Evaluating the Mixed Fraction Expression

The problem requires calculating the value of the expression $7\frac{2}{3} \times 5\frac{3}{4} + 9\frac{1}{2} \times 6\frac{3}{4}$. This involves converting mixed fractions to improper fractions, performing multiplications, and then adding the results.

Step 1: Convert Mixed Fractions to Improper Fractions

• $7\frac{2}{3} = \frac{(7 \times 3) + 2}{3} = \frac{23}{3}$
• $5\frac{3}{4} = \frac{(5 \times 4) + 3}{4} = \frac{23}{4}$
• $9\frac{1}{2} = \frac{(9 \times 2) + 1}{2} = \frac{19}{2}$
• $6\frac{3}{4} = \frac{(6 \times 4) + 3}{4} = \frac{27}{4}$

Step 2: Perform the Multiplications

Multiply the pairs of improper fractions:

• First multiplication: $\frac{23}{3} \times \frac{23}{4} = \frac{23 \times 23}{3 \times 4} = \frac{529}{12}$
• Second multiplication: $\frac{19}{2} \times \frac{27}{4} = \frac{19 \times 27}{2 \times 4} = \frac{513}{8}$

Step 3: Add the Results

Add the products obtained in Step 2. First, find a common denominator for 12 and 8, which is 24.

• Convert $\frac{529}{12}$ to an equivalent fraction with denominator 24: $\frac{529 \times 2}{12 \times 2} = \frac{1058}{24}$
• Convert $\frac{513}{8}$ to an equivalent fraction with denominator 24: $\frac{513 \times 3}{8 \times 3} = \frac{1539}{24}$
• Add the fractions: $\frac{1058}{24} + \frac{1539}{24} = \frac{1058 + 1539}{24} = \frac{2597}{24}$

Step 4: Convert the Result to a Mixed Fraction

Divide the numerator 2597 by the denominator 24 to express the result as a mixed fraction.

$2597 \div 24 = 108$ with a remainder of $5$. ($108 \times 24 = 2592$; $2597 - 2592 = 5$).

Therefore, the value is $108\frac{5}{24}$.