This question asks us to evaluate the value of a given mathematical expression, which is a fraction. To solve this, we need to follow the order of operations, often remembered by the acronyms BODMAS or PEMDAS.

• B/P: Brackets or Parentheses first
• O/E: Orders or Exponents
• D/M: Division and Multiplication (from left to right)
• A/S: Addition and Subtraction (from left to right)

The expression is \(\frac{46+\frac{3}{4} \ \text{of}\ 32-6}{37-\frac{3}{4} \ \text{of}\ (34+6)}\). The term 'of' means multiplication.

Evaluating the Numerator of the Fraction

The numerator is \(46+\frac{3}{4} \ \text{of}\ 32-6\). Let's break it down:

First, calculate the 'of' part (multiplication):

\(\frac{3}{4} \ \text{of}\ 32 = \frac{3}{4} \times 32\)

\(= \frac{3 \times 32}{4}\)

\(= 3 \times 8\)

\(= 24\)

Now substitute this value back into the numerator expression:

\(46 + 24 - 6\)

Perform addition and subtraction from left to right:

\(46 + 24 = 70\)

\(70 - 6 = 64\)

So, the value of the numerator is 64.

Evaluating the Denominator of the Fraction

The denominator is \(37-\frac{3}{4} \ \text{of}\ (34+6)\). Let's break it down following BODMAS/PEMDAS:

First, evaluate the expression inside the parentheses:

\(34 + 6 = 40\)

Now substitute this value back into the denominator expression:

\(37 - \frac{3}{4} \ \text{of}\ 40\)

Next, calculate the 'of' part (multiplication):

\(\frac{3}{4} \ \text{of}\ 40 = \frac{3}{4} \times 40\)

\(= \frac{3 \times 40}{4}\)

\(= 3 \times 10\)

\(= 30\)

Now substitute this value back into the remaining denominator expression:

\(37 - 30\)

\(= 7\)

So, the value of the denominator is 7.

Calculating the Final Value of the Expression

The expression is the numerator divided by the denominator:

Value \( = \frac{\text{Numerator}}{\text{Denominator}}\)

Value \( = \frac{64}{7}\)

The value of the given expression is \(\frac{64}{7}\).

Verification with Options

Let's compare our calculated value with the given options:

Our calculated value \(\frac{64}{7}\) matches Option 2.

Revision Table: BODMAS/PEMDAS Order of Operations

Additional Information on Mathematical Expression Evaluation

Evaluating mathematical expressions correctly is fundamental in mathematics. It ensures everyone arrives at the same unique answer for a given expression. The order of operations, like BODMAS or PEMDAS, provides a clear set of rules for the sequence in which calculations should be performed.

• The 'of' operator is equivalent to multiplication but is often performed before standard multiplication and division when it appears with fractions or percentages (e.g., 3/4 of 32). In the strict BODMAS rule, it falls under 'Orders'. However, in practical calculations like this one, it's often treated as multiplication done just after brackets.
• When operations are on the same level (like multiplication and division, or addition and subtraction), you perform them from left to right.
• Understanding how to simplify parts of an expression, like performing calculations inside parentheses or evaluating terms with 'of', step-by-step makes complex expressions manageable.