Question

two thirds of a number decreased by six is two. what is the number?

Answer 1

Answer:  The number is:  " 12 ". 

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  Let "x" represent "the unknown number" (for which we wish to solve.

The expression:

$\frac{2}{3}$ x  − 6  =  2  ;   Solve for "x" ;  
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Method 1) 

   Add "6" to EACH SIDE of the equation;
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       →   $\frac{2}{3}$ x  − 6  + 6 =  2 + 6 ;

to get:

      →   $\frac{2}{3}$ x = 8 ;
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Multiply each side of the equation by "$\frac{3}{2}$" ; to isolate "x" on one side of the equation ; and to solve for "x" ;
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     → $\frac{3}{2}$ * $\frac{2}{3}$ x = 8 * $\frac{3}{2}$ ;

       →  x = 8 * $\frac{3}{2}$ ;

                = $\frac{8}{1}$ * $\frac{3}{2}$ ;

                = $\frac{8*3}{1*2}$ ;
       
                = $\frac{24}{2}$ ;
 
                = 12 .
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  x =  12 .
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Method 2)
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$\frac{2}{3}$ x  − 6  =  2  ;   Solve for "x" ; 

   Add "6" to EACH SIDE of the equation;
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       →   $\frac{2}{3}$ x  − 6  + 6 =  2 + 6 ;

to get:
      →   $\frac{2}{3}$ x = 8 ;
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Multiply each side of the equation by "3" ; to get rid of the "fraction" ;
               → 3 * $\frac{2}{3}$ x = 8 * 3  ;
               → $\frac{3}{1}$ * $\frac{2}{3}$ x = 8 * 3 ;
               → $\frac{3*2}{1*3}$  x = 8 * 3 
               → $\frac{6}{3}$ x = 24 ; 

                → 2x = 24 ;

 →  Divide each side of the equation by "2" ; to isolate "x" on one side of the equation; & to solve for "x" : 
 
                    2x / 2 = 24 / 2  ;

                        x = 12 .
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Method 3).
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$\frac{2}{3}$ x  − 6  =  2  ;   Solve for "x" ;  
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Add "6" to EACH SIDE of the equation;
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       →   $\frac{2}{3}$ x  − 6  + 6 =  2 + 6 ;

to get:

      →   $\frac{2}{3}$ x = 8 ;
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Now, divide each side of the equation by " $\frac{2}{3}$ " ;
  to isolate "x" on one side of the equation; & to solve for "x" ;
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{$\frac{2}{3}$ x }  /  {$\frac{2}{3}$}  =  8 / {$\frac{2}{3}$} ;

to get:  x =  8 / {$\frac{2}{3}$} ;

                =  8 * ($\frac{3}{2}$ ;

                =  $\frac{8}{1}$  *  $\frac{3}{2}$ ;

                =  $\frac{8*3}{1*2}$ ;

                =  $\frac{24}{2}$ ;

                = 12 ; 
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                         x = 12 .
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NOTE:  Variant:  (in "Methods 2 & 3") :
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At the point where:
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 =  8 * ($\frac{3}{2}$) ;

  =  $\frac{8}{1}$  *  $\frac{3}{2}$ ;
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  We can cancel out the "2" to a "1" ; and we can cancel out the "8" to a "4" ;
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  {since: "8÷2 = 4" ; and since:  "2÷2 =1" } ;
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and we can rewrite the expression:
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 $\frac{8}{1}$  *  $\frac{3}{2}$ ;
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as:   $\frac{4}{1}$  *  $\frac{3}{1}$ ; 
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which equals:
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→  $\frac{4*3}{1*1}$ ; 

   =   $\frac{12}{1}$ ;

            =  12 .
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         x = 12 . 
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Answer:  The number is:  " 12 ". 
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