Simplify the following complex fractions

\(\textbf{1)}\) \(\displaystyle\frac{\frac{x}{5}+\frac{1}{3}}{\frac{1}{5}-\frac{1}{6}}\)

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The answer is \(6x+10\)

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\(\,\,\,\,\,\,\displaystyle\frac{\frac{x}{5}+\frac{1}{3}}{\frac{1}{5}-\frac{1}{6}}\)

\(\,\,\,\,\,\,\displaystyle\frac{\frac{3 \cdot x}{3\cdot 5}+\frac{1\cdot5}{3\cdot 5}}{\frac{6\cdot 1}{6\cdot 5}-\frac{1\cdot5}{6\cdot5}}\) \(\,\,\,\,\,\,\displaystyle\frac{\frac{3x}{15}+\frac{5}{15}}{\frac{6}{30}-\frac{5}{30}}\) \(\,\,\,\,\,\,\displaystyle\frac{\frac{3x+5}{15}}{\frac{1}{30}}\) \(\,\,\,\,\,\,\displaystyle\frac{3x+5}{15} \div \frac{1}{30}\) \(\,\,\,\,\,\,\displaystyle\frac{3x+5}{15} \times \frac{30}{1}\) \(\,\,\,\,\,\,\displaystyle\frac{(3x+5)\cdot(30)}{15}\) \(\,\,\,\,\,\,\displaystyle\frac{(3x+5)\cdot(2)}{1}\)

\(\,\,\,\,\,\,\)The answer is \(6x+10\)

 

\(\textbf{2)}\) \(\displaystyle\frac{\frac{4x^3}{2y}}{\frac{x^2}{8y}}\)

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The answer is \(16x\)

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\(\,\,\,\,\,\displaystyle\frac{\frac{4x^3}{2y}}{\frac{x^2}{8y}}\)
\(\,\,\,\,\,\displaystyle\frac{4x^3}{2y}\div\frac{x^2}{8y}\)
\(\,\,\,\,\,\displaystyle\frac{4x^3}{2y}\cdot\frac{8y}{x^2}\)
\(\,\,\,\,\,\displaystyle\frac{32x^3y}{2x^2y}\)
\(\,\,\,\,\,\displaystyle\frac{16x}{1}\)
\(\,\,\,\,\,\,\)The answer is \(16x\)

 

\(\textbf{3)}\) \(\displaystyle\frac{1+\frac{1}{x}}{1-\frac{1}{x^2}}\)

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The answer is \(\displaystyle\frac{x}{x-1}\)

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\(\,\,\,\,\,\displaystyle\frac{1+\frac{1}{x}}{1-\frac{1}{x^2}}\)
\(\,\,\,\,\,\displaystyle\frac{\frac{x+1}{x}}{\frac{x^2-1}{x^2}}\)
\(\,\,\,\,\,\displaystyle\frac{x+1}{x}\div\frac{x^2-1}{x^2}\)
\(\,\,\,\,\,\displaystyle\frac{x+1}{x}\cdot\frac{x^2}{x^2-1}\)
\(\,\,\,\,\,\displaystyle\frac{(x+1)x^2}{x(x^2-1)}\)
\(\,\,\,\,\,\displaystyle\frac{(x+1)x^2}{x(x-1)(x+1)}\)
\(\,\,\,\,\,\displaystyle\frac{x}{x-1}\)
\(\,\,\,\,\,\,\)The answer is \(\displaystyle\frac{x}{x-1}\)

 

\(\textbf{4)}\) \(\displaystyle\frac{\frac{5}{x}-\frac{5}{4x}}{\frac{1}{x}-\frac{5}{8x}}\)

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The answer is \(10\)

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\(\,\,\,\,\,\displaystyle\frac{\frac{5}{x}-\frac{5}{4x}}{\frac{1}{x}-\frac{5}{8x}}\)
\(\,\,\,\,\,\displaystyle\frac{\frac{20}{4x}-\frac{5}{4x}}{\frac{8}{8x}-\frac{5}{8x}}\)
\(\,\,\,\,\,\displaystyle\frac{\frac{15}{4x}}{\frac{3}{8x}}\)
\(\,\,\,\,\,\displaystyle\frac{15}{4x}\div\frac{3}{8x}\)
\(\,\,\,\,\,\displaystyle\frac{15}{4x}\cdot\frac{8x}{3}\)
\(\,\,\,\,\,\displaystyle\frac{120x}{12x}\)
\(\,\,\,\,\,\displaystyle\frac{10}{1}\)
\(\,\,\,\,\,\,\)The answer is \(10\)

 

\(\textbf{5)}\) \(\displaystyle\frac{\frac{8x^4}{3y^3}}{\frac{2x^2}{6y^3}}\)

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The answer is \(8x^2\)

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\(\,\,\,\,\,\displaystyle\frac{\frac{8x^4}{3y^3}}{\frac{2x^2}{6y^3}}\)
\(\,\,\,\,\,\displaystyle\frac{8x^4}{3y^3} \div \frac{2x^2}{6y^3}\)
\(\,\,\,\,\,\displaystyle\frac{8x^4}{3y^3} \cdot \frac{6y^3}{2x^2}\)
\(\,\,\,\,\,\displaystyle\frac{48x^4y^3}{6x^2y^3}\)
\(\,\,\,\,\,\displaystyle\frac{8x^2}{1}\)
\(\,\,\,\,\,\,\)The answer is \(8x^2\)

