This question asks us to find the value of a mathematical expression involving repeating decimals. We need to perform the operations in the correct order (following the BODMAS/PEMDAS rule: Brackets, Orders (of), Division, Multiplication, Addition, Subtraction).

Converting Repeating Decimals to Fractions

First, let's convert the repeating decimals into fractions. A repeating decimal of the form \(0.\overline{a}\) can be written as the fraction \( \frac{a}{9} \). A repeating decimal of the form \(0.\overline{ab}\) can be written as \( \frac{ab}{99} \).

• \( 0.\overline{3} = \frac{3}{9} = \frac{1}{3} \)
• \( 0.\overline{5} = \frac{5}{9} \)
• \( 0.\overline{7} = \frac{7}{9} \)
• \( 0.\overline{4} = \frac{4}{9} \)

Evaluating the Expression Step-by-Step

The expression is: \(0.\overline {3{\rm{\;}}} {\rm{of\;}}0.\bar 5 \div 0.\bar 5{\rm{\;of}}\left( {0.\bar 7 - 0.\bar 4} \right)\)

Substitute the fractional equivalents:

$$ \frac{1}{3} \text{ of } \frac{5}{9} \div \frac{5}{9} \text{ of } \left( \frac{7}{9} - \frac{4}{9} \right) $$

Step 1: Solve the expression inside the brackets (parentheses).

$$ \left( \frac{7}{9} - \frac{4}{9} \right) = \frac{7 - 4}{9} = \frac{3}{9} = \frac{1}{3} $$

The expression now becomes:

$$ \frac{1}{3} \text{ of } \frac{5}{9} \div \frac{5}{9} \text{ of } \frac{1}{3} $$

Step 2: Perform the 'of' operations (multiplication) from left to right.

• First 'of': \( \frac{1}{3} \times \frac{5}{9} = \frac{1 \times 5}{3 \times 9} = \frac{5}{27} \)
• Second 'of': \( \frac{5}{9} \times \frac{1}{3} = \frac{5 \times 1}{9 \times 3} = \frac{5}{27} \)

The expression simplifies to:

$$ \frac{5}{27} \div \frac{5}{27} $$

Step 3: Perform the division.

Dividing a number by itself always results in 1.

$$ \frac{5}{27} \div \frac{5}{27} = \frac{5}{27} \times \frac{27}{5} = 1 $$

Therefore, the value of the given expression is 1.