Question 77989
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what rational expression divided by 3a over 4 equals 1 over three
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Let the rational expression be R

Then R divided by 3a over 4 equals 1 over three

     {{{R}}} ÷ {{{(3a)/4}}} = {{{1/3}}}
          
Put the R over 1

     {{{R/1}}} ÷ {{{(3a)/4}}} = {{{1/3}}}

Invert the {{{(3a)/4}}} and change the ÷ to ×

     {{{R/1}}} × {{{4/(3a)}}} = {{{1/3}}}

Multiply numerators and denominators on the left:

     {{{(4R)/(3a)}}} = {{{1/3}}}

Multiply both sides by the LCD, {{{((3a)/1)}}}

     {{{((3a)/1)}}}{{{(4R)/(3a)}}} = {{{((3a)/1)}}}{{{1/3}}}

The {{{3a}}}'s cancel on the left, leaving just {{{4R}}}
The {{{3}}}'s cancel on the right, leaving just {{{a}}}.

          {{{4R}}} = {{{a}}}

Divide both sides by 4:

          {{{R}}}={{{a/4}}}

That's the rational expression.

Let's check and see:

     {{{R}}} ÷ {{{(3a)/4}}} = {{{1/3}}}

   {{{a/4}}} ÷ {{{(3a)/4}}} = {{{1/3}}}

Invert the second fraction on the left and change the ÷ to ×:

   {{{a/4}}} × {{{4/(3a)}}} = {{{1/3}}}

The {{{4}}}'s  and the {{{a}}}'s cancel on the left, leaving 
just {{{1/3}}}

                  {{{1/3}}} = {{{1/3}}}

So it checks.

Edwin</pre>