Simplifying Complex Mathematical Expressions

To simplify the given complex fraction, we need to evaluate the numerator and the denominator separately following the order of operations (BODMAS/PEDMAS).

Understanding the Order of Operations (BODMAS)

BODMAS stands for:

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

The expression is:

\(\frac{{6\frac{1}{2} + 2\frac{5}{7} \times \frac{{14}}{{19}} - \frac{1}{2} \div 2\ of\frac{1}{4}}}{{11 \times 12 \div 12 + 12}}\)

Simplifying the Numerator

Let's evaluate the numerator first: \(6\frac{1}{2} + 2\frac{5}{7} \times \frac{{14}}{{19}} - \frac{1}{2} \div 2\ of\frac{1}{4}\)

First, convert the mixed numbers to improper fractions:

• \(6\frac{1}{2} = \frac{(6 \times 2) + 1}{2} = \frac{12 + 1}{2} = \frac{13}{2}\)
• \(2\frac{5}{7} = \frac{(2 \times 7) + 5}{7} = \frac{14 + 5}{7} = \frac{19}{7}\)

The numerator becomes: \(\frac{13}{2} + \frac{19}{7} \times \frac{14}{19} - \frac{1}{2} \div 2\ of\frac{1}{4}\)

Next, evaluate the 'of' operation:

• \(2\ of\frac{1}{4} = 2 \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2}\)

The numerator becomes: \(\frac{13}{2} + \frac{19}{7} \times \frac{14}{19} - \frac{1}{2} \div \frac{1}{2}\)

Now, perform multiplication and division from left to right:

• Multiplication: \(\frac{19}{7} \times \frac{14}{19} = \frac{\cancel{19}}{7} \times \frac{2 \times 7}{\cancel{19}} = \frac{2 \times \cancel{7}}{\cancel{7}} = 2\)
• Division: \(\frac{1}{2} \div \frac{1}{2} = \frac{1}{2} \times \frac{2}{1} = 1\)

The numerator becomes: \(\frac{13}{2} + 2 - 1\)

Finally, perform addition and subtraction from left to right:

• \(\frac{13}{2} + 2 = \frac{13}{2} + \frac{4}{2} = \frac{17}{2}\)
• \(\frac{17}{2} - 1 = \frac{17}{2} - \frac{2}{2} = \frac{15}{2}\)

So, the simplified numerator is \(\frac{15}{2}\).

Simplifying the Denominator

Now, let's evaluate the denominator: \(11 \times 12 \div 12 + 12\)

Perform multiplication and division from left to right:

• Multiplication: \(11 \times 12 = 132\)
• The expression becomes: \(132 \div 12 + 12\)
• Division: \(132 \div 12 = 11\)
• The expression becomes: \(11 + 12\)

Finally, perform addition:

• \(11 + 12 = 23\)

So, the simplified denominator is \(23\).

Combining Numerator and Denominator

The simplified fraction is the numerator divided by the denominator:

\(\frac{\text{Numerator}}{\text{Denominator}} = \frac{\frac{15}{2}}{23}\)

Dividing by a number is the same as multiplying by its reciprocal. The reciprocal of \(23\) is \(\frac{1}{23}\).

\(\frac{15}{2} \div 23 = \frac{15}{2} \times \frac{1}{23}\)

Multiply the numerators and the denominators:

\(\frac{15 \times 1}{2 \times 23} = \frac{15}{46}\)

The simplified expression is \(\frac{15}{46}\).

Comparing this result with the given options, we find that the simplified value matches Option 1.

Revision Table: Key Steps in Simplifying Expressions

Additional Information: Working with Fractions and Operations

When working with fractions, remember these key points:

• Mixed Numbers: Always convert mixed numbers to improper fractions before performing multiplication or division. They can be used directly in addition/subtraction sometimes, but improper fractions are generally safer for complex expressions.
• 'Of' vs. Multiplication: 'Of' signifies multiplication, but it is typically evaluated before standard multiplication and division in the BODMAS/PEDMAS order.
• Division by Fraction: Dividing by a fraction is equivalent to multiplying by its reciprocal. For example, \(a \div \frac{b}{c} = a \times \frac{c}{b}\).
• Division by Whole Number: Dividing a fraction by a whole number (like \(\frac{a}{b} \div c\)) is equivalent to \(\frac{a}{b} \times \frac{1}{c}\).

Paying close attention to the order of operations and performing each step carefully is crucial for simplifying mathematical expressions correctly.