Simplify. \[ \frac{\frac{x+7}{35 x}}{\frac{3 x+21}{5}} \]

To simplify the expression \[ \frac{\frac{x+7}{35 x}}{\frac{3 x+21}{5}}, \] we start by rewriting the division of fractions as multiplication by the reciprocal: \[ \frac{x+7}{35 x} \div \frac{3 x + 21}{5} = \frac{x+7}{35 x} \cdot \frac{5}{3 x + 21}. \] Next, we simplify \(3x + 21\): \[ 3x + 21 = 3(x + 7). \] Now we can substitute this back into the expression: \[ \frac{x+7}{35 x} \cdot \frac{5}{3(x + 7)}. \] Notice that \(x + 7\) appears in both the numerator and the denominator: \[ = \frac{x + 7}{35 x} \cdot \frac{5}{3(x + 7)} = \frac{5}{35x \cdot 3} = \frac{5}{105x}. \] Now we simplify \(\frac{5}{105x}\). We can reduce the fraction \(5/105\) by dividing both the numerator and the denominator by 5: \[ \frac{5 \div 5}{105 \div 5} = \frac{1}{21}. \] Thus, the simplified expression is: \[ \frac{1}{21x}. \]