Understanding the Problem
To solve the expression \( 6 \times \frac{3}{4} - \frac{31}{3} \), we will follow the order of operations, which is multiplication and division before addition and subtraction.

Step 1: Multiplication
First, we will multiply \( 6 \) by \( \frac{3}{4} \):
- Multiply the whole number by the numerator:
\( 6 \times 3 = 18 \)
- Divide by the denominator:
\( \frac{18}{4} = 4.5 \) or \( \frac{9}{2} \)
So, \( 6 \times \frac{3}{4} = \frac{9}{2} \).

Step 2: Finding a Common Denominator
Next, we need to subtract \( \frac{31}{3} \) from \( \frac{9}{2} \). To do this, we need a common denominator. The least common multiple of \( 2 \) and \( 3 \) is \( 6 \).
- Convert \( \frac{9}{2} \) to a denominator of \( 6 \):
\( \frac{9}{2} = \frac{9 \times 3}{2 \times 3} = \frac{27}{6} \)
- Convert \( \frac{31}{3} \) to a denominator of \( 6 \):
\( \frac{31}{3} = \frac{31 \times 2}{3 \times 2} = \frac{62}{6} \)

Step 3: Performing the Subtraction
Now we can subtract the two fractions:
- \( \frac{27}{6} - \frac{62}{6} = \frac{27 - 62}{6} = \frac{-35}{6} \)

Final Answer
The final result of \( 6 \times \frac{3}{4} - \frac{31}{3} \) is:
- \( \frac{-35}{6} \)
This means the answer is negative, indicating that the subtraction resulted in a value less than zero.