We need to evaluate the expression $1\frac{1}{5} \times [(-1\frac{1}{4}) + (-3\frac{3}{4})]$.

1. Problem Description

We need to evaluate the expression $1\frac{1}{5} \times [(-1\frac{1}{4}) + (-3\frac{3}{4})]$.

2. Solution Steps

First, we convert the mixed numbers to improper fractions:

$1\frac{1}{5} = \frac{1 \times 5 + 1}{5} = \frac{6}{5}$

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

$-3\frac{3}{4} = -\frac{3 \times 4 + 3}{4} = -\frac{15}{4}$

Now we substitute these values into the expression:

$\frac{6}{5} \times [(-\frac{5}{4}) + (-\frac{15}{4})]$

We simplify the expression inside the brackets:

$-\frac{5}{4} + (-\frac{15}{4}) = -\frac{5}{4} - \frac{15}{4} = -\frac{5+15}{4} = -\frac{20}{4} = -5$

Now we substitute this back into the expression:

$\frac{6}{5} \times (-5) = \frac{6}{5} \times \frac{-5}{1} = \frac{6 \times (-5)}{5 \times 1} = \frac{-30}{5} = -6$

3. Final Answer

-6