The problem is to evaluate the expression $\frac{5\frac{2}{7} + \frac{1}{14} \times \frac{2}{3} - 1\frac{1}{4}}{\frac{3}{8} \div \frac{1}{16}}$

1. Problem Description

The problem is to evaluate the expression

$\frac{5\frac{2}{7} + \frac{1}{14} \times \frac{2}{3} - 1\frac{1}{4}}{\frac{3}{8} \div \frac{1}{16}}$

2. Solution Steps

First, we evaluate the expression in the numerator.

We can rewrite the mixed numbers as improper fractions:

$5\frac{2}{7} = \frac{5 \times 7 + 2}{7} = \frac{35 + 2}{7} = \frac{37}{7}$

$1\frac{1}{4} = \frac{1 \times 4 + 1}{4} = \frac{4 + 1}{4} = \frac{5}{4}$

The expression in the numerator is

$\frac{37}{7} + \frac{1}{14} \times \frac{2}{3} - \frac{5}{4}$

$= \frac{37}{7} + \frac{2}{42} - \frac{5}{4}$

$= \frac{37}{7} + \frac{1}{21} - \frac{5}{4}$

We need to find the least common multiple of 7, 21, and 4, which is

8

4. $\frac{37}{7} \times \frac{12}{12} + \frac{1}{21} \times \frac{4}{4} - \frac{5}{4} \times \frac{21}{21}$

$= \frac{444}{84} + \frac{4}{84} - \frac{105}{84}$

$= \frac{444 + 4 - 105}{84} = \frac{343}{84}$

Now, we evaluate the expression in the denominator.

$\frac{3}{8} \div \frac{1}{16} = \frac{3}{8} \times \frac{16}{1} = \frac{3 \times 16}{8 \times 1} = \frac{48}{8} = 6$

The entire expression is

$\frac{\frac{343}{84}}{6} = \frac{343}{84} \div 6 = \frac{343}{84} \times \frac{1}{6} = \frac{343}{504}$

We check if the fraction can be simplified.

$343 = 7^3$

$504 = 2^3 \times 3^2 \times 7$

$\frac{343}{504} = \frac{7^3}{2^3 \times 3^2 \times 7} = \frac{7^2}{2^3 \times 3^2} = \frac{49}{8 \times 9} = \frac{49}{72}$

3. Final Answer

$\frac{49}{72}$