Simplify the denominator

Add the fractions in the denominator: $\frac{3}{4} + \frac{4}{6} = \frac{9}{12} + \frac{8}{12} = \frac{17}{12}$

$\frac{3}{4} + \frac{4}{6} = \frac{17}{12}$

Evaluate the main expression

Now, perform the division: $1 \div \frac{17}{12} = \frac{12}{17}$

$1 \div \frac{17}{12} = \frac{12}{17}$

Perform the final division

Divide the result by $\frac{2}{15}$ by multiplying by its reciprocal $\frac{15}{2}$: $\frac{12}{17} \times \frac{15}{2} = \frac{6 \times 15}{17} = \frac{90}{17}$

$\frac{12}{17} \times \frac{15}{2} = \frac{90}{17}$

Convert to mixed fraction

Divide $90$ by $17$: $90 = 17 \times 5 + 5$. Thus, $\frac{90}{17} = 5 \frac{5}{17}$

$5 \frac{5}{17}$