Questions: Simplify (1/(30 x) + 1/15) / (1/3 + 1/(6 x)) 1/5 1/(6 x) 5 1/(5 x) I don't know.

Solution Steps

We start with the numerator of the expression:

\[ \frac{1}{30 x} + \frac{1}{15} \]

To combine these fractions, we find a common denominator, which is \(30x\):

\[ \frac{1}{30 x} + \frac{2x}{30 x} = \frac{1 + 2x}{30x} \]

Next, we simplify the denominator:

\[ \frac{1}{3} + \frac{1}{6 x} \]

The common denominator here is \(6x\):

\[ \frac{2x}{6x} + \frac{1}{6x} = \frac{2x + 1}{6x} \]

Now we can rewrite the original expression as:

\[ \frac{\frac{1 + 2x}{30x}}{\frac{2x + 1}{6x}} \]

This can be simplified by multiplying by the reciprocal of the denominator:

\[ \frac{1 + 2x}{30x} \cdot \frac{6x}{2x + 1} \]

Now we simplify the expression:

Factoring gives us:

• Numerator: \(6x(2x + 1)\)
• Denominator: \(30x(2x + 1)\)

Thus, the expression simplifies to:

\[ \frac{6x(2x + 1)}{30x(2x + 1)} = \frac{6}{30} = \frac{1}{5} \]

Final Answer