4
  ———— -  ——
   3      4 
 Step  2  :
            r3
 Simplify   ——
            3 

Equation at the end of step  2  :

  r3    r4
  —— -  ——
  3     4 

Step  3  :

Calculating the Least Common Multiple :

 3.1    Find the Least Common Multiple

      The left denominator is :       3 

      The right denominator is :       4 

      Least Common Multiple:
      12 

Calculating Multipliers :

 3.2    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 = 4

   Right_M = L.C.M / R_Deno = 3

Making Equivalent Fractions :

 3.3      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.      r3 • 4
   ——————————————————  =   ——————
         L.C.M               12  

   R. Mult. • R. Num.      r4 • 3
   ——————————————————  =   ——————
         L.C.M               12  

Adding fractions that have a common denominator :

 3.4       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:

 r3 • 4 - (r4 • 3)     4r3 - 3r4
 —————————————————  =  —————————
        12                12    

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   4r³ - 3r⁴  =   -r³ • (3r - 4) 

Final result :

  +r3 • (3r + 4)
  ——————————————
        12