If $$\frac{1}{5\frac{1}{3}} + \frac{1}{3\frac{3}{7}} + \frac{1}{4\frac{4}{7}} + \frac{1}{?} = \frac{77}{96}$$, find what should be filled in the place marked.

Converting each mixed number to an improper fraction and taking reciprocals gives $$\frac{3}{16}+\frac{7}{24}+\frac{7}{32}+\frac{1}{x} = \frac{77}{96}$$. Finding a common denominator of 96, the first three terms sum to $$\frac{18+28+21}{96} = \frac{67}{96}$$, so $$\frac{1}{x} = \frac{77}{96}-\frac{67}{96} = \frac{10}{96}$$. Solving gives $$x = \frac{96}{10} = 9.6$$.