If \(\rm \frac{15}{8}+\left[\left\{\frac{4}{5}\ of\ \frac{5}{12}\div \left(1\frac{1}{15}\right)\right\}\times 1\frac{7}{12}\right]\) is expressed in the form of \(\rm \frac{a}{b}\) where a and b are integers and relatively prime to each other, the value 4b — a is: 

Given:

\(\frac{15}{8}+\left[\left\{\frac{4}{5}\ of\ \frac{5}{12}\div \left(1\frac{1}{15}\right)\right\}\times 1\frac{7}{12}\right]\)

Formula used:

Mixed fraction: \(\rm a\frac{b}{c} = \frac{ac+b}{c}\)

Calculations:

Step 1: Simplify \(\rm 1\frac{1}{15} = \frac{16}{15}\)

Step 2: \(\frac{4}{5} \ of \ \frac{5}{12} \div \frac{16}{15} = \frac{4}{5} \times \frac{5}{12} \times \frac{15}{16} = \frac{4 \times 5 \times 15}{5 \times 12 \times 16} = \frac{300}{960} = \frac{5}{16}\)

Step 3: \(\frac{5}{16} \times 1\frac{7}{12} = \frac{5}{16} \times \frac{19}{12} = \frac{95}{192}\)

Step 4: Add \(\frac{15}{8} + \frac{95}{192} = \frac{360}{192} + \frac{95}{192} = \frac{455}{192}\)

Step 5: \(\frac{455}{192}\) is in its simplest form, where a = 455 and b = 192.

Step 6: Calculate \(\rm 4b - a = 4 \times 192 - 455 = 768 - 455 = 313\)

∴ The value of 4b - a is 313.