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:


`(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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→
`(2)/(3)` x − 6 + 6 = 2 + 6 ;

to get:

→
`(2)/(3)` x = 8 ;
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Multiply each side of the equation by "
`(3)/(2)`" ; to isolate "x" on one side of the equation ; and to solve for "x" ;
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→
`(3)/(2)` *
`(2)/(3)` x = 8 *
`(3)/(2)` ;

→ x = 8 *
`(3)/(2)` ;

=
`(8)/(1)` *
`(3)/(2)` ;

=
`(8*3)/(1*2)` ;

=
`(24)/(2)` ;

= 12 .
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x = 12 .
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Method 2)
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`(2)/(3)` x − 6 = 2 ; Solve for "x" ;

Add "6" to EACH SIDE of the equation;
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→
`(2)/(3)` x − 6 + 6 = 2 + 6 ;

to get:
→
`(2)/(3)` x = 8 ;
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Multiply each side of the equation by "3" ; to get rid of the "fraction" ;
→ 3 *
`(2)/(3)` x = 8 * 3 ;
→
`(3)/(1)` *
`(2)/(3)` x = 8 * 3 ;
→
`(3*2)/(1*3)` x = 8 * 3
→
`(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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`(2)/(3)` x − 6 = 2 ; Solve for "x" ;
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Add "6" to EACH SIDE of the equation;
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→
`(2)/(3)` x − 6 + 6 = 2 + 6 ;

to get:

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

to get: x = 8 / {
`(2)/(3)`} ;

= 8 * (
`(3)/(2)` ;

=
`(8)/(1)` *
`(3)/(2)` ;

=
`(8*3)/(1*2)` ;

=
`(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 * (
`(3)/(2)`) ;

=
`(8)/(1)` *
`(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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`(8)/(1)` *
`(3)/(2)` ;
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as:
`(4)/(1)` *
`(3)/(1)` ;
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which equals:
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→
`(4*3)/(1*1)` ;

=
`(12)/(1)` ;

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