Solving the Complex Fraction Expression

This problem asks us to find the value of a complex mathematical expression involving fractions, mixed numbers, and different operations like division and multiplication. To solve this, we need to follow the correct order of operations, often remembered by acronyms like BODMAS or PEMDAS.

Understanding the Order of Operations (BODMAS/PEMDAS)

The order of operations dictates the sequence in which we perform calculations in a mathematical expression:

• B/P: Brackets or Parentheses
• O/E: Orders or Exponents (and roots)
• D/M: Division and Multiplication (from left to right)
• A/S: Addition and Subtraction (from left to right)

The expression also includes the 'of' operation, which is a form of multiplication but is typically performed after brackets and before division/multiplication.

Breaking Down the Expression

The given expression is:

\(\frac{5}{8}÷ (\frac{8}{11}\times2\frac{3}{4}÷\frac{4}{9})\) +  \(5\frac{1}{3}\)  ÷  \((5\frac{1}{4}\div\frac{3}{8}\times\frac{3}{7}) \)  of  \(1\frac{7}{9}\)

Let's convert all mixed numbers to improper fractions first:

• \(2\frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}\)
• \(5\frac{1}{3} = \frac{(5 \times 3) + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3}\)
• \(5\frac{1}{4} = \frac{(5 \times 4) + 1}{4} = \frac{20 + 1}{4} = \frac{21}{4}\)
• \(1\frac{7}{9} = \frac{(1 \times 9) + 7}{9} = \frac{9 + 7}{9} = \frac{16}{9}\)

Substituting these into the expression, we get:

\(\frac{5}{8}÷ (\frac{8}{11}\times\frac{11}{4}÷\frac{4}{9})\)  +  \(\frac{16}{3}\)  ÷  \((\frac{21}{4}\div\frac{3}{8}\times\frac{3}{7}) \)  of  \(\frac{16}{9}\)

Let's evaluate the expression in two parts connected by the addition sign.

Evaluating the First Part

The first part is: \(\frac{5}{8}÷ (\frac{8}{11}\times\frac{11}{4}÷\frac{4}{9})\)

First, calculate the expression inside the parentheses:

\((\frac{8}{11}\times\frac{11}{4}÷\frac{4}{9})\)

Perform multiplication and division from left to right:

Step 1: Multiplication — \(\frac{8}{11}\times\frac{11}{4}\)

\(\frac{8}{11} \times \frac{11}{4} = \frac{8 \times 11}{11 \times 4}\)

Cancel out common factors:

\(= \frac{\cancel{11} \times 8}{\cancel{11} \times 4} = \frac{8}{4} = 2\)

Step 2: Division — Now we have \(2 \div \frac{4}{9}\)

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

\(2 \div \frac{4}{9} = 2 \times \frac{9}{4}\)

\(= \frac{2 \times 9}{4} = \frac{18}{4}\)

Simplify the fraction:

\(= \frac{9}{2}\)

So, the value inside the first set of parentheses is \(\frac{9}{2}\).

Now, the first part of the expression becomes:

\(\frac{5}{8} \div \frac{9}{2}\)

Divide by multiplying by the reciprocal:

\(\frac{5}{8} \times \frac{2}{9} = \frac{5 \times 2}{8 \times 9} = \frac{10}{72}\)

Simplify the fraction:

\(= \frac{5}{36}\)

The value of the first part is \(\frac{5}{36}\).

Evaluating the Second Part

The second part is: \(\frac{16}{3}\)  ÷  \((\frac{21}{4}\div\frac{3}{8}\times\frac{3}{7}) \)  of  \(\frac{16}{9}\)

First, calculate the expression inside the parentheses:

\((\frac{21}{4}\div\frac{3}{8}\times\frac{3}{7})\)

Perform division and multiplication from left to right:

Step 1: Division — \(\frac{21}{4}\div\frac{3}{8}\)

\(\frac{21}{4} \div \frac{3}{8} = \frac{21}{4} \times \frac{8}{3}\)

Cancel out common factors:

\(= \frac{21}{3} \times \frac{8}{4} = 7 \times 2 = 14\)

Step 2: Multiplication — Now we have \(14 \times \frac{3}{7}\)

\(= \frac{14 \times 3}{7}\)

Cancel out common factors:

\(= \frac{\cancel{14}^2 \times 3}{\cancel{7}^1} = 2 \times 3 = 6\)

So, the value inside the second set of parentheses is \(6\).

The second part of the expression now becomes:

\(\frac{16}{3}\)  ÷  \(6 \)  of  \(\frac{16}{9}\)

According to BODMAS/PEMDAS, 'of' is calculated before division.

Step 3: 'of' operation — \(6 \text{ of } \frac{16}{9}\)

\(6 \text{ of } \frac{16}{9} = 6 \times \frac{16}{9}\)

Cancel out common factors:

\(= \frac{\cancel{6}^2 \times 16}{\cancel{9}^3} = \frac{2 \times 16}{3} = \frac{32}{3}\)

Step 4: Division — Now we have \(\frac{16}{3} \div \frac{32}{3}\)

Divide by multiplying by the reciprocal:

\(\frac{16}{3} \div \frac{32}{3} = \frac{16}{3} \times \frac{3}{32}\)

Cancel out common factors:

\(= \frac{\cancel{16}^1 \times \cancel{3}^1}{\cancel{3}^1 \times \cancel{32}^2} = \frac{1}{2}\)

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

Combining Both Parts

The original expression is the sum of the first part and the second part:

\(\text{First Part} + \text{Second Part} = \frac{5}{36} + \frac{1}{2}\)

To add these fractions, we need a common denominator. The least common multiple of 36 and 2 is 36.

Convert \(\frac{1}{2}\) to a fraction with denominator 36:

\(\frac{1}{2} = \frac{1 \times 18}{2 \times 18} = \frac{18}{36}\)

Now add the fractions:

\(\frac{5}{36} + \frac{18}{36} = \frac{5 + 18}{36} = \frac{23}{36}\)

The final value of the expression is \(\frac{23}{36}\).

Summary of Calculation Steps

The final answer obtained is \(\frac{23}{36}\).

Revision Table: Order of Operations & Fraction Arithmetic

Additional Information on Fraction Operations

When dealing with expressions involving fractions and mixed numbers, careful attention to the order of operations is crucial. Mistakes often happen when the 'of' operation is not handled correctly relative to division and multiplication, or when simplifying fractions incorrectly during intermediate steps.

Remember these key points:

• Always convert mixed numbers to improper fractions before performing multiplication or division.
• Division by a fraction is equivalent to multiplication by its reciprocal.
• Inside parentheses, follow BODMAS/PEMDAS.
• Between division, multiplication, and 'of', the order is typically Brackets > Of > Division/Multiplication (left to right) > Addition/Subtraction (left to right).
• Simplify fractions whenever possible to make calculations easier.
• When adding or subtracting fractions, a common denominator is necessary.

Practicing different types of fraction problems with varying operations will help solidify your understanding and improve accuracy.