Evaluating Mathematical Expressions Using BODMAS

This question requires us to find the value of a mathematical expression by correctly applying the order of operations. The widely accepted rule for the order of operations is BODMAS or PEDMAS.

Understanding the BODMAS/PEDMAS Rule

The BODMAS rule dictates the sequence in which mathematical operations should be performed:

• B - Brackets (Parentheses)
• O - Order (Powers and Square Roots) or E - Exponents
• D - Division and M - Multiplication (from left to right)
• A - Addition and S - Subtraction (from left to right)

The operation 'of' is also handled early, typically treated like multiplication but often evaluated before standard multiplication and division, especially when it involves fractions or percentages acting on a number. In this specific expression, '5 of 5' means $5 \times 5$.

Step-by-Step Calculation

Let's evaluate the given expression step by step:

The expression is: $\{5 - 5 \div (10 - 12) \times 8 + 9\} \times 3 + 5 + 5 \times 5 \div 5 \text{ of } 5$

Step 1: Solve Operations inside Brackets/Parentheses

First, evaluate the expression inside the parentheses:

$(10 - 12) = -2$

The expression becomes: $\{5 - 5 \div (-2) \times 8 + 9\} \times 3 + 5 + 5 \times 5 \div 5 \text{ of } 5$

Step 2: Evaluate 'of' operation

Next, evaluate '5 of 5':

$5 \text{ of } 5 = 5 \times 5 = 25$

The expression becomes: $\{5 - 5 \div (-2) \times 8 + 9\} \times 3 + 5 + 5 \times 5 \div 25$

Step 3: Solve Operations inside the Curly Braces {}

Now, evaluate the expression inside the curly braces following BODMAS (Division and Multiplication first, then Addition and Subtraction):

• Perform Division: $5 \div (-2) = -2.5$

The expression inside braces is now: $\{5 - (-2.5) \times 8 + 9\}$

• Perform Multiplication: $-2.5 \times 8 = -20$

The expression inside braces is now: $\{5 - (-20) + 9\}$

• Perform Subtraction (which becomes Addition): $5 - (-20) = 5 + 20 = 25$

The expression inside braces is now: $\{25 + 9\}$

• Perform Addition: $25 + 9 = 34$

So, the value inside the curly braces is 34.

The overall expression is now: $34 \times 3 + 5 + 5 \times 5 \div 25$

Step 4: Perform Multiplication and Division (from left to right) outside the braces

Evaluate the multiplications and divisions in the remaining expression:

• Perform Multiplication: $34 \times 3 = 102$

The expression is now: $102 + 5 + 5 \times 5 \div 25$

• Perform Multiplication: $5 \times 5 = 25$

The expression is now: $102 + 5 + 25 \div 25$

• Perform Division: $25 \div 25 = 1$

The expression is now: $102 + 5 + 1$

Step 5: Perform Addition (from left to right)

Finally, perform the additions:

• $102 + 5 = 107$
• $107 + 1 = 108$

The final value of the expression is 108.

Summary of steps and results:

Thus, the value of the expression is 108.

Revision Table: Order of Operations

Additional Information: Handling Negative Numbers

When applying the BODMAS rule, special attention must be paid to negative numbers. For example, in the step $5 - (-20)$, subtracting a negative number is the same as adding the corresponding positive number, resulting in $5 + 20 = 25$. Also, division involving a negative number ($5 \div (-2)$) results in a negative quotient ($-2.5$). Correctly handling these signs is crucial for accurate calculations.

The term 'of' often appears in questions involving fractions or percentages (e.g., "half of 10", "20% of 50"). In these cases, 'of' signifies multiplication and is usually evaluated before standard multiplication and division in the same expression.