Simplify (1-x/5)/((x-8)/15+1/x)

$\frac{1 - \frac{x}{5}}{\frac{x - 8}{15} + \frac{1}{x}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $15 x$ .

Multiply $\frac{1 - \frac{x}{5}}{\frac{x - 8}{15} + \frac{1}{x}}$ by $\frac{15 x}{15 x}$ .

$\frac{15 x}{15 x} \cdot \frac{1 - \frac{x}{5}}{\frac{x - 8}{15} + \frac{1}{x}}$

Combine.

$\frac{15 x (1 - \frac{x}{5})}{15 x (\frac{x - 8}{15} + \frac{1}{x})}$

Step 2

Apply the distributive property.

$\frac{15 x \cdot 1 + 15 x (- \frac{x}{5})}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Step 3

Simplify by cancelling.

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{x}{5}$ into the numerator.

$\frac{15 x \cdot 1 + 15 x \frac{- x}{5}}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Factor $5$ out of $15 x$ .

$\frac{15 x \cdot 1 + 5 (3 x) \frac{- x}{5}}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Cancel the common factor.

$\frac{15 x \cdot 1 + 5 (3 x) \frac{- x}{5}}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Rewrite the expression.

$\frac{15 x \cdot 1 + 3 x (- x)}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Multiply $- 1$ by $3$ .

$\frac{15 x \cdot 1 - 3 x \cdot x}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Raise $x$ to the power of $1$ .

$\frac{15 x \cdot 1 - 3 (x^{1} x)}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Raise $x$ to the power of $1$ .

$\frac{15 x \cdot 1 - 3 (x^{1} x^{1})}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{15 x \cdot 1 - 3 x^{1 + 1}}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Add $1$ and $1$ .

$\frac{15 x \cdot 1 - 3 x^{2}}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Cancel the common factor of $15$ .

Factor $15$ out of $15 x$ .

$\frac{15 x \cdot 1 - 3 x^{2}}{15 (x) \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Cancel the common factor.

$\frac{15 x \cdot 1 - 3 x^{2}}{15 x \frac{x - 8}{15} + 15 x \frac{1}{x}}$

Rewrite the expression.

$\frac{15 x \cdot 1 - 3 x^{2}}{x (x - 8) + 15 x \frac{1}{x}}$

Cancel the common factor of $x$ .

Factor $x$ out of $15 x$ .

$\frac{15 x \cdot 1 - 3 x^{2}}{x (x - 8) + x \cdot 15 \frac{1}{x}}$

Cancel the common factor.

$\frac{15 x \cdot 1 - 3 x^{2}}{x (x - 8) + x \cdot 15 \frac{1}{x}}$

Rewrite the expression.

$\frac{15 x \cdot 1 - 3 x^{2}}{x (x - 8) + 15}$

Step 4

Simplify the numerator.

Factor $3 x$ out of $15 x \cdot 1 - 3 x^{2}$ .

Factor $3 x$ out of $15 x \cdot 1$ .

$\frac{3 x (5 \cdot 1) - 3 x^{2}}{x (x - 8) + 15}$

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

$\frac{3 x (5 \cdot 1) + 3 x (- x)}{x (x - 8) + 15}$

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

$\frac{3 x (5 \cdot 1 - x)}{x (x - 8) + 15}$

Multiply $5$ by $1$ .

$\frac{3 x (5 - x)}{x (x - 8) + 15}$

Step 5

Simplify the denominator.

Apply the distributive property.

$\frac{3 x (5 - x)}{x \cdot x + x \cdot - 8 + 15}$

Multiply $x$ by $x$ .

$\frac{3 x (5 - x)}{x^{2} + x \cdot - 8 + 15}$

Move $- 8$ to the left of $x$ .

$\frac{3 x (5 - x)}{x^{2} - 8 \cdot x + 15}$

Factor $x^{2} - 8 x + 15$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $15$ and whose sum is $- 8$ .

$- 5, - 3$

Write the factored form using these integers.

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

Step 6

Reduce the expression by cancelling the common factors.

Cancel the common factor of $5 - x$ and $x - 5$ .

Rewrite $5$ as $- 1 (- 5)$ .

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

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

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

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

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

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the expression.

Multiply $- 1$ by $3$ .

$\frac{- 3 x}{x - 3}$

Move the negative in front of the fraction.

$- \frac{3 x}{x - 3}$

Simplify (1-x/5)/((x-8)/15+1/x)

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