Expression Value Calculation

The problem asks us to evaluate the following mathematical expression:

$ 6\frac{4}{11} \div 3 \frac{2}{11} \text{ of } 2 + 5 \frac{1}{3} \text{ of } \frac{3}{8} \div 4 + 5 \times 2 $

To solve this correctly, we need to follow the standard order of operations, commonly remembered by the acronym BODMAS (Brackets, Orders (powers/roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right)). The term 'of' in mathematics typically implies multiplication and is usually treated as having higher precedence than division or multiplication unless specified otherwise by brackets.

Step-by-Step Calculation using BODMAS

Mixed Fraction Conversion

First, let's convert all the mixed fractions into improper fractions to make calculations easier.

• $6\frac{4}{11} = \frac{(6 \times 11) + 4}{11} = \frac{66 + 4}{11} = \frac{70}{11}$
• $3 \frac{2}{11} = \frac{(3 \times 11) + 2}{11} = \frac{33 + 2}{11} = \frac{35}{11}$
• $5 \frac{1}{3} = \frac{(5 \times 3) + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3}$

Substituting these back into the expression, we get:

$ \frac{70}{11} \div \frac{35}{11} \text{ of } 2 + \frac{16}{3} \text{ of } \frac{3}{8} \div 4 + 5 \times 2 $

BODMAS 'of' Operation Evaluation

According to BODMAS, 'of' operations are performed before division and multiplication. We need to carefully interpret the structure. The expression "$A \div B$ of $C$" is typically interpreted as $A \div (B \times C)$.

• First 'of' part: $ \frac{35}{11} \text{ of } 2 = \frac{35}{11} \times 2 = \frac{70}{11} $
• Second 'of' part: $ \frac{16}{3} \text{ of } \frac{3}{8} = \frac{16}{3} \times \frac{3}{8} = \frac{16 \times 3}{3 \times 8} = \frac{16}{8} = 2 $

Now, let's substitute these results back into the expression, applying the division rule for the first term:

$ \frac{70}{11} \div \left( \frac{35}{11} \times 2 \right) + \left( \frac{16}{3} \times \frac{3}{8} \right) \div 4 + 5 \times 2 $

$ \frac{70}{11} \div \frac{70}{11} + 2 \div 4 + 5 \times 2 $

Division and Multiplication Calculation

Now, we perform the division and multiplication operations from left to right.

The expression is currently: $ \frac{70}{11} \div \frac{70}{11} + 2 \div 4 + 5 \times 2 $

• Calculate the first division: $ \frac{70}{11} \div \frac{70}{11} = \frac{70}{11} \times \frac{11}{70} = 1 $.
• Calculate the next division: $ 2 \div 4 = \frac{2}{4} = \frac{1}{2} = 0.5 $.
• Calculate the multiplication: $ 5 \times 2 = 10 $.

Substituting these results, the expression becomes:

$ 1 + 0.5 + 10 $

Addition Calculation

Finally, we perform the addition.

$ 1 + 0.5 + 10 = 11.5 $

Final Result

The value of the given expression $6\frac{4}{11} \div 3 \frac{2}{11}$ of $2 + 5 \frac{1}{3}$ of $\frac{3}{8} \div 4 + 5 \times 2$ is 11.5.