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Divide (using Complex Fractions ((((5)/(6)x+3)/((1)/(6)-y)))/((3x)/(4y))

Divide (using Complex Fractions\newline(56x+316−y)3x4y \frac{\left(\frac{\frac{5}{6} x+3}{\frac{1}{6}-y}\right)}{\frac{3 x}{4 y}}

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Q. Divide (using Complex Fractions\newline(56x+316−y)3x4y \frac{\left(\frac{\frac{5}{6} x+3}{\frac{1}{6}-y}\right)}{\frac{3 x}{4 y}}
  1. Simplify Division by Reciprocal: Simplify the division of two fractions by multiplying by the reciprocal of the divisor.\newline(56x+316−y)÷(3x4y)=(56x+316−y)⋅(4y3x) \left(\frac{\frac{5}{6}x + 3}{\frac{1}{6} - y}\right) \div \left(\frac{3x}{4y}\right) = \left(\frac{\frac{5}{6}x + 3}{\frac{1}{6} - y}\right) \cdot \left(\frac{4y}{3x}\right)
  2. Distribute Multiplication: Distribute the multiplication across the numerator.\newline(56x+3)⋅(4y3x)=(56x⋅4y3x)+(3⋅4y3x) \left(\frac{5}{6}x + 3\right) \cdot \left(\frac{4y}{3x}\right) = \left(\frac{5}{6}x \cdot \frac{4y}{3x}\right) + \left(3 \cdot \frac{4y}{3x}\right)
  3. Simplify Numerator Terms: Simplify each term in the numerator.\newline56x⋅4y3x=20y18=10y9 \frac{5}{6}x \cdot \frac{4y}{3x} = \frac{20y}{18} = \frac{10y}{9} \newline3⋅4y3x=12y3x=4yx 3 \cdot \frac{4y}{3x} = \frac{12y}{3x} = \frac{4y}{x}
  4. Combine Simplified Terms: Combine the simplified terms over the common denominator.\newline10y9+4yx16−y \frac{\frac{10y}{9} + \frac{4y}{x}}{\frac{1}{6} - y}
  5. Find Common Denominator: Find a common denominator for the terms in the numerator.\newline10yx9x+36y9x16−y=46y9x16−y \frac{\frac{10yx}{9x} + \frac{36y}{9x}}{\frac{1}{6} - y} = \frac{\frac{46y}{9x}}{\frac{1}{6} - y}
  6. Multiply by Reciprocal: Multiply by the reciprocal of the denominator.\newline46y9x16−y⋅116−y−1=46y9x⋅(61−6y) \frac{\frac{46y}{9x}}{\frac{1}{6} - y} \cdot \frac{1}{\frac{1}{6} - y}^{-1} = \frac{46y}{9x} \cdot \left(\frac{6}{1 - 6y}\right)
  7. Final Expression Simplification: Simplify the final expression.\newline46y⋅69x⋅(1−6y)=276y9x(1−6y) \frac{46y \cdot 6}{9x \cdot (1 - 6y)} = \frac{276y}{9x(1 - 6y)}

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