To evaluate the expression \(6 \div (-4 - -1 \frac{1}{2} + 3 \frac{3}{4})\), we will first simplify the expression inside the parentheses step-by-step.
- Convert the mixed numbers to improper fractions:
- \(-1 \frac{1}{2} = -\frac{3}{2}\)
- \(3 \frac{3}{4} = \frac{15}{4}\)
Now the expression inside the parentheses becomes: \[ -4 - -\frac{3}{2} + \frac{15}{4} \]
- Rewrite \(-4\) as a fraction: \(-4 = -\frac{16}{4}\).
Now we have: \[ -\frac{16}{4} + \frac{3}{2} + \frac{15}{4} \]
-
To add and subtract the fractions, we will need a common denominator. The least common denominator of \(4\) and \(2\) is \(4\). Convert \(\frac{3}{2}\) to have this denominator: \(\frac{3}{2} = \frac{6}{4}\).
-
Now substitute that back into the expression: \[ -\frac{16}{4} + \frac{6}{4} + \frac{15}{4} \]
-
Combine the fractions: \[ (-16 + 6 + 15) / 4 = (-16 + 21) / 4 = 5 / 4 \]
Now we have simplified the expression inside the parentheses to \(\frac{5}{4}\).
-
Now evaluate the initial expression: \[ 6 \div \frac{5}{4} \]
-
Dividing by a fraction is the same as multiplying by its reciprocal: \[ 6 \times \frac{4}{5} = \frac{6 \times 4}{5} = \frac{24}{5} \]
So, the final result is: \[ \frac{24}{5} \]
This can also be expressed as a mixed number: \[ 4 \frac{4}{5} \]
Thus, the answer is \(\frac{24}{5}\) or \(4 \frac{4}{5}\).