Simplifying Mathematical Expressions Using BODMAS/PEMDAS

To simplify the given mathematical expression, we must follow the standard order of operations, which is commonly known as BODMAS or PEMDAS. This rule dictates the sequence in which operations should be performed to arrive at the correct result.

Understanding the BODMAS/PEMDAS Rule for Simplification

The BODMAS acronym represents the following order:

• Brackets (Parentheses)
• Orders (Powers, Square Roots, etc.) or Of
• Division and Multiplication (Performed from left to right)
• Addition and Subtraction (Performed from left to right)

PEMDAS is an equivalent rule: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). The key is to handle 'of' as multiplication performed before standard division and multiplication.

Step-by-Step Simplification of the Expression

The expression we need to simplify is:

\(8 \div 4 \text{ of } 2 - 15 \div 2 \text{ of } 5 - 6 \div 5 \times (-7 + 5) \text{ of } 2\)

Step 1: Address Brackets

First, we solve the operation inside the brackets:

\((-7 + 5) = -2\)

The expression now becomes:

\(8 \div 4 \text{ of } 2 - 15 \div 2 \text{ of } 5 - 6 \div 5 \times (-2) \text{ of } 2\)

Step 2: Handle 'Of' Operations

'Of' signifies multiplication and is evaluated after brackets but before division and multiplication. We have three 'of' terms:

• \(4 \text{ of } 2 = 4 \times 2 = 8\)
• \(2 \text{ of } 5 = 2 \times 5 = 10\)
• \((-2) \text{ of } 2 = -2 \times 2 = -4\)

Substitute these results back into the expression:

\(8 \div 8 - 15 \div 10 - 6 \div 5 \times (-4)\)

Step 3: Perform Division and Multiplication (Left to Right)

Now, we execute division and multiplication operations in the order they appear from left to right:

• \(8 \div 8 = 1\)
• \(15 \div 10 = \frac{15}{10} = \frac{3}{2}\)
• \(6 \div 5 \times (-4) = \frac{6}{5} \times (-4) = -\frac{24}{5}\)

The expression is simplified to:

\(1 - \frac{3}{2} - (-\frac{24}{5})\)

Which simplifies further to:

\(1 - \frac{3}{2} + \frac{24}{5}\)

Step 4: Perform Addition and Subtraction (Left to Right)

Finally, we perform the addition and subtraction. To combine these fractions, we need a common denominator. The denominators are 1 (for the integer 1), 2, and 5. The least common multiple (LCM) of 1, 2, and 5 is 10.

Convert each term into an equivalent fraction with a denominator of 10:

• \(1 = \frac{1 \times 10}{1 \times 10} = \frac{10}{10}\)
• \(\frac{3}{2} = \frac{3 \times 5}{2 \times 5} = \frac{15}{10}\)
• \(\frac{24}{5} = \frac{24 \times 2}{5 \times 2} = \frac{48}{10}\)

Substitute these fractions back into the expression:

\(\frac{10}{10} - \frac{15}{10} + \frac{48}{10}\)

Combine the numerators over the common denominator:

\(\frac{10 - 15 + 48}{10}\)

\(\frac{-5 + 48}{10}\)

\(\frac{43}{10}\)

The result is the improper fraction \(\frac{43}{10}\). To express this as a mixed number, divide 43 by 10. The quotient is 4 and the remainder is 3. So, the mixed number is \(4 \frac{3}{10}\).

Result of Simplifying the Expression

The simplified value of the given expression \(8 \div 4 \text{ of } 2 - 15 \div 2 \text{ of } 5 - 6 \div 5 \times (-7 + 5) \text{ of } 2\) is \(4 \frac{3}{10}\).

Revision Table: Order of Mathematical Operations (BODMAS)

Additional Information: Handling Negative Numbers and Fractions

This problem involved both negative numbers and fractions, which are common in mathematical expressions. Here are some key points:

• Negative Numbers: Be careful with signs, especially during multiplication and subtraction. Subtracting a negative number is equivalent to adding a positive number (e.g., \(a - (-b) = a + b\)). Multiplying a positive and a negative number gives a negative result (e.g., \(5 \times -4 = -20\)).
• Fractions: Adding or subtracting fractions requires finding a common denominator. Multiplying fractions involves multiplying the numerators together and the denominators together (\(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)). Dividing fractions involves multiplying by the reciprocal of the second fraction (\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)). Converting between improper fractions and mixed numbers is useful for different representations of the same value.
• 'Of' vs. Multiplication/Division: Remember that 'of' acts like multiplication but takes precedence over the division and multiplication performed in step 3 of BODMAS/PEMDAS. Calculate 'of' terms in step 2.