The problem is to evaluate the expression $3\frac{3}{4} \times (-1\frac{1}{5}) + 5\frac{1}{2}$.

1. Problem Description

The problem is to evaluate the expression $3\frac{3}{4} \times (-1\frac{1}{5}) + 5\frac{1}{2}$.

2. Solution Steps

First, convert the mixed numbers to improper fractions.

$3\frac{3}{4} = \frac{3\times4 + 3}{4} = \frac{12+3}{4} = \frac{15}{4}$

$-1\frac{1}{5} = -\frac{1\times5 + 1}{5} = -\frac{5+1}{5} = -\frac{6}{5}$

$5\frac{1}{2} = \frac{5\times2 + 1}{2} = \frac{10+1}{2} = \frac{11}{2}$

Now, substitute the improper fractions into the expression:

$\frac{15}{4} \times (-\frac{6}{5}) + \frac{11}{2}$

Multiply the first two fractions:

$\frac{15}{4} \times (-\frac{6}{5}) = -\frac{15\times6}{4\times5} = -\frac{90}{20} = -\frac{9}{2}$

Now the expression is:

$-\frac{9}{2} + \frac{11}{2}$

Add the fractions:

$-\frac{9}{2} + \frac{11}{2} = \frac{-9+11}{2} = \frac{2}{2} = 1$

3. Final Answer

1