Simplify (1/5-1/(15x))/(1/x-1/(3x^2))

$\frac{\frac{1}{5} - \frac{1}{15 x}}{\frac{1}{x} - \frac{1}{3 x^{2}}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $15 x^{2}$ .

Multiply $\frac{\frac{1}{5} - \frac{1}{15 x}}{\frac{1}{x} - \frac{1}{3 x^{2}}}$ by $\frac{15 x^{2}}{15 x^{2}}$ .

$\frac{15 x^{2}}{15 x^{2}} \cdot \frac{\frac{1}{5} - \frac{1}{15 x}}{\frac{1}{x} - \frac{1}{3 x^{2}}}$

Combine.

$\frac{15 x^{2} (\frac{1}{5} - \frac{1}{15 x})}{15 x^{2} (\frac{1}{x} - \frac{1}{3 x^{2}})}$

Step 2

Apply the distributive property.

$\frac{15 x^{2} \frac{1}{5} + 15 x^{2} (- \frac{1}{15 x})}{15 x^{2} \frac{1}{x} + 15 x^{2} (- \frac{1}{3 x^{2}})}$

Step 3

Simplify by cancelling.

Cancel the common factor of $5$ .

Factor $5$ out of $15 x^{2}$ .

$\frac{5 (3 x^{2}) \frac{1}{5} + 15 x^{2} (- \frac{1}{15 x})}{15 x^{2} \frac{1}{x} + 15 x^{2} (- \frac{1}{3 x^{2}})}$

Cancel the common factor.

$\frac{5 (3 x^{2}) \frac{1}{5} + 15 x^{2} (- \frac{1}{15 x})}{15 x^{2} \frac{1}{x} + 15 x^{2} (- \frac{1}{3 x^{2}})}$

Rewrite the expression.

$\frac{3 x^{2} + 15 x^{2} (- \frac{1}{15 x})}{15 x^{2} \frac{1}{x} + 15 x^{2} (- \frac{1}{3 x^{2}})}$

Cancel the common factor of $15 x$ .

Move the leading negative in $- \frac{1}{15 x}$ into the numerator.

$\frac{3 x^{2} + 15 x^{2} \frac{- 1}{15 x}}{15 x^{2} \frac{1}{x} + 15 x^{2} (- \frac{1}{3 x^{2}})}$

Factor $15 x$ out of $15 x^{2}$ .

$\frac{3 x^{2} + 15 x (x) \frac{- 1}{15 x}}{15 x^{2} \frac{1}{x} + 15 x^{2} (- \frac{1}{3 x^{2}})}$

Cancel the common factor.

$\frac{3 x^{2} + 15 x \cdot x \frac{- 1}{15 x}}{15 x^{2} \frac{1}{x} + 15 x^{2} (- \frac{1}{3 x^{2}})}$

Rewrite the expression.

$\frac{3 x^{2} + x \cdot - 1}{15 x^{2} \frac{1}{x} + 15 x^{2} (- \frac{1}{3 x^{2}})}$

Cancel the common factor of $x$ .

Factor $x$ out of $15 x^{2}$ .

$\frac{3 x^{2} + x \cdot - 1}{x (15 x) \frac{1}{x} + 15 x^{2} (- \frac{1}{3 x^{2}})}$

Cancel the common factor.

$\frac{3 x^{2} + x \cdot - 1}{x (15 x) \frac{1}{x} + 15 x^{2} (- \frac{1}{3 x^{2}})}$

Rewrite the expression.

$\frac{3 x^{2} + x \cdot - 1}{15 x + 15 x^{2} (- \frac{1}{3 x^{2}})}$

Cancel the common factor of $3 x^{2}$ .

Move the leading negative in $- \frac{1}{3 x^{2}}$ into the numerator.

$\frac{3 x^{2} + x \cdot - 1}{15 x + 15 x^{2} \frac{- 1}{3 x^{2}}}$

Factor $3 x^{2}$ out of $15 x^{2}$ .

$\frac{3 x^{2} + x \cdot - 1}{15 x + 3 x^{2} (5) \frac{- 1}{3 x^{2}}}$

Cancel the common factor.

$\frac{3 x^{2} + x \cdot - 1}{15 x + 3 x^{2} \cdot 5 \frac{- 1}{3 x^{2}}}$

Rewrite the expression.

$\frac{3 x^{2} + x \cdot - 1}{15 x + 5 \cdot - 1}$

Multiply $5$ by $- 1$ .

$\frac{3 x^{2} + x \cdot - 1}{15 x - 5}$

Step 4

Simplify terms.

Factor $x$ out of $3 x^{2} + x \cdot - 1$ .

Factor $x$ out of $3 x^{2}$ .

$\frac{x (3 x) + x \cdot - 1}{15 x - 5}$

Factor $x$ out of $x \cdot - 1$ .

$\frac{x (3 x) + x (- 1)}{15 x - 5}$

Factor $x$ out of $x (3 x) + x (- 1)$ .

$\frac{x (3 x - 1)}{15 x - 5}$

Factor $5$ out of $15 x - 5$ .

Factor $5$ out of $15 x$ .

$\frac{x (3 x - 1)}{5 (3 x) - 5}$

Factor $5$ out of $- 5$ .

$\frac{x (3 x - 1)}{5 (3 x) + 5 (- 1)}$

Factor $5$ out of $5 (3 x) + 5 (- 1)$ .

$\frac{x (3 x - 1)}{5 (3 x - 1)}$

Cancel the common factor of $3 x - 1$ .

Cancel the common factor.

$\frac{x (3 x - 1)}{5 (3 x - 1)}$

Rewrite the expression.

$\frac{x}{5}$

Simplify (1/5-1/(15x))/(1/x-1/(3x^2))

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