Evaluating the Arithmetic Expression Step-by-Step

The problem requires evaluating the value of the expression $43 \frac{2}{3} \div \left[ 35 + \frac{3}{4} \text{ of } 24 + \left( 42 \div 7 - 5 \frac{1}{3} \right) \right]$. We follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Orders/Of, Division/Multiplication, Addition/Subtraction.

1. Innermost Parentheses Calculation

First, solve the expression inside the innermost parentheses: $ \left( 42 \div 7 - 5 \frac{1}{3} \right) $.

• Division: $42 \div 7 = 6$.
• Convert the mixed number $5 \frac{1}{3}$ to an improper fraction: $5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{16}{3}$.
• Subtract: $6 - \frac{16}{3} = \frac{6 \times 3}{3} - \frac{16}{3} = \frac{18}{3} - \frac{16}{3} = \frac{2}{3}$.

The value of the innermost parentheses is $\frac{2}{3}$.

2. 'Of' Operation Calculation

Next, calculate the 'of' part: $ \frac{3}{4} \text{ of } 24 $.

• Multiplication: $\frac{3}{4} \times 24 = 3 \times \frac{24}{4} = 3 \times 6 = 18$.

The value of the 'of' operation is $18$.

3. Brackets Calculation

Now, evaluate the expression inside the square brackets: $ \left[ 35 + 18 + \frac{2}{3} \right] $.

• Add the whole numbers: $35 + 18 = 53$.
• Add the fraction: $53 + \frac{2}{3} = \frac{53 \times 3}{3} + \frac{2}{3} = \frac{159}{3} + \frac{2}{3} = \frac{161}{3}$.

The value inside the brackets is $\frac{161}{3}$.

4. Final Division

Finally, perform the main division. Convert the mixed number $43 \frac{2}{3}$ to an improper fraction:

• $43 \frac{2}{3} = \frac{43 \times 3 + 2}{3} = \frac{129 + 2}{3} = \frac{131}{3}$.

Now divide this by the value of the brackets:

• $\frac{131}{3} \div \frac{161}{3}$
• To divide fractions, multiply by the reciprocal of the divisor: $\frac{131}{3} \times \frac{3}{161}$.
• Simplify: $\frac{131 \times 3}{3 \times 161} = \frac{131}{161}$.

The value of the expression is $\frac{131}{161}$.