Solving the Mathematical Expression Using BODMAS

To find the value of the given expression, we must follow the order of operations, commonly known as BODMAS or PEMDAS. This rule dictates the sequence of operations: Brackets, Of (Orders/Exponents), Division, Multiplication, Addition, and Subtraction.

The given expression is: \(3\frac{1}{5} \div 4\frac{1}{2}\;of\;5\frac{1}{3} + \frac{1}{8} \div \frac{1}{2}of\frac{1}{4} - \frac{1}{4}\left( {\frac{1}{2} \div \frac{1}{8} \times \frac{1}{4}} \right)\)

First, let's convert all mixed fractions into improper fractions:

• \(3\frac{1}{5} = \frac{(3 \times 5) + 1}{5} = \frac{16}{5}\)
• \(4\frac{1}{2} = \frac{(4 \times 2) + 1}{2} = \frac{9}{2}\)
• \(5\frac{1}{3} = \frac{(5 \times 3) + 1}{3} = \frac{16}{3}\)

Now, substitute these back into the expression:

\(\frac{16}{5} \div \frac{9}{2}\;of\;\frac{16}{3} + \frac{1}{8} \div \frac{1}{2}of\frac{1}{4} - \frac{1}{4}\left( {\frac{1}{2} \div \frac{1}{8} \times \frac{1}{4}} \right)\)

Let's break down the calculation into parts following BODMAS.

Step 1: Solve the 'Of' operations

The 'of' operation means multiplication. We have two 'of' terms:

• First 'of' term: \(\frac{9}{2}\;of\;\frac{16}{3} = \frac{9}{2} \times \frac{16}{3} = \frac{\cancel{9}^3}{\cancel{2}^1} \times \frac{\cancel{16}^8}{\cancel{3}^1} = 3 \times 8 = 24\)
• Second 'of' term: \(\frac{1}{2}of\frac{1}{4} = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}\)

Substitute these values back into the expression:

\(\frac{16}{5} \div 24 + \frac{1}{8} \div \frac{1}{8} - \frac{1}{4}\left( {\frac{1}{2} \div \frac{1}{8} \times \frac{1}{4}} \right)\)

Step 2: Solve the Bracketed Expression

Next, we solve the expression inside the brackets: \(\left( {\frac{1}{2} \div \frac{1}{8} \times \frac{1}{4}} \right)\)

Inside the bracket, we follow the order of Division then Multiplication:

• Division: \(\frac{1}{2} \div \frac{1}{8} = \frac{1}{2} \times \frac{8}{1} = \frac{8}{2} = 4\)
• Multiplication: \(4 \times \frac{1}{4} = 1\)

So, the value inside the bracket is \(1\).

Substitute this back into the main expression:

\(\frac{16}{5} \div 24 + \frac{1}{8} \div \frac{1}{8} - \frac{1}{4}\left( 1 \right)\)

\(\frac{16}{5} \div 24 + \frac{1}{8} \div \frac{1}{8} - \frac{1}{4}\)

Step 3: Solve Divisions

We have two divisions to perform:

• First division: \(\frac{16}{5} \div 24 = \frac{16}{5} \times \frac{1}{24} = \frac{\cancel{16}^2}{5} \times \frac{1}{\cancel{24}^3} = \frac{2}{15}\)
• Second division: \(\frac{1}{8} \div \frac{1}{8} = \frac{1}{8} \times \frac{8}{1} = 1\)

Substitute these results into the expression:

\(\frac{2}{15} + 1 - \frac{1}{4}\)

Step 4: Solve Addition and Subtraction

Finally, we perform the addition and subtraction from left to right. To do this, find a common denominator for the fractions \(\frac{2}{15}\), \(1\) (which is \(\frac{1}{1}\)), and \(\frac{1}{4}\). The least common multiple (LCM) of 15, 1, and 4 is 60.

• Convert \(\frac{2}{15}\): \(\frac{2}{15} = \frac{2 \times 4}{15 \times 4} = \frac{8}{60}\)
• Convert \(1\): \(1 = \frac{60}{60}\)
• Convert \(\frac{1}{4}\): \(\frac{1}{4} = \frac{1 \times 15}{4 \times 15} = \frac{15}{60}\)

Now, rewrite the expression with the common denominator:

\(\frac{8}{60} + \frac{60}{60} - \frac{15}{60}\)

Combine the numerators:

\(\frac{8 + 60 - 15}{60} = \frac{68 - 15}{60} = \frac{53}{60}\)

Thus, the value of the expression is \(\frac{53}{60}\).

Summary of Steps:

Revision Table: BODMAS Rules and Fraction Operations

Understanding the order of operations is crucial for correctly solving mathematical expressions. The BODMAS rule provides this order:

When dealing with fractions:

• Mixed to Improper: \(a\frac{b}{c} = \frac{a \times c + b}{c}\)
• Multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)
• Division: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\) (Multiply by the reciprocal of the second fraction)
• Addition/Subtraction: Find a common denominator (LCM of denominators), convert fractions, then add/subtract numerators.

Additional Information: Importance of Order of Operations

The order of operations is not just a convention; it's a fundamental principle in mathematics that ensures everyone gets the same result when evaluating an expression. Without a standard order, an expression like \(2 + 3 \times 4\) could be interpreted as \((2+3) \times 4 = 5 \times 4 = 20\) or \(2 + (3 \times 4) = 2 + 12 = 14\). BODMAS/PEMDAS standardizes this, telling us to do multiplication before addition, resulting in \(14\). This is particularly important in algebra, calculus, and computer programming where complex expressions are common.

Errors in applying the order of operations are frequent sources of mistakes in mathematical calculations. Always remember to work systematically through the BODMAS steps to achieve the correct result.