Solution - Adding, subtracting and finding the least common multiple

Step  1  :

 1.1     900 = 22•32•52

(900)² = (2²•3²•5²)² = 2⁴ • 3⁴ • 5⁴

Equation at the end of step  1  :

    1          1    
  ———— -  ——————————
  1600    (24•34•54) 

Step  2  :

                 1    
 Simplify   ——————————
            (24•34•54) 

Equation at the end of step  2  :

    1          1    
  ———— -  ——————————
  1600    (24•34•54) 

Step  3  :

              1 
 Simplify   ————
            1600

Equation at the end of step  3  :

    1          1    
  ———— -  ——————————
  1600    (24•34•54) 

Step  4  :

 4.1      Finding a Common Denominator   The left  1600 
 The right  24 • 34 • 54 

 The product of any two denominators can be used
 as a common denominator. 

 Said product is not necessarily the least common 
 denominator.

 As a matter of fact, whenever the two denominators
have a common factor, their product will be bigger
 than the least common denominator. 

Anyway, the product is a fine common denominator and
 can perfectly be used for 
 calculating multipliers, as well as for generating
 equivalent fractions. 

 1600 • 24 • 34 • 54   will be used as a common denominator.

Calculating Multipliers :

 4.2    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 = 2⁴ • 3⁴ • 5⁴

   Right_M = L.C.M / R_Deno = 1600

Making Equivalent Fractions :

 4.3      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.                      (24•34•54)   
   ——————————————————  =               —————————————————
             Common denominator        1600 • (24•34•54)

   R. Mult. • R. Num.                         1600      
   ——————————————————  =               —————————————————
             Common denominator        1600 • (24•34•54) 

Adding fractions that have a common denominator :

 4.4       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:

 (24•34•54) - (1600)      24•34•54 - 1600 
 ———————————————————  =  —————————————————
  1600 • (24•34•54)      1600 • (24•34•54)

Final result :

       1 - 1600    
  —————————————————
  1600 • (24•34•54)