Expression Value Calculation

The problem requires calculating the value of a mathematical expression involving several operations: division, multiplication, and addition, along with mixed numbers. To solve this accurately, we need to follow the order of operations, known as BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) or PEMDAS.

Step-by-Step BODMAS Breakdown

The expression given is:

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

1. Convert Mixed Numbers to Improper Fractions

First, let's convert all the mixed numbers into improper fractions:

• $ 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 original 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 $

2. Calculate 'Of' Operations

According to BODMAS, the 'of' operation (which means multiplication) is performed before division and multiplication.

• Calculate $ \frac{35}{11} \text{ of } 2 $: $ \frac{35}{11} \times 2 = \frac{70}{11} $
• Calculate $ \frac{16}{3} \text{ of } \frac{3}{8} $: $ \frac{16}{3} \times \frac{3}{8} = \frac{16 \times 3}{3 \times 8} = \frac{48}{24} = 2 $

The expression now simplifies to:

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

3. Perform Division and Multiplication

Next, we perform all divisions and multiplications from left to right.

• First term: $ \frac{70}{11} \div \frac{70}{11} $. Dividing a number by itself results in 1. Alternatively, $ \frac{70}{11} \times \frac{11}{70} = 1 $.
• Second term (division): $ 2 \div 4 $. This equals $ \frac{2}{4} $, which simplifies to $ \frac{1}{2} $ or $ 0.5 $.
• Third term (multiplication): $ 5 \times 2 = 10 $.

Substituting these results, the expression becomes:

$ 1 + 0.5 + 10 $

4. Perform Addition

Finally, we perform the addition operation.

$ 1 + 0.5 + 10 = 11.5 $

Final Result Value

Therefore, the value of the given mathematical expression is 11.5.