Understanding the Problem: Simplifying a Complex Fractional Expression

The question asks us to find the value of a complex mathematical expression involving fractions and mixed numbers, requiring us to follow the correct order of operations.

The expression is: \(1\frac{1}{8} \div \left( {4\frac{1}{4} \div \frac{3}{5}of\;8\frac{1}{2}} \right) - \frac{2}{5} \times 1\frac{1}{3} \div \frac{4}{5}of\;1\frac{2}{3} + \frac{{11}}{{20}}\)

Applying the Order of Operations (BODMAS/PEMDAS)

To correctly evaluate this expression, we must apply the BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) rule. Operations are performed from left to right within each level (Division/Multiplication, Addition/Subtraction). The 'of' operation is treated as multiplication and is typically done after brackets but before standard multiplication/division.

Step 1: Convert Mixed Numbers to Improper Fractions

First, let's convert all the mixed numbers into improper fractions to make calculations easier:

• \(1\frac{1}{8} = \frac{(1 \times 8) + 1}{8} = \frac{9}{8}\)
• \(4\frac{1}{4} = \frac{(4 \times 4) + 1}{4} = \frac{17}{4}\)
• \(8\frac{1}{2} = \frac{(8 \times 2) + 1}{2} = \frac{17}{2}\)
• \(1\frac{1}{3} = \frac{(1 \times 3) + 1}{3} = \frac{4}{3}\)
• \(1\frac{2}{3} = \frac{(1 \times 3) + 2}{3} = \frac{5}{3}\)

Substituting these improper fractions into the original expression, we get:

\(\frac{9}{8} \div \left( \frac{17}{4} \div \frac{3}{5}\;of\;\frac{17}{2} \right) - \frac{2}{5} \times \frac{4}{3} \div \frac{4}{5}\;of\;\frac{5}{3} + \frac{11}{20}\)

Step 2: Evaluate 'of' Operations

Next, we calculate the terms involving 'of', which signifies multiplication:

• For the first 'of' term: \(\frac{3}{5}\;of\;\frac{17}{2} = \frac{3}{5} \times \frac{17}{2} = \frac{3 \times 17}{5 \times 2} = \frac{51}{10}\)
• For the second 'of' term: \(\frac{4}{5}\;of\;\frac{5}{3} = \frac{4}{5} \times \frac{5}{3} = \frac{4 \times 5}{5 \times 3} = \frac{20}{15}\). This fraction can be simplified by dividing the numerator and denominator by 5: \(\frac{20 \div 5}{15 \div 5} = \frac{4}{3}\)

The expression now looks like this:

\(\frac{9}{8} \div \left( \frac{17}{4} \div \frac{51}{10} \right) - \frac{2}{5} \times \frac{4}{3} \div \frac{4}{3} + \frac{11}{20}\)

Step 3: Evaluate the Expression within Brackets

According to BODMAS/PEMDAS, we evaluate the expression inside the brackets next:

\(\frac{17}{4} \div \frac{51}{10}\)

Dividing by a fraction is the same as multiplying by its reciprocal:

\(\frac{17}{4} \times \frac{10}{51}\)

We can simplify before multiplying. \(17\) is a factor of \(51\) (\(51 = 17 \times 3\)), and \(4\) and \(10\) share a common factor of \(2\):

\(\frac{\cancel{17}^1}{\cancel{4}_2} \times \frac{\cancel{10}^5}{\cancel{51}_3} = \frac{1 \times 5}{2 \times 3} = \frac{5}{6}\)

The expression is simplified to:

\(\frac{9}{8} \div \frac{5}{6} - \frac{2}{5} \times \frac{4}{3} \div \frac{4}{3} + \frac{11}{20}\)

Step 4: Perform Multiplication and Division (from Left to Right)

Now, we perform all multiplication and division operations in the order they appear from left to right.

First term: \(\frac{9}{8} \div \frac{5}{6}\)

Multiply by the reciprocal:

\(\frac{9}{8} \times \frac{6}{5}\)

Simplify (8 and 6 share a factor of 2):

\(\frac{9}{\cancel{8}_4} \times \frac{\cancel{6}^3}{5} = \frac{9 \times 3}{4 \times 5} = \frac{27}{20}\)

Second group of terms: \(\frac{2}{5} \times \frac{4}{3} \div \frac{4}{3}\)

First, multiplication:

\(\frac{2}{5} \times \frac{4}{3} = \frac{2 \times 4}{5 \times 3} = \frac{8}{15}\)

Then, division:

\(\frac{8}{15} \div \frac{4}{3}\)

Multiply by the reciprocal:

\(\frac{8}{15} \times \frac{3}{4}\)

Simplify (8 and 4 share a factor of 4; 15 and 3 share a factor of 3):

\(\frac{\cancel{8}^2}{\cancel{15}_5} \times \frac{\cancel{3}^1}{\cancel{4}_1} = \frac{2 \times 1}{5 \times 1} = \frac{2}{5}\)

Substitute these results back into the expression:

\(\frac{27}{20} - \frac{2}{5} + \frac{11}{20}\)

Step 5: Perform Addition and Subtraction (from Left to Right)

Finally, we perform the addition and subtraction. We need a common denominator for \(20\), \(5\), and \(20\). The least common multiple is \(20\).

Convert \(\frac{2}{5}\) to an equivalent fraction with a denominator of \(20\):

\(\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}\)

The expression becomes:

\(\frac{27}{20} - \frac{8}{20} + \frac{11}{20}\)

Combine the numerators since the denominators are the same:

\(\frac{27 - 8 + 11}{20}\)

Perform subtraction and then addition:

\(\frac{19 + 11}{20} = \frac{30}{20}\)

Step 6: Simplify the Result

The fraction \(\frac{30}{20}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is \(10\):

\(\frac{30 \div 10}{20 \div 10} = \frac{3}{2}\)

The improper fraction \(\frac{3}{2}\) can be converted back to a mixed number:

\(\frac{3}{2} = 1\frac{1}{2}\)

The value of the expression is \(1\frac{1}{2}\).

Revision Table: Key Fraction Simplification Steps

Additional Information on Fraction Arithmetic and Order of Operations

Successfully solving expressions like this relies on a solid understanding of fraction arithmetic and strict adherence to the order of operations. Here are some key points:

• Mixed Numbers: Represent a whole number and a fractional part. Converting them to improper fractions (where the numerator is greater than or equal to the denominator) is often the first step in complex calculations.
• Fraction Multiplication: Multiply numerators together and denominators together. Simplifying fractions before multiplying can prevent large numbers.
• Fraction Division: The rule is "keep, change, flip". Keep the first fraction, change the division sign to multiplication, and flip the second fraction (use its reciprocal).
• Fraction Addition and Subtraction: Require a common denominator. Find the LCM of the denominators, convert each fraction to an equivalent fraction with this LCM as the denominator, and then add or subtract the numerators.
• BODMAS/PEMDAS: This acronym ensures calculations are performed in the correct sequence: Brackets/Parentheses first, then Orders/Exponents, followed by Division and Multiplication (from left to right), and finally Addition and Subtraction (from left to right). 'Of' falls under multiplication but is sometimes done before standard multiplication/division within the multiplication/division stage.

Practicing problems with mixed operations and fractions helps build confidence and accuracy.