Mathematics, 15.07.2019 10:00, leethiel1

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

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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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The number is: " 12 ".
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