 

\(\textbf{6)}\) \(\displaystyle\frac{\frac{x+3}{x-2}-1}{\frac{x+2}{x^2-4}}\)

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The answer is \(5\)

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\(\,\,\,\,\,\displaystyle\frac{\frac{x+3}{x-2}-1}{\frac{x+2}{x^2-4}}\)
\(\,\,\,\,\,\displaystyle\frac{\frac{x+3}{x-2}-\frac{x-2}{x-2}}{\frac{x+2}{x^2-4}}\)
\(\,\,\,\,\,\displaystyle\frac{\frac{x+3-(x-2)}{x-2}}{\frac{x+2}{x^2-4}}\)
\(\,\,\,\,\,\displaystyle\frac{\frac{5}{x-2}}{\frac{x+2}{x^2-4}}\)
\(\,\,\,\,\,\displaystyle\frac{5}{x-2}\div\frac{x+2}{x^2-4}\)
\(\,\,\,\,\,\displaystyle\frac{5}{x-2}\cdot\frac{x^2-4}{x+2}\)
\(\,\,\,\,\,\displaystyle\frac{5}{x-2}\cdot\frac{(x-2)(x+2)}{x+2}\)
\(\,\,\,\,\,\displaystyle\frac{5}{1}\)
\(\,\,\,\,\,\,\)The answer is \(5\)

 

\(\textbf{7)}\) \(\displaystyle\frac{\frac{1}{5x}+\frac{1}{6x}}{\frac{1}{5x}-\frac{1}{6x}}\)

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The answer is \(11\)

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\(\,\,\,\,\,\displaystyle\frac{\frac{1}{5x}+\frac{1}{6x}}{\frac{1}{5x}-\frac{1}{6x}}\)
\(\,\,\,\,\,\displaystyle\frac{\frac{6}{30x}+\frac{5}{30x}}{\frac{6}{30x}-\frac{5}{30x}}\)
\(\,\,\,\,\,\displaystyle\frac{\frac{11}{30x}}{\frac{1}{30x}}\)
\(\,\,\,\,\,\displaystyle\frac{11}{30x}\div\frac{1}{30x}\)
\(\,\,\,\,\,\displaystyle\frac{11}{30x}\cdot\frac{30x}{1}\)
\(\,\,\,\,\,\displaystyle\frac{330x}{30x}\)
\(\,\,\,\,\,\displaystyle\frac{11}{1}\)
\(\,\,\,\,\,\,\)The answer is \(11\)

 

\(\textbf{8)}\) \(\displaystyle\frac{x^{-1}-y^{-1}}{x^{-2}-y^{-2}}\)

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The answer is \(\displaystyle\frac{xy}{x+y}\)

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\(\,\,\,\,\,\displaystyle\frac{x^{-1}-y^{-1}}{x^{-2}-y^{-2}}\)
\(\,\,\,\,\,\displaystyle\frac{\frac{1}{x}-\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}}\)
\(\,\,\,\,\,\displaystyle\frac{\frac{y-x}{xy}}{\frac{y^2-x^2}{x^2y^2}}\)
\(\,\,\,\,\,\displaystyle\frac{y-x}{xy}\div\frac{y^2-x^2}{x^2y^2}\)
\(\,\,\,\,\,\displaystyle\frac{y-x}{xy}\cdot\frac{x^2y^2}{y^2-x^2}\)
\(\,\,\,\,\,\displaystyle\frac{(y-x)x^2y^2}{xy(y^2-x^2)}\)
\(\,\,\,\,\,\displaystyle\frac{xy(y-x)}{(y-x)(y+x)}\)
\(\,\,\,\,\,\displaystyle\frac{xy}{x+y}\)
\(\,\,\,\,\,\,\)The answer is \(\displaystyle\frac{xy}{x+y}\)

 

See Related Pages\(\)

\(\bullet\text{ Complex Fraction Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)

\(\bullet\text{ Ratios and Proportions}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{4}{3}=\frac{d-4}{12}…\)

\(\bullet\text{ Rational Expressions- Multiplying and Dividing}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{x^2+3x-4}{(x+4)(x+5)}\cdot \displaystyle\frac{x+5}{x-1}…\)

\(\bullet\text{ Rational Expressions- Adding and Subtracting}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{x-5}{x+3}+\frac{x+2}{x^2+5x+6}…\)

\(\bullet\text{ Direct, Inverse, and Joint Variation}\)
\(\,\,\,\,\,\,\,\,\) \(…\)

\(\bullet\text{ Complex Fractions}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\frac{x}{5}+\frac{1}{3}}{\frac{1}{5}-\frac{1}{6}}…\)

\(\bullet\text{ Partial Fraction Decomposition}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{8x+10}{x^2+2x}=\displaystyle\frac{5}{x} + \frac{3}{x+2}…\)

\(\bullet\text{ Andymath Homepage}\)