$A$ and $B$ are acute angles such that $\sin(A - B) = \frac{1}{4}$ and $\sin A \cos B = \frac{3}{5}$. Without using a calculator, find the value of

1. Apply the compound angle formula for $\sin(A - B)$

$\sin(A - B) = \sin A \cos B - \cos A \sin B$

2. Substitute the given values and solve for $\cos A \sin B$

$\frac{1}{4} = \frac{3}{5} - \cos A \sin B$

$\cos A \sin B = \frac{3}{5} - \frac{1}{4}$

$\cos A \sin B = \frac{12}{20} - \frac{5}{20} = \frac{7}{20}$

1. Find the value of $\cos A \sin B$ from the given information (result of part i)

$\sin(A - B) = \sin A \cos B - \cos A \sin B$
$\frac{1}{4} = \frac{3}{5} - \cos A \sin B$
$\cos A \sin B = \frac{3}{5} - \frac{1}{4} = \frac{7}{20}$

2. Use the addition formula for sine to find $\sin(A + B)$

$\sin(A + B) = \sin A \cos B + \cos A \sin B$
$\sin(A + B) = \frac{3}{5} + \frac{7}{20}$
$\sin(A + B) = \frac{12}{20} + \frac{7}{20} = \frac{19}{20}$

1. Express the given expression in terms of sine and cosine

$2 \tan A \cot B = 2 \left( \frac{\sin A}{\cos A} \right) \left( \frac{\cos B}{\sin B} \right) = 2 \left( \frac{\sin A \cos B}{\cos A \sin B} \right)$

2. Substitute known values to evaluate the expression

From part (i), we found $\cos A \sin B = \frac{7}{20}$. We are given $\sin A \cos B = \frac{3}{5}$.

$2 \left( \frac{\sin A \cos B}{\cos A \sin B} \right) = 2 \left( \frac{\frac{3}{5}}{\frac{7}{20}} \right)$
$= 2 \left( \frac{3}{5} \times \frac{20}{7} \right)$
$= 2 \left( \frac{12}{7} \right) = \frac{24}{7}$