Solve (xy/6):[2/3div(x/6-3x/4)]

$$( \frac { x y } { 6 } ) : [ \frac { 2 } { 3 } \div ( \frac { x } { 6 } - \frac { 3 x } { 4 } ) ]$$

Evaluate

$$-\frac{7yx^{2}}{48}$$

Short Solution Steps

Divide \frac{xy}{6} by \frac{\frac{2}{3}}{\frac{x}{6}-\frac{3x}{4}} by multiplying \frac{xy}{6} by the reciprocal of \frac{\frac{2}{3}}{\frac{x}{6}-\frac{3x}{4}}.

$$\frac{xy\left(\frac{x}{6}-\frac{3x}{4}\right)}{6\times \frac{2}{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6 and 4 is 12. Multiply \frac{x}{6} times \frac{2}{2}. Multiply \frac{3x}{4} times \frac{3}{3}.

$$\frac{xy\left(\frac{2x}{12}-\frac{3\times 3x}{12}\right)}{6\times \frac{2}{3}}$$

Since \frac{2x}{12} and \frac{3\times 3x}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{xy\times \frac{2x-3\times 3x}{12}}{6\times \frac{2}{3}}$$

Do the multiplications in 2x-3\times 3x.

$$\frac{xy\times \frac{2x-9x}{12}}{6\times \frac{2}{3}}$$

Combine like terms in 2x-9x.

$$\frac{xy\times \frac{-7x}{12}}{6\times \frac{2}{3}}$$

Express x\times \frac{-7x}{12} as a single fraction.

$$\frac{\frac{x\left(-7\right)x}{12}y}{6\times \frac{2}{3}}$$

Express 6\times \frac{2}{3} as a single fraction.

$$\frac{\frac{x\left(-7\right)x}{12}y}{\frac{6\times 2}{3}}$$

Multiply 6 and 2 to get 12.

$$\frac{\frac{x\left(-7\right)x}{12}y}{\frac{12}{3}}$$

Divide 12 by 3 to get 4.

$$\frac{\frac{x\left(-7\right)x}{12}y}{4}$$

Multiply x and x to get x^{2}.

$$\frac{\frac{x^{2}\left(-7\right)}{12}y}{4}$$

Express \frac{x^{2}\left(-7\right)}{12}y as a single fraction.

$$\frac{\frac{x^{2}\left(-7\right)y}{12}}{4}$$

Express \frac{\frac{x^{2}\left(-7\right)y}{12}}{4} as a single fraction.

$$\frac{x^{2}\left(-7\right)y}{12\times 4}$$

Multiply 12 and 4 to get 48.

$$\frac{x^{2}\left(-7\right)y}{48}$$

Differentiate w.r.t. x

$$-\frac{7xy}{24}$$