Let's simplify the given expression step by step:

\(

\frac{\frac{1}{10} \times \frac{2}{3} + \frac{1}{4}}{\frac{\frac{1}{2}}{\frac{3}{5}} - \frac{1}{4}}

\)

First, perform the multiplications within the numerator:

\(

\frac{\frac{2}{30} + \frac{1}{4}}{\frac{1}{2} \div \frac{3}{5} - \frac{1}{4}}

\)

Now, convert the divisions into multiplications by taking the reciprocal of the divisor:

\(

\frac{\frac{2}{30} + \frac{1}{4}}{\frac{1}{2} \times \frac{5}{3} - \frac{1}{4}}

\)

Simplify the multiplications:

\(

\frac{\frac{1}{15} + \frac{1}{4}}{\frac{5}{6} - \frac{1}{4}}

\)

Find a common denominator for the fractions in the numerator:

\(

\frac{\frac{4}{60} + \frac{15}{60}}{\frac{10}{12} - \frac{3}{12}}

\)

Combine the fractions in the numerator:

\(

\frac{\frac{19}{60}}{\frac{7}{12}}

\)

Now, divide by a fraction by multiplying by its reciprocal:

\(

\frac{19}{60} \times \frac{12}{7} = \frac{19 \times 12}{60 \times 7} = \frac{228}{420} = \frac{19}{35}

\)

Therefore, the value of the given expression is \(\frac{19}{35}\).