Evaluating the Mathematical Expression Step-by-Step

The problem requires calculating the value of the expression $\frac{81}{7} \times \frac{21}{3} - 4\frac{2}{3} \div \frac{7}{9} \times \frac{121}{6}$. We follow the order of operations (BODMAS/PEMDAS): Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number $4\frac{2}{3}$ into an improper fraction:

$4\frac{2}{3} = \frac{(4 \times 3) + 2}{3} = \frac{12 + 2}{3} = \frac{14}{3}$

Step 2: Perform Division

Next, perform the division operation:

$\frac{14}{3} \div \frac{7}{9} = \frac{14}{3} \times \frac{9}{7}$

Simplify the terms:

$\frac{14}{7} \times \frac{9}{3} = 2 \times 3 = 6$

Step 3: Perform Multiplications

Now, perform the multiplication operations from left to right.

First multiplication:

$\frac{81}{7} \times \frac{21}{3} = \frac{81}{3} \times \frac{21}{7} = 27 \times 3 = 81$

Second multiplication (using the result from Step 2):

$6 \times \frac{121}{6}$

Simplify the terms:

$6 \times \frac{121}{6} = 121$

Step 4: Perform Subtraction

Finally, perform the subtraction using the results from the multiplication steps:

$81 - 121 = -40$

Final Expression Value

The value of the expression $\frac{81}{7} \times \frac{21}{3} - 4\frac{2}{3} \div \frac{7}{9} \times \frac{121}{6}$ is -40.