Evaluating Complex Mathematical Expressions Using BODMAS

This question asks us to find the value of a given mathematical expression which is presented as a fraction. To solve this, we need to evaluate the numerator and the denominator separately, following the order of operations, commonly known as BODMAS or PEDMAS.

The expression is:

\(\frac{{11 \times 11 - 24 \times 6\ of\ 2 + 12 \times 12}}{{4 \times 5 + 6 \times 2 \div 4}}\)

Let's break down the evaluation process step-by-step for both the numerator and the denominator.

Understanding the BODMAS/PEDMAS Rule

BODMAS/PEDMAS dictates the sequence in which operations should be performed:

• Brackets / Parentheses
• Orders (powers, square roots, etc.) / Exponents
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

The term 'of' in mathematical expressions typically means multiplication and is usually evaluated before multiplication and division.

Evaluating the Numerator

The numerator is: \(11 \times 11 - 24 \times 6\ of\ 2 + 12 \times 12\)

1. First, evaluate the 'of' operation: \(6\ of\ 2 = 6 \times 2 = 12\).
2. Substitute this back into the numerator expression: \(11 \times 11 - 24 \times 12 + 12 \times 12\).
3. Next, perform all multiplications from left to right:
   • \(11 \times 11 = 121\)
   • \(24 \times 12 = 288\)
   • \(12 \times 12 = 144\)
4. Substitute these values back: \(121 - 288 + 144\).
5. Finally, perform addition and subtraction from left to right:
   • \(121 - 288 = -167\)
   • \(-167 + 144 = -23\)

So, the value of the numerator is \(-23\).

Evaluating the Denominator

The denominator is: \(4 \times 5 + 6 \times 2 \div 4\)

1. Perform multiplication and division from left to right:
   • \(4 \times 5 = 20\)
   • \(6 \times 2 = 12\)
   • \(12 \div 4 = 3\)
2. Substitute these values back: \(20 + 3\).
3. Finally, perform the addition: \(20 + 3 = 23\).

So, the value of the denominator is \(23\).

Calculating the Final Value of the Fraction

Now we have the values for both the numerator and the denominator. We can substitute these back into the original fraction expression:

\(\frac{\text{Numerator}}{\text{Denominator}} = \frac{-23}{23}\)

Performing the division:

\(\frac{-23}{23} = -1\)

Therefore, the value of the given expression is \(-1\).

Let's quickly verify the calculations:

The final value is \(-1\), which corresponds to Option 4.

Revision Table: Key Math Operations

Additional Information on Order of Operations

The order of operations is crucial in mathematics to ensure that everyone gets the same result when evaluating an expression. Without these rules, an expression could have multiple possible values. The BODMAS/PEDMAS rule provides a standard convention.

It's important to remember that multiplication and division have equal priority, as do addition and subtraction. When operations of the same priority appear in an expression, you should perform them from left to right.

The term 'of' is sometimes included in BODMAS (often as part of 'Orders' or before Division/Multiplication) and represents multiplication, particularly used with fractions (e.g., "half of 10") or percentages.