To evaluate the expression \( 0.75 \div \frac{11}{8} \cdot \frac{8}{3} \), we will follow the order of operations.
First, let's convert \( 0.75 \) to a fraction. We know that: \[ 0.75 = \frac{75}{100} = \frac{3}{4} \]
Now, we can rewrite the expression as: \[ \frac{3}{4} \div \frac{11}{8} \cdot \frac{8}{3} \]
Next, we can handle the division first: \[ \frac{3}{4} \div \frac{11}{8} = \frac{3}{4} \cdot \frac{8}{11} = \frac{3 \times 8}{4 \times 11} = \frac{24}{44} \]
Now let's simplify \(\frac{24}{44}\): \[ \frac{24}{44} = \frac{12}{22} = \frac{6}{11} \quad (\text{dividing both numerator and denominator by 2}) \]
Next, we multiply this result by \(\frac{8}{3}\): \[ \frac{6}{11} \cdot \frac{8}{3} = \frac{6 \times 8}{11 \times 3} = \frac{48}{33} \]
Now we simplify \(\frac{48}{33}\). The greatest common divisor of 48 and 33 is 3, thus: \[ \frac{48}{33} = \frac{16}{11} \quad (\text{dividing both numerator and denominator by 3}) \]
Therefore, the final answer in simplest form is: \[ \boxed{\frac{16}{11}} \]