Simplifying Complex Fraction Expressions with BODMAS

To solve the given mathematical expression, we need to follow the order of operations, often remembered by the acronym BODMAS or PEMDAS. This rule dictates the sequence for performing calculations:

• Brackets (or Parentheses)
• Orders (or Exponents/Powers)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Also, the term 'of' in mathematics indicates multiplication and is usually performed after solving brackets but before standard multiplication and division.

The expression is: \(\left[ {\frac{4}{7}\rm \;of\;2\frac{4}{5} \times 1\frac{2}{3} - \left( {3\frac{1}{2} - 2\frac{1}{6}} \right)} \right] \div \left( {3\frac{1}{5} \div 4\frac{1}{2}\;\rm of\;\;5\frac{1}{3}} \right)\)

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert all mixed numbers in the expression into improper fractions:

• \(2\frac{4}{5} = \frac{(2 \times 5) + 4}{5} = \frac{10+4}{5} = \frac{14}{5}\)
• \(1\frac{2}{3} = \frac{(1 \times 3) + 2}{3} = \frac{3+2}{3} = \frac{5}{3}\)
• \(3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{6+1}{2} = \frac{7}{2}\)
• \(2\frac{1}{6} = \frac{(2 \times 6) + 1}{6} = \frac{12+1}{6} = \frac{13}{6}\)
• \(3\frac{1}{5} = \frac{(3 \times 5) + 1}{5} = \frac{15+1}{5} = \frac{16}{5}\)
• \(4\frac{1}{2} = \frac{(4 \times 2) + 1}{2} = \frac{8+1}{2} = \frac{9}{2}\)
• \(5\frac{1}{3} = \frac{(5 \times 3) + 1}{3} = \frac{15+1}{3} = \frac{16}{3}\)

The expression now becomes:

\(\left[ {\frac{4}{7}\rm \;of\;\frac{14}{5} \times \frac{5}{3} - \left( {\frac{7}{2} - \frac{13}{6}} \right)} \right] \div \left( {\frac{16}{5} \div \frac{9}{2}\;\rm of\;\;\frac{16}{3}} \right)\)

Step 2: Evaluate the First Bracket \(\left[ {\frac{4}{7}\rm \;of\;\frac{14}{5} \times \frac{5}{3} - \left( {\frac{7}{2} - \frac{13}{6}} \right)} \right]\)

First, solve the inner parenthesis:

• \(\frac{7}{2} - \frac{13}{6}\). Find a common denominator, which is 6.
• \(\frac{7}{2} = \frac{7 \times 3}{2 \times 3} = \frac{21}{6}\)
• So, \(\frac{21}{6} - \frac{13}{6} = \frac{21 - 13}{6} = \frac{8}{6} = \frac{4}{3}\)

The first bracket simplifies to: \(\left[ {\frac{4}{7}\rm \;of\;\frac{14}{5} \times \frac{5}{3} - \frac{4}{3}} \right]\)

Next, calculate the 'of' part:

• \(\frac{4}{7}\rm \;of\;\frac{14}{5} = \frac{4}{7} \times \frac{14}{5}\)
• \(= \frac{4 \times 14}{7 \times 5} = \frac{56}{35}\). Simplify this fraction by dividing numerator and denominator by their greatest common divisor, 7.
• \(\frac{56 \div 7}{35 \div 7} = \frac{8}{5}\)

The bracket is now: \(\left[ {\frac{8}{5} \times \frac{5}{3} - \frac{4}{3}} \right]\)

Perform the multiplication:

• \(\frac{8}{5} \times \frac{5}{3} = \frac{8 \times 5}{5 \times 3} = \frac{40}{15}\). Simplify by dividing by 5.
• \(\frac{40 \div 5}{15 \div 5} = \frac{8}{3}\)

The bracket is now: \(\left[ {\frac{8}{3} - \frac{4}{3}} \right]\)

Finally, perform the subtraction:

• \(\frac{8}{3} - \frac{4}{3} = \frac{8 - 4}{3} = \frac{4}{3}\)

The value of the first bracket is \(\frac{4}{3}\).

Step 3: Evaluate the Second Parenthesis \(\left( {\frac{16}{5} \div \frac{9}{2}\;\rm of\;\;\frac{16}{3}} \right)\)

First, calculate the 'of' part:

• \(\frac{9}{2}\;\rm of\;\;\frac{16}{3} = \frac{9}{2} \times \frac{16}{3}\)
• \(= \frac{9 \times 16}{2 \times 3} = \frac{144}{6}\)
• Simplify: \(\frac{144}{6} = 24\)

The parenthesis simplifies to: \(\left( {\frac{16}{5} \div 24} \right)\)

Perform the division. Dividing by a whole number is the same as multiplying by its reciprocal (\(\frac{1}{24}\)):

• \(\frac{16}{5} \div 24 = \frac{16}{5} \times \frac{1}{24}\)
• \(= \frac{16 \times 1}{5 \times 24} = \frac{16}{120}\)
• Simplify this fraction by dividing numerator and denominator by their greatest common divisor, 8.
• \(\frac{16 \div 8}{120 \div 8} = \frac{2}{15}\)

The value of the second parenthesis is \(\frac{2}{15}\).

Step 4: Perform the Final Division

Now we divide the result of the first bracket by the result of the second parenthesis:

• \(\frac{4}{3} \div \frac{2}{15}\)

To divide by a fraction, multiply by its reciprocal:

• \(\frac{4}{3} \times \frac{15}{2}\)
• \(= \frac{4 \times 15}{3 \times 2} = \frac{60}{6}\)

Simplify the final fraction:

• \(\frac{60}{6} = 10\)

The value of the entire expression is 10.

Final Answer

The calculated value of the expression is 10, which matches option 1.

Revision Table: Key Concepts for Simplifying Expressions

Understanding the following concepts is crucial for solving such problems:

Additional Information on Fraction Operations

Working with fractions and mixed numbers requires careful application of arithmetic rules. Here are some additional points:

• Adding/Subtracting Fractions: Fractions must have a common denominator before adding or subtracting their numerators. Find the least common multiple (LCM) of the denominators.
• Multiplying Fractions: Multiply the numerators together and the denominators together. Simplify the resulting fraction if possible. \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\).
• Simplifying Fractions: Divide both the numerator and denominator by their greatest common divisor (GCD) to reduce the fraction to its simplest form. This makes calculations easier.
• Reciprocal: The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). The product of a fraction and its reciprocal is always 1 (\(\frac{a}{b} \times \frac{b}{a} = 1\)).

Practicing these steps and rules will help you confidently solve complex fraction problems involving mixed operations.