To evaluate \((g \div h) \div k\) with the given values \(g = 5 \frac{3}{4}\), \(h = -\frac{2}{3}\), and \(k = -0.5\), we'll proceed with the calculations step by step.
Step 1: Convert \(g\) to an improper fraction
First, convert \(g = 5 \frac{3}{4}\) to an improper fraction:
\[
5 \frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{20 + 3}{4} = \frac{23}{4}
\]
Step 2: Calculate \(g \div h\)
Now, we compute \(g \div h\):
\[
\frac{23}{4} \div \left(-\frac{2}{3}\right) = \frac{23}{4} \times \left(-\frac{3}{2}\right)
\]
Now perform the multiplication:
\[
\frac{23 \times (-3)}{4 \times 2} = \frac{-69}{8}
\]
Step 3: Calculate \((g \div h) \div k\)
Next, we will divide our result by \(k = -0.5\). First, convert \(-0.5\) to a fraction:
\[
-0.5 = -\frac{1}{2}
\]
Now, perform the calculation:
\[
\left(-\frac{69}{8}\right) \div \left(-\frac{1}{2}\right) = \left(-\frac{69}{8}\right) \times \left(-2\right)
\]
This results in:
\[
\frac{69 \times 2}{8} = \frac{138}{8}
\]
Step 4: Simplify \(\frac{138}{8}\)
Now, simplify \(\frac{138}{8}\):
The GCD of 138 and 8 is 2, so we divide both the numerator and denominator by 2:
\[
\frac{138 \div 2}{8 \div 2} = \frac{69}{4}
\]
Step 5: Convert to a mixed number
To express \(\frac{69}{4}\) as a mixed number:
- Divide \(69\) by \(4\):
- \(69 \div 4 = 17\) with a remainder of \(1\).
- Write as a mixed number:
\[
17 \frac{1}{4}
\]
Final Answer
Thus, the value of \((g \div h) \div k\) is:
\[
\mathbf{17 \frac{1}{4}}
\]
