Evaluate: (2-(3)/(4))/(-(3)/(4))-((5)/(6)-(2)/(3))/2 * (2)/(3)

Expression: $\left( 2-\frac{ 3 }{ 4 } \right)\div\left( -\frac{ 3 }{ 4 } \right)-\left( \frac{ 5 }{ 6 }-\frac{ 2 }{ 3 } \right)\div2 \times \frac{ 2 }{ 3 }$

Calculate the difference

$\frac{ 5 }{ 4 }\div\left( -\frac{ 3 }{ 4 } \right)-\left( \frac{ 5 }{ 6 }-\frac{ 2 }{ 3 } \right)\div2 \times \frac{ 2 }{ 3 }$

Subtract the fractions

$\frac{ 5 }{ 4 }\div\left( -\frac{ 3 }{ 4 } \right)-\frac{ 1 }{ 6 }\div2 \times \frac{ 2 }{ 3 }$

Dividing is equivalent to multiplying by the reciprocal

$\frac{ 5 }{ 4 }\div\left( -\frac{ 3 }{ 4 } \right)-\frac{ 1 }{ 6 } \times \frac{ 1 }{ 2 } \times \frac{ 2 }{ 3 }$

Dividing a positive and a negative equals a negative: $\left( + \right)\div\left( - \right)=\left( - \right)$

$-\frac{ 5 }{ 4 }\div\frac{ 3 }{ 4 }-\frac{ 1 }{ 6 } \times \frac{ 1 }{ 2 } \times \frac{ 2 }{ 3 }$

To divide by a fraction, multiply by the reciprocal of that fraction

$-\frac{ 5 }{ 4 } \times \frac{ 4 }{ 3 }-\frac{ 1 }{ 6 } \times \frac{ 1 }{ 2 } \times \frac{ 2 }{ 3 }$

Cancel out the greatest common factor $2$

$-\frac{ 5 }{ 4 } \times \frac{ 4 }{ 3 }-\frac{ 1 }{ 6 } \times \frac{ 1 }{ 3 }$

Cancel out the greatest common factor $4$

$-5 \times \frac{ 1 }{ 3 }-\frac{ 1 }{ 6 } \times \frac{ 1 }{ 3 }$

Multiply the fractions

$-5 \times \frac{ 1 }{ 3 }-\frac{ 1 }{ 18 }$

Calculate the product

$-\frac{ 5 }{ 3 }-\frac{ 1 }{ 18 }$

Calculate the difference

\begin{align*}&-\frac{ 31 }{ 18 } \\&\begin{array} { l }-1 \frac{ 13 }{ 18 },& -1.7\overset{ \cdot }{ 2 } \end{array}\end{align*}

